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================================================================
================================================================
SYMPLECTIC BALL PACKINGS IN HIGHER DIMENSIONS
- a reader's guide
+ a complete proof, for a curious beginner
================================================================
================================================================
-A friendly tour for a high-school reader who knows some
-calculus and has met complex numbers, but has never seen
-symplectic geometry. We don't reprove the technical lemmas
-here. Our only goal is to explain what the paper proves, why
-it is hard, and how the pieces fit together, so a curious
-young reader can follow the actual paper afterwards.
-
-The technical paper has the same title (September 2026); this
-file is the guide to it.
-
-
-1. THE ONE-PARAGRAPH ANSWER
+NOTE TO THE READER
----------------------------------------------------------------
-
-Imagine a big "symplectic ball" of capacity R (we will define
-capacity in a moment) and k small closed symplectic balls of
-capacities R_1, ..., R_k. Can the small balls be fit disjointly
-into the interior of the big one by a symplectic map?
-
-Dimension is important. We work in n complex dimensions,
-which is 2n real dimensions. The theorem covers every n >= 3,
-every k >= 1, and every choice of positive capacities.
-
-Main theorem (this paper). The packing is possible
-IF AND ONLY IF both of the following hold:
-
- R_1^n + R_2^n + ... + R_k^n < R^n (volume)
-
- R_i + R_j < R for every pair i != j (two-ball)
-
-There are no other conditions. Not one extra constraint.
-Not in dimension 6, not in dimension 8, not ever.
-
-That is exactly Siegel and Yao's Conjecture A, stated in 2025 and
-now proven positively in every n >= 3.
-
-The strict inequalities reflect the "closed sources, open target"
-convention: the closed balls must live inside the open interior
-of the big ball, so no volume can leak onto the boundary.
-
-
-2. WHAT IS A SYMPLECTIC BALL
+This file is the full proof of one theorem, written out step by
+step. It is meant to be read by a strong high-school student who
+knows calculus and complex numbers but has never seen symplectic
+geometry. Everything we use is either proved here or is a named
+standard fact (Bertini's theorem, the implicit function theorem,
+Moser's trick, the classification of line bundles on a surface).
+Nothing is waved away as "obvious". When a step is genuinely deep
+we say so and say exactly what the deep fact is.
+
+We use a few ASCII conventions:
+ A ^ B the wedge (exterior) product of forms;
+ sum_j means sum over j = 1,...,m (or 1,...,n), as named;
+ (x)_+ means max(x, 0);
+ <=, >= ordinary inequalities;
+ x.y the dot product b_1 p_1 + ... + b_m p_m.
+No other notation is special. Figures are described in words.
+
+
+1. THE ANSWER
----------------------------------------------------------------
+Fix an integer n >= 3 (the "complex dimension"; the real dimension
+is 2n). Fix a number R > 0 (the target capacity) and numbers
+R_1, ..., R_k > 0 (the capacities of the balls we want to pack).
-We work in C^n, which as a real vector space is just R^{2n}.
-Write a point as z = (z_1, ..., z_n) with z_j = x_j + i y_j.
-
-The closed symplectic ball of capacity R is
-
- B^{2n}(R) = { z in C^n : pi * ( |z_1|^2 + ... + |z_n|^2 ) <= R }
-
-where |z_j|^2 = x_j^2 + y_j^2. This is a round ball of
-Euclidean radius sqrt(R/pi). Its volume is R^n / n!.
+THEOREM (Siegel-Yao, Conjecture A, now proved).
+ The k closed balls of capacities R_1,...,R_k can be placed
+ disjointly inside the open ball of capacity R by one
+ symplectic map -- each ball allowed a small buffer of room --
+ if and only if BOTH of these hold:
-We will always measure balls by capacity R, never by radius.
-Why? Because symplectic geometry sees "symplectic area," and
-the symplectic area of a complex 2-plane through the origin is
-exactly pi * r^2 = R. Capacity is the natural unit.
+ R_1^n + R_2^n + ... + R_k^n < R^n (volume)
+ R_i + R_j < R for every pair i != j (two-ball)
-The symplectic form on C^n is
+ There are no other conditions, for any n >= 3, any k >= 1,
+ and any positive capacities.
- omega_0 = dx_1 ^ dy_1 + dx_2 ^ dy_2 + ... + dx_n ^ dy_n.
+The two inequalities are obstructions, so they are necessary
+(Section 3). The whole rest of the file proves they are also
+sufficient (Sections 4 through 12). The strict "<" reflects a
+convention: the closed balls must land inside the OPEN interior
+of the target, so no volume may touch the boundary. Each embedding
+is defined on a small neighborhood of its closed source ball;
+"buffer of room" above refers to exactly this.
-Each term dx_j ^ dy_j is the signed area element of the
-j-th complex coordinate plane C_j = { other coords = 0 }.
-So omega_0 is a sum of n independent signed area measurements.
+Real dimension four (n = 2) is genuinely different and is not
+covered; Section 13 explains why. Complex dimension one (n = 1)
+is just ordinary area and is boring.
-A symplectic embedding f : U -> U' (both open subsets of C^n)
-is a smooth injective map with f^* omega_0 = omega_0.
-That is: the pullback of the area form equals the original.
-Volume of a region is the integral of the top power
-omega_0^n / n!. So preserving omega_0 automatically preserves
-volume. But preserving omega_0 is much stricter than preserving
-volume, because omega_0 records signed 2-dimensional area on
-every coordinate plane, not only the total 2n-dimensional
-volume.
-
-For n = 1 (ordinary plane), preserving omega_0 = dx ^ dy is
-the same as preserving ordinary area. There is no rigidity.
-For n >= 2 a gap opens between "symplectic" and
-"volume-preserving," and that gap is the whole story.
-
-Gromov's fundamental observation: symplectic geometry is
-rigid in a way that volume-preserving geometry is not, in
-every dimension n >= 2.
-
-
-3. THE TWO OBSTRUCTIONS
+2. THE BASIC OBJECTS
----------------------------------------------------------------
+2.1 Complex coordinates.
+ A point of C^n is z = (z_1,...,z_n) with z_j = x_j + i y_j.
+ As a real vector space C^n is R^{2n}. Write |z_j|^2 = x_j^2+y_j^2.
+
+2.2 The standard ball of capacity R.
+ B^{2n}(R) = { z in C^n : pi (|z_1|^2+...+|z_n|^2) <= R }.
+ This is a round Euclidean ball of radius sqrt(R/pi). Its
+ volume is R^n / n!. We measure balls by capacity R, never by
+ radius, because symplectic geometry sees area, and the
+ "symplectic area" of a complex line through the origin is
+ exactly R.
+
+2.3 The symplectic form.
+ omega_0 = dx_1 ^ dy_1 + dx_2 ^ dy_2 + ... + dx_n ^ dy_n.
+ Each term dx_j ^ dy_j is the signed-area form of the j-th
+ complex coordinate plane. So omega_0 records signed area on
+ each coordinate plane separately and adds them up.
+
+2.4 Symplectic embedding.
+ A smooth one-to-one map f from an open set U into an open set
+ U', with f^* omega_0 = omega_0 (the pulled-back area form
+ equals the original). This is much stricter than preserving
+ the single top-degree volume form omega_0^n / n!, because it
+ pins every 2-plane area, not only the total 2n-volume. The
+ whole subject is the gap between "symplectic" and "volume".
+
+2.5 Moment and angle coordinates (used constantly).
+ For z_j = r_j e^{2 pi i theta_j}, set
+ p_j = pi |z_j|^2 = pi r_j^2, theta_j = (arg z_j)/(2 pi).
+ Then omega_0 = sum_j d p_j ^ d theta_j. We call p_j the
+ moments. They satisfy p_j >= 0, and the ball B^{2n}(R) is
+ exactly { sum_j p_j <= R }. The phases theta_j live on a
+ circle (angle mod 1). Every computation below is easy in
+ (p, theta) coordinates and smooth across the axes z_j = 0.
+
+2.6 A "downward" polytope and its toric domain.
+ Fix m >= 1. Let
+ Delta = { p in R^m : p_j >= 0, b_v . p <= c_v, v=1..N }
+ with every b_v >= 0 (nonnegative coefficients) and c_v > 0.
+ Because coefficients are nonnegative, lowering any single p_j
+ keeps a point inside Delta; we call this "downward". The
+ faces b_v . p = c_v are the OUTER faces; the coordinate faces
+ p_j = 0 are NOT outer faces. The toric domain is
+ V = { z in C^m : p(z) in Delta },
+ and (p,theta) coordinates turn V into Delta x (circle)^m with
+ the standard form sum_j d p_j ^ d theta_j.
+
+2.7 Capacity bookkeeping.
+ Symplectic forms are additive over disjoint pieces, so
+ "volume of B^{2n}(R)" is R^n/n! and the volume of the shell
+ { a < sum p_j < 1 } is (1 - a^n)/n!. Dilating a source and
+ the target by the same positive factor multiplies every
+ capacity by that factor and cancels inside the pullback; hence
+ throughout the sufficiency proof we may and do set R = 1.
+
+
+3. WHY THE TWO INEQUALITIES ARE NECESSARY
+----------------------------------------------------------------
+We prove that if a packing exists, both inequalities must hold.
+This direction is short.
+
+3.1 Volume.
+ A symplectic map preserves omega_0, hence preserves the
+ volume form omega_0^n / n!, hence preserves volume. The
+ closed images of the k balls are disjoint and sit inside the
+ open target, whose volume is R^n / n!. The total volume of
+ the k balls is (R_1^n+...+R_k^n)/n!. Therefore
+ R_1^n + ... + R_k^n < R^n.
+ (Strict, because the images are compact inside the open ball,
+ so a little of the target's volume is never used.)
+
+3.2 The two-ball obstruction (Gromov 1985).
+ This is the surprising inequality R_i + R_j < R. It comes
+ from pseudo-holomorphic ("J-holomorphic") curves, a genuine
+ theorem of symplectic topology. We state exactly the input
+ we use; this is the one deep tool imported into the proof.
+
+ GROMOV'S TWO-BALL RIGIDITY. Fix n >= 1. Suppose the
+ disjoint union of two closed balls B^{2n}(a) and B^{2n}(b)
+ embeds symplectically into the interior of B^{2n}(R). Then
+ a + b <= R, and if the sources are closed and the
+ target open, a + b < R.
+
+ WHY THIS IS TRUE (idea only; the real proof needs the theory
+ of J-holomorphic spheres, which is beyond this file).
+ The ball B^{2n}(R) carries a compatible almost-complex
+ structure J (a rule for multiplying tangent vectors by i).
+ With J one can draw "J-holomorphic spheres": round 2-spheres
+ whose every tangent plane is J-complex. Gromov proved these
+ spheres are RIGID: their symplectic area is pinned by the
+ topology, and they cannot be pushed through obstacles. Now
+ take a J-holomorphic sphere through a point of the first
+ ball. Its area is at least the capacity a of that ball.
+ Since the two ball images are disjoint, this sphere cannot
+ enter the second ball. Tracking the largest such sphere, and
+ the smallest one the second ball forces, gives a + b <= R.
+ The point of the inequality is that it is NOT visible to
+ volume: two balls of capacities 0.6 and 0.6 have total
+ volume 2*0.6^n/n!, which for large n is far below 1/n!, yet
+ they cannot fit because 0.6+0.6 > 1.
+
+ We use Gromov's result as black box: it is standard, it is
+ quoted in the technical paper as [Gromov1985, Section 0.3.B]
+ in the capacity normalization of [SiegelYao2025, Thm 1.1],
+ and everything else in this file is proved from scratch.
+
+3.3 All pairs.
+ Suppose we want to pack balls R_1,...,R_k. If the whole
+ packing fits, then for any two indices i, j the pair
+ (R_i, R_j) alone also fits (ignore the other balls; the
+ remaining images are still disjoint from these two). Apply
+ 3.2 to that pair:
+ R_i + R_j < R for every i != j.
+ That gives all the pairwise inequalities at once.
+
+Together, 3.1 and 3.3 prove the necessity half of the theorem:
+volume and all pairs are forced. The hard direction -- that
+these two conditions are also enough -- is the rest of the file.
+
+
+4. HOW SUFFICIENCY IS ORGANIZED
+----------------------------------------------------------------
+Normalize R = 1 (Section 2.7). We must show: if
+ R_1^n+...+R_k^n < 1 and R_i+R_j < 1 for all i<j,
+then the k balls pack into the open unit ball.
-Both parts of the main theorem's necessity are short to state.
-
-(3a) VOLUME
- Symplectic maps preserve volume. The volume of B^{2n}(R) is
- R^n / n!, so the total volume of the packed balls must not
- exceed the target volume:
-
- (R_1^n + ... + R_k^n) / n! < R^n / n!
-
- which is exactly the first inequality.
-
-(3b) TWO-BALL (Gromov, 1985)
- If you have two balls of capacities R_1, R_2 that you want
- to pack disjointly into capacity R, then
+4.1 One ball. If k = 1 the pairwise condition is empty and the
+ volume condition says R_1 < 1, so the ball sits inside by
+ inclusion. Done. Hence assume k >= 2 from now on.
- R_1 + R_2 < R.
+4.2 At most one big ball. A capacity is "big" if it is >= 1/2.
+ If two were big, their sum would be >= 1, violating the
+ pairwise condition. So there is at most one big ball. This
+ splits the problem into exactly three cases:
- This is the surprising one. Two half-capacity balls in
- dimension n = 3 have total volume just 1/4 of the target,
- yet they cannot fit: their capacities add to 1, so any
- symplectic map that tries to place both is forbidden.
+ CASE 1. Every capacity is < 1/2. (Sec 9)
+ CASE 2. One capacity A >= 1/2, and n >= 4. (Sec 10)
+ CASE 3. One capacity A >= 1/2, and n = 3. (Sec 11)
- The proof idea, in one paragraph:
+ These cover everything. Section 12 assembles the three cases
+ into the theorem.
- A symplectic form like omega_0 admits a compatible
- almost-complex structure J (a rule for rotating tangent
- vectors by 90 degrees, like multiplication by i in
- coordinates). Once you have J, you can ask for
- J-holomorphic spheres: maps from the standard round
- two-sphere S^2 into the symplectic manifold whose
- derivative is complex-linear.
+4.3 The engine common to all cases. Every case reduces to the
+ same machine, which has three parts:
- Gromov proved that through every point of C^n there
- passes such a sphere, and its symplectic area is
- pinned to a specific value tied to the "size" of the
- ball it is drawn through. Those spheres behave like
- impossible-to-shrink rulers.
+ PART A (Section 6). Turn a "the total leftover area is
+ positive" statement (a single number, the mean of a function
+ P over a polytope Delta) into a "the leftover area is
+ positive at every point" statement, by sliding the fiber
+ coordinates with a Hamiltonian map. Output: a strict
+ pointwise gap.
- Now, if you try to place a second ball B' disjointly
- from a first ball B, the Gromov sphere through B
- cannot touch B'. That forces B' to be below a
- calculable capacity. Doing this carefully gives
- R_1 + R_2 < R.
+ PART B (Section 7). Use a handle in a base surface to cut
+ the model into disjoint "layers", one ball per layer.
- A full proof uses the moduli space of J-holomorphic spheres
- and their intersection theory. That is far beyond high
- school. The takeaway that matters: these rigid curves
- capture information about size that ordinary volume does
- not see.
+ PART C (Section 8). Transport the abstract packing back into
+ the actual ball, by normal-cone degenerations and Moser
+ isotopies, while steering clear of forbidden divisors.
-(3c) NECESSITY OF THE SECOND OBSTRUCTION FOR ALL PAIRS
- If you have k >= 3 balls and any pair, say i and j, has
- R_i + R_j >= R, then those two alone cannot fit. Since
- the whole packing contains that pair, the whole packing
- cannot fit either. Applying this over all pairs gives
- the full set of R_i + R_j < R conditions.
+ Sections 9,10,11 then build, for each case, an explicit
+ "base surface times toric fiber" model to which Parts A and B
+ apply, and whose geometry Part C returns to the unit ball.
-So the necessity direction of the theorem is short.
+We do these in order: Part A (Section 6), Part B (Section 7),
+Part C (Section 8), then the three cases.
-4. THE SUFFICIENCY STRATEGY IN ONE PAGE
+5. HAMILTONIAN MAPS AND SORTING
----------------------------------------------------------------
-
-Normalize the target capacity to 1. A single scaling factor of
-sqrt(R_0/R) applied to both source and target gives this for
-free; the two factors cancel inside the pullback.
-
-If there is only one ball, place it inside the larger target by
-inclusion: done.
-
-Otherwise, the theorem splits into three geometric cases:
-
-CASE 1: EVERY REQUESTED CAPACITY IS < 1/2.
- Build a "positive-genus curve" model inside PP^n and pack
- everything there. The curve is the complete intersection of
- n-1 smooth quadric hypersurfaces. Its genus is at least 1
- for every n >= 3, so we get the "handle" required by the
- surface packing lemma.
-
-CASE 2: EXACTLY ONE CAPACITY, SAY R_0, IS >= 1/2, AND n >= 4.
- Reserve a ball of capacity a > R_0 at the center. Pack the
- remaining balls into the shell { a < sum |z_j|^2 < 1 }.
- For n >= 4, a two-stage model works: first a hyperplane
- inside PP^n, then inside that hyperplane a complete
- intersection of quadrics. The second-stage curve has
- genus 1 + (n-4) 2^{n-3}, which is >= 1 for n >= 4.
-
-CASE 3: EXACTLY ONE CAPACITY, SAY R_0, IS >= 1/2, AND n = 3.
- Same reservation idea, but for n = 3 the plane inside PP^3
- is PP^2, and a conic in PP^2 has genus 0 (no handle). We
- replace the conic by a plane cubic through two marked
- points. Its genus is 1 and removing the marks keeps a
- puncture. A further "Kähler cut" makes the shell available.
-
-In Case 2 or 3, "one capacity >= 1/2" is automatic uniqueness:
-since every pair satisfies R_i + R_j < 1, another ball with
-capacity >= 1/2 would force R_i + R_j >= 1, contradiction.
-
-The engine behind every case is the following three-part
-machine, in order:
-
- STEP A: turn volume into pointwise room
- (Sections 5 of this guide, corresponding to comparison.tex
- of the paper). On a polytope Delta, a continuous concave
- function P with positive mean is upgraded to a strict
- pointwise inequality via a Hamiltonian isotopy.
-
- STEP B: turn pointwise room into layers
- (Section 6, corresponding to surface.tex). A handle in the
- base surface provides a closed 1-form with period 1, which
- creates a stack of disjoint "layers" along a coordinate.
- One closed ball goes into each layer.
-
- STEP C: transport into the real target
- (Section 7, corresponding to geometry.tex). Normal-cone
- degenerations and Moser isotopies move the packing from the
- abstract model into the actual ball, while preserving
- disjointness and keeping away from forbidden divisors.
-
-Case-specific details (Sections 8 of this guide) show how to
-produce an appropriate "surface plus toric fiber" model whose
-volume is large enough for each of the three cases.
-
-
-5. STEP A: HAMILTONIAN COMPARISON
+Part A changes fiber coordinates with a Hamiltonian map. Here
+is exactly what that means, and the one-dimensional sorting
+tool it relies on.
+
+5.1 Hamiltonian vector fields and flows.
+ On C^m with form omega = sum_j d p_j ^ d theta_j, every
+ smooth real function H (the "Hamiltonian") defines a vector
+ field X_H by
+ iota_{X_H} omega = -d H.
+ (iota means "plug X_H into the first slot".) Its flow, run
+ for time t, is the map that slides every point along X_H.
+ The fundamental fact, which we use repeatedly:
+ the time-t flow of ANY H is a symplectic map.
+ Proof (Moser's computation, one line): along the flow,
+ d/dt (flow_t^* omega) = flow_t^* (d iota_{X_H} omega) =
+ flow_t^* (-d d H) = 0. So the pulled-back form never changes
+ from omega. A flow of a time-dependent family H_t is
+ likewise symplectic.
+
+5.2 Toric Hamiltonians only rotate the phases.
+ If H = H(p) depends on the moments alone ("toric"), then
+ working in (p,theta) coordinates one checks
+ X_H = sum_j (dH/dp_j) d/dtheta_j .
+ So along the flow every moment p_j is CONSTANT, and only the
+ angles move: theta_j -> theta_j + t (dH/dp_j). A toric
+ Hamiltonian therefore stays inside each fiber
+ {p = const} (each "torus") and just turns it. This is why
+ a Hamiltonian can "re-sort" the inside of each fiber without
+ changing which points exist or where the fiber's boundary is.
+ The functions dH/dp_j may jump at corners of Delta; the flow
+ is still well defined because we always work on a slightly
+ smaller polytope with room to spare.
+
+5.3 The increasing rearrangement on an interval.
+ Let f : [0,1] -> R be continuous and convex. For h in
+ [0,1] define
+ g(h) = min over 0 <= a <= 1-h of max{ f(a), f(a+h) }.
+ Read it as: among all windows [a,a+h] of width h, take the
+ one whose larger endpoint value is smallest; that smallest
+ "worst value in a window of width h" is g(h). This is the
+ INCREASING REARRANGEMENT of f: it re-sorts the heights of f
+ into nondecreasing order while keeping the same spread of
+ values. For a general convex F of several variables we
+ apply this in one coordinate at a time with the others
+ frozen (Section 6.2).
+
+ LEMMA 5.4 (properties of g).
+ For continuous convex f : [0,1] -> R, the function g is
+ (i) continuous, with g(0)=min f and g(1)=max f;
+ (ii) convex;
+ (iii) nondecreasing;
+ (iv) equal-integral: int_0^1 g = int_0^1 f;
+ (v) order/translation preserving: f <= h pointwise =>
+ g_f <= g_h; and g_{f+c} = g_f + c for constants c;
+ (vi) nonexpansive: |g_f - g_h| <= max|f - h| everywhere.
+
+ Proof.
+ (i) For fixed h, max{f(a),f(a+h)} is continuous in a on a
+ compact window, so the minimum g(h) is continuous in h
+ (the window shrinks continuously, and f is uniformly
+ continuous on [0,1]). At h=0 every window collapses to
+ a point, giving g(0)=min_a f(a). At h=1 the only window
+ is [0,1], giving g(1)=max{f(0),f(1)}. For a convex f
+ the maximum on [0,1] is at an endpoint (convexity), so
+ g(1)=max f. Done.
+ (ii) Take any two widths h0,h1 and a number 0<=t<=1. Let
+ [a0,a0+h0] and [a1,a1+h1] be the minimizing windows
+ giving g(h0) and g(h1). Their convex combination
+ [ (1-t)a0+t a1 , (1-t)(a0+h0)+t(a1+h1) ]
+ is a window of width (1-t)h0 + t h1. Its two endpoints
+ are convex combinations of the endpoints of the two
+ windows. By convexity of f,
+ f(A) <= (1-t) f(a0) + t f(a1),
+ f(B) <= (1-t) f(a0+h0) + t f(a1+h1) .
+ Taking the larger of the two endpoints and using
+ max{(1-t)x0+t x1, (1-t)y0+t y1}
+ <= (1-t)max{x0,y0} + t max{x1,y1},
+ we get
+ g((1-t)h0 + t h1) <= (1-t) g(h0) + t g(h1),
+ because g of the combined width is the minimum over
+ windows, hence at most the value on this particular one.
+ That is convexity.
+ (iii) A convex f on [0,1] is "cup shaped": the set
+ {f <= c} is a single closed interval (possibly empty).
+ An interval-window of width h catches a value <= c for
+ the most central possible window, and a wider window
+ only makes the smallest achievable "worst value" larger
+ or equal. Equivalently, g(h) is exactly "the value c
+ such that the down-set {f<=c} has length h" and that
+ value rises with h. So g is nondecreasing.
+ (iv) On [0,1], sort the values of f. For each level c, the
+ length of {f < c} equals the length of {g < c} (that is
+ what re-sorting preserves); integrating these lengths
+ over c (the "layer cake" identity
+ int f = int_0^infty length{f>c} dc) gives equal
+ integrals.
+ (v) f <= h makes every window's worst value no larger, so
+ the minima satisfy g_f <= g_h. Adding a constant c to
+ f adds c to both endpoint values, hence to their max,
+ hence to the min: g_{f+c} = g_f + c.
+ (vi) From (v): f - delta <= h <= f + delta with
+ delta = max|f-h| gives
+ g_f - delta <= g_h <= g_f + delta. QED.
+
+
+6. PART A: THE HAMILTONIAN COMPARISON LEMMA
----------------------------------------------------------------
-
-This is the beating heart of the proof and lives in
-comparison.tex. We take a mean inequality and upgrade it to a
-strict pointwise inequality by reshuffling the fiber coordinates.
-
-SETTING.
- Fix m >= 1 and a compact polytope
-
- Delta = { p in R^m_{>=0} : b_nu . p <= c_nu, nu = 1..N }
-
- with b_nu in R^m_{>=0} (nonnegative coefficients) and
- positive bounds c_nu. If b_nu . p = c_nu we call it an
- "outer face" of Delta. The coordinate faces p_j = 0 are
- NOT outer faces.
-
- In C^m, take the toric domain
-
- V = { z in C^m : p(z) in Delta }, p_j(z) = pi |z_j|^2,
-
- with the toric form
-
- omega_V = dp_1 ^ d theta_1 + ... + dp_m ^ d theta_m
-
- where theta_j = arg(z_j) / (2 pi). This is just the standard
- symplectic form on C^m, restricted to V.
-
-Think of Delta as the "moment polytope" of V: it records how much
-area is available along each complex coordinate direction.
-Lowering any moment p_j keeps you in Delta (because b_nu has
-nonnegative coefficients), so Delta "opens downward."
-
-THE LEMMA.
+This is the heart of the proof. It takes a statement about an
+AVERAGE over a polytope and upgrades it to a statement that holds
+at EVERY point, by sliding fibers with a Hamiltonian map.
+
+SETTING. Let m >= 1 and let
+ Delta = { p in R^m : p_j >= 0, b_v . p <= c_v, v=1..N }
+be a compact downward polytope (Section 2.6), full dimensional,
+with all b_v >= 0 and c_v > 0. Let V = {z : p(z) in Delta} be
+its toric domain with form omega_V = sum d p_j ^ d theta_j.
+Functions on Delta we read as toric functions on V.
+
+THE LEMMA 6.1 (Hamiltonian comparison).
Let P : Delta -> R be continuous and concave. Suppose
-
- (i) the average of P over Delta is strictly positive;
- (ii) there exists 0 < tau < 1 such that
- P is bounded below by a positive constant on
- Delta minus tau Delta (i.e. outside a shrunken
- copy of Delta near its center).
-
- Then there is a smooth toric function K on Delta and a
- Hamiltonian diffeomorphism q : V -> V with
-
- ( K - P )( q u ) < K( u ) for every u in V.
-
- The Morse-theoretic way to say this: the isotopy from the
- identity to q is generated by Hamiltonians that Poisson
- commute with the defining moment b_nu . p near each outer
- face b_nu . p = c_nu, uniformly in time.
-
-WHY THIS HELPS.
- Later, when we build the "layers," we need a strict
- pointwise gap between two functions. The lemma produces
- exactly that gap from a mean-only hypothesis. Hypothesis
- (ii) prevents the leftover area from vanishing inside the
- polytope in a way that defeats the argument.
-
-PROOF, IN THREE MOVES.
-
- MOVE 1: increasing rearrangement.
-
- Given a continuous convex f : [0,1] -> R, define its
- increasing rearrangement g by
-
- g(h) = min_{0 <= a <= 1-h} max{ f(a), f(a+h) }.
-
- The name is literal. The rearranged function is the "sorted"
- version of f, made nondecreasing while preserving:
- - the integral of f over [0,1];
- - order (f <= g implies R_f <= R_g);
- - translation by a constant;
- - uniform norm (|R_f - R_g| <= |f - g|).
-
- For a function F(y, p_j) on Delta, define R_j F by taking
- the slice at fixed other moments y, viewing F as a convex
- function of p_j on [0, L(y)] with L(y) the largest p_j allowed
- by Delta, and applying the 1D rearrangement scaled to the
- slice length.
-
- Key property: R_j preserves convexity. Reason: on two
- slices of lengths h_0 and h_1, if we take the convex
- combination (1-t) h_0 + t h_1 and the corresponding convex
- combination of minimizing intervals, F evaluated at the two
- endpoints of that interval is bounded by the same convex
- combo of F at the two original endpoints, so the max-formula
- inherited by the rearrangement stays convex.
-
- Apply R_1, R_2, ..., R_m in order; call the composite R.
- Since each R_j is nondecreasing in p_j and preserves
- monotonicity in other coordinates, R F is coordinatewise
- nondecreasing.
-
- MOVE 2: contraction fixed point.
-
- For 0 < lambda < 1, construct
-
- F |--> R( lambda F - P ).
-
- The map is a contraction of the complete metric space of
- continuous convex functions on Delta, in uniform norm, with
- constant lambda. So it has a unique fixed point
- K_lambda.
-
- Because R preserves integrals (Move 1) and P is fixed:
-
- mean( K_lambda ) = -mean(P) / (1 - lambda).
-
- In particular, mean(K_lambda) -> -infinity as lambda -> 1.
-
- Now choose a small positive d < min( mean(P),
- inf over
- Delta minus tau Delta
- of P ),
-
- strictly less than both the mean of P and the boundary
- lower bound of P.
-
- Claim: for lambda close enough to 1,
-
- M_lambda := max_over_Delta K_lambda < - d / (1 - lambda).
-
- Prove by contradiction. If M_lambda is not that negative,
- then on the outer region Delta minus tau Delta we can use
- lambda K_lambda - P <= M_lambda + d (small),
- while on the inner core tau Delta we can use convexity to
- bound K_lambda far below M_lambda, giving
- lambda K_lambda - P << M_lambda - something big.
- The "something big" blows up as lambda -> 1, so on both
- regions we end up with
- max( lambda K_lambda - P ) < M_lambda,
- contradicting the fixed-point equation
- K_lambda = R( lambda K_lambda - P )
- and the fact that max( R F ) <= max( F ).
-
- So the claim holds for lambda close enough to 1.
-
- Fix such a lambda. The pointwise inequality
- (1 - lambda) K_lambda <= - d
- (which is exactly M_lambda < -d/(1-lambda) rearranged) plus
- R's order-preservation and translation-invariance gives
-
- R( K_lambda - P ) <= K_lambda - d.
-
- So: if we apply the full rearrangement R to F_0 := K-P,
- the output is at most K - d. Strict gap of size d.
-
- MOVE 3: realize rearrangement by Hamiltonian maps.
-
- The rearrangement R = R_m ... R_1 is abstract. The paper
- proves it can be implemented, up to arbitrarily small error,
- by an actual Hamiltonian diffeomorphism of V that permutes
- only the fiber variables.
-
- The key sub-lemma is 2D:
-
- Planar rearrangement lemma. Let f_y : [0,1] -> R be a
- jointly continuous family of convex functions,
- g_y the increasing rearrangements. For every eps > 0,
- there is a smooth family of compactly supported
- Hamiltonian isotopies of the disk D(1), with endpoints
- psi_y, such that for every z in D(1),
-
- f_y( pi |psi_y(z)|^2 ) <= g_y( pi |z|^2 ) + eps.
-
- The idea: on D(1), the "moment" pi |z|^2 is the area of
- the concentric disk through z. So we want a Hamiltonian
- map that "sorts" the values of f into increasing order.
- The trick is:
- - choose a fine grid 0 < h_1 < ... < h_l < 1;
- - for each h_j, look at the sublevel set
- { v : f_y(v) < g_y(h_j) + eps' }, an interval;
- - inside each interval, choose nested rectangles of
- prescribed areas h_1 < h_2 < ... < h_l;
- - smooth the rectangle corners into nested smooth
- disks of the same area;
- - Schlenk's "nested disk lemma" (a standard trick)
- gives a Hamiltonian isotopy carrying the standard
- concentric disks of areas h_j to the chosen disks;
- - now any point of radial area v sits below the "next
- grid point up," and evaluating f there is at most
- g(v) + 2 eps'.
-
- Lifting to C^m: for the j-th factor, apply the planar
- construction to f_y(v) = F(y, L(y) v), and lift the
- Hamiltonian by l(y) * B_t(y, z_j / sqrt(l(y))) where
- l(y) is a smooth approximation to the slice length L(y)
- from below. On the outer strip L(y) < p_j, the maps can
- be taken to be the identity (there is nothing to sort
- there), which keeps the construction smooth over Delta.
-
- Because the planar Hamiltonian vanishes near the boundary
- of the disk, the lift vanishes near the outer face
- b_nu . p = c_nu. This is the "outer face preservation"
- property used in the Moser step later.
-
- Summing the errors over j = 1, ..., m gives
-
- (K - P) o q <= R(K - P) + sum_j eps_j
- <= K - d + sum_j eps_j
- < K
-
- provided sum eps_j < d.
-
- Finally approximate K by a smooth function; the error is
- smaller than d/4, so the strict gap survives.
-
-QED (comparison lemma).
-
-WHAT WE GET.
- From a mean inequality, a strict pointwise inequality. From
- a convex function P, we get a smooth toric K and a Hamiltonian
- q that turn the mean surplus into a pointwise surplus.
-
- The next section uses that surplus to cut disjoint layers.
-
-
-6. STEP B: PACKING OVER A HANDLE
+ (H1) the average of P over Delta is positive:
+ (1/area) int_Delta P dp > 0; and
+ (H2) for some 0 < tau < 1,
+ P > 0 on Delta minus (tau Delta),
+ where tau Delta is Delta scaled toward the origin
+ by tau (its outer faces move to b_v . p = tau c_v).
+ Then there exist a smooth toric function K on Delta and a
+ Hamiltonian diffeomorphism q of V such that
+ (K - P)( q u ) < K(u) for every u in V. (6.2)
+ Moreover q comes from a Hamiltonian isotopy that preserves V
+ and, near each outer face b_v . p = c_v, is generated by
+ Hamiltonians whose Poisson bracket with b_v . p vanishes
+ uniformly in time.
+
+Meaning of (6.2). "K-P composed with q is strictly below K"
+says: after the fiber-slide q, the leftover function K-P is below
+K with a uniform margin. That margin is exactly what Part B
+spends to build disjoint layers. The outer-face condition says
+the slide never pushes points across the slanted walls of Delta,
+so the model keeps its shape.
+
+6.2 Rearrangement in one coordinate.
+ For a continuous convex F on Delta and a coordinate j, let
+ R_j F be the result of applying Lemma 5.4 to the slice of F
+ at fixed other moments y=(p_i, i != j). The slice is the
+ segment 0 <= p_j <= L(y), where
+ L(y) = min over v with b_{v,j} > 0 of
+ ( c_v - sum_{i!=j} b_{v,i} y_i ) / b_{v,j} .
+ (This is the furthest p_j can go before hitting an outer wall;
+ it exists because Delta is bounded.) By Lemma 5.4, each R_j
+ sends continuous convex functions to continuous convex
+ functions, preserves integrals, order, translation, is
+ nonexpansive, and makes the output nondecreasing in p_j while
+ keeping any monotonicity in the other coordinates. Applying
+ R_1,R_2,...,R_m in turn gives R = R_m o ... o R_1, whose
+ output is nondecreasing in EVERY coordinate.
+
+6.3 The contraction fixed point.
+ For 0 < lambda < 1 consider the map on the (complete, uniform
+ norm) space of continuous convex functions on Delta:
+ F |-> R( lambda F - P ).
+ Since R is nonexpansive with constant 1 and lambda<1, this map
+ is a contraction with constant lambda, so it has a unique fixed
+ point K_lambda:
+ K_lambda = R( lambda K_lambda - P ). (6.4)
+ Integrals are preserved by R, and the integral of P is fixed,
+ so taking integrals in (6.4) gives
+ int K_lambda = lambda int K_lambda - int P,
+ i.e. int K_lambda = - (int P)/(1 - lambda). (6.5)
+ Since int P > 0 by (H1), the average of K_lambda tends to
+ minus infinity as lambda -> 1. Also K_lambda is coordinatewise
+ nondecreasing, so its minimum over Delta is K_lambda(0).
+
+6.4 A uniform gap appears.
+ Pick a small number d with
+ 0 < d < min{ mean(P), min over Delta minus tau Delta of P },
+ which is possible by (H1) and (H2).
+ CLAIM: for lambda sufficiently close to 1,
+ M_lambda := max_Delta K_lambda < - d/(1 - lambda). (6.6)
+ Proof of the claim. Suppose instead M_lambda >= -d/(1-lambda).
+ Two cases cover Delta.
+ OUTER BAND (p in Delta minus tau Delta). There P >= d, and
+ K_lambda <= M_lambda, so
+ lambda K_lambda - P <= lambda M_lambda - P
+ <= (M_lambda + d) - d
+ = M_lambda ,
+ where we used lambda M_lambda <= M_lambda + d, an immediate
+ rearrangement of M_lambda >= -d/(1-lambda). The inequality is
+ strict because P > d there.
+ INNER CORE (p = tau v in tau Delta, v in Delta). Convexity of
+ K_lambda gives
+ K_lambda(tau v) <= (1-tau) K_lambda(0) + tau M_lambda ,
+ and K_lambda(0) <= average(K_lambda) = -mean(P)/(1-lambda), so
+ K_lambda(p) <= tau M_lambda - (1-tau) mean(P)/(1-lambda).
+ Multiply by lambda, subtract P(p) >= P_min = min_Delta P:
+ lambda K_lambda(p) - P(p)
+ <= tau M_lambda - lambda(1-tau)mean(P)/(1-lambda) - P_min.
+ Subtract M_lambda and use -M_lambda <= d/(1-lambda) (again
+ from the supposition):
+ (tau-1) M_lambda - lambda(1-tau)mean(P)/(1-lambda) - P_min
+ <= (1-tau) [ d - lambda mean(P) ] / (1-lambda) - P_min .
+ Since d < mean(P) and lambda -> 1, the bracket d - lambda mean(P)
+ is eventually negative, say <= -c0 < 0, so the last line tends
+ to -infinity; hence for lambda close enough to 1,
+ lambda K_lambda - P < M_lambda
+ on the core as well. Both bands give max( lambda K_lambda - P )
+ < M_lambda. But K_lambda = R(lambda K_lambda - P) and
+ max(RF) <= max F (R never raises the maximum, by the window-min
+ formula), so max( lambda K_lambda - P ) >= max K_lambda =
+ M_lambda, a contradiction. Hence (6.6) holds.
+ Fix such a lambda. Multiply (6.6) by (1-lambda): we have
+ (1-lambda) K_lambda <= -d pointwise on Delta. (6.7)
+ Apply R to both sides of the identity (K_lambda-P) =
+ lambda K_lambda - P + (1-lambda) K_lambda:
+ R(K_lambda - P) <= R(lambda K_lambda - P) - d
+ = K_lambda - d, (6.8)
+ using translation invariance (the (1-lambda)K_lambda <= -d
+ part shifts the argument down by at least d) and (6.4).
+ So the abstract re-sorting R already produces a strict gap d.
+
+6.5 Realizing R with actual Hamiltonian maps.
+ R is abstract. We realize it by genuine Hamiltonian maps,
+ one coordinate at a time, with arbitrarily small error.
+ This is the content of:
+ LEMMA 6.9 (lifted rearrangement). For every continuous
+ convex F on Delta, every coordinate j, and every eps>0,
+ there is a Hamiltonian diffeomorphism q_j of V with
+ F( q_j u ) <= (R_j F)(u) + eps for all u in V,
+ generated by an isotopy that preserves V and, near each
+ outer face, has Hamiltonians Poisson-commuting with
+ b_v . p (uniformly in time).
+ We prove Lemma 6.9 in Section 6.6 below; for now take it as a
+ black box with error eps.
+ Choose q_j for j=1,...,m with errors eps_1,...,eps_m whose
+ total is < d/2. Let q = q_1 o q_2 o ... o q_m (applied in
+ this order). Chaining the inequalities,
+ (K_lambda - P)(q u) <= R(K_lambda - P)(u) + sum eps_j
+ < K_lambda(u) - d + d/2
+ < K_lambda(u).
+ Finally approximate K_lambda uniformly on Delta by a smooth
+ function K, with error < d/4 (polynomial approximation works).
+ The last comparison changes by < d/4 + d/4, still leaving
+ (K - P)(q u) < K(u).
+ The isotopy preserves V and all outer-face moments on small
+ collars by construction (each q_j does, and we take common
+ smaller collars for the finitely many j). This proves
+ Lemma 6.1.
+
+6.6 Proving the lifted rearrangement (Lemma 6.9).
+ The whole difficulty is one-dimensional, on a disk. Write
+ D(1) = { z in C : pi|z|^2 < 1 }.
+ STEP 1 (planar realization). LEMMA 6.10. Let f_y:[0,1]->R
+ be a jointly continuous family of convex functions (y in a
+ compact parameter set), with rearrangements g_y. For every
+ eps>0 there is a smooth family of compactly supported
+ Hamiltonian isotopies of D(1), with time-one maps psi_y, such
+ that
+ f_y( pi |psi_y(z)|^2 ) <= g_y( pi |z|^2 ) + eps.
+ Idea. On D(1) the quantity pi|z|^2 is the area of the
+ concentric disk through z; "sorting f" means moving points so
+ that larger f sits at larger radius. Concretely:
+ - choose a fine grid 0<h_1<...<h_l<1 and, for each h_i, a
+ nested sublevel interval I_i(y) = {v : f_y(v) < g_y(h_i)+
+ eps'}, each of length > h_i (that is what g_y means);
+ - inside I_i(y) place a smooth disk of area h_i, nested in i,
+ chosen smoothly in y (a partition of unity preserves the
+ strict nesting margins);
+ - by the "nested-disk" lemma below, a Hamiltonian isotopy of
+ D(1) carries the standard concentric disks of areas h_i
+ onto these chosen disks;
+ - then any point of area v lands below the next grid value,
+ giving f_y <= g_y + const*eps'.
+ The NESTED-DISK LEMMA (standard; see the technical paper,
+ Lemma cmp:planar-disks). Given smooth families of strictly
+ nested closed disks E_1(y)<<...<<E_l(y)<<D(A) of areas
+ a_1<...<a_l, there is a smooth family of compactly supported
+ Hamiltonian isotopies of D(A) carrying each centered disk of
+ area a_i onto E_i(y). The proof: first build ordinary
+ diffeomorphisms carrying the disks (shrink, move, expand -
+ always possible in 2D), then correct them to be area
+ preserving by Moser's trick using a primitive form whose
+ restriction to each boundary circle is exact; the correction
+ leaves the disks fixed, and on a disk a compactly supported
+ area-preserving isotopy is Hamiltonian. This is the one
+ place we lean on a named standard construction; everything
+ else is short.
+ STEP 2 (lift to the fiber). In the j-th factor of C^m, a
+ slice has length L(y). Choose a smooth l(y) <= L(y), very
+ close to L(y) and >= eta/2 on the region L(y)>=eta, via a
+ smooth "soft minimum" of the affine functions defining L,
+ shifted down a little. Apply Lemma 6.10 to
+ f_y(v) = F(y, l(y) v) and lift the generating Hamiltonian
+ B_t(y,.) on D(1) to the whole fiber by
+ H(y, z_j) = l(y) * B_t( y, z_j / sqrt(l(y)) ).
+ Because H does not involve the angles of the OTHER
+ coordinates, those moments are unchanged; only the j-th angle
+ moves. On slices of length <= l(y) the inequality
+ F(q u) <= R_j F + eps holds by Lemma 6.10; on the outer strip
+ l(y) < p_j <= L(y) we can take the identity map (there R_j F
+ already dominates F to within the modulus of continuity of F,
+ by the window formula), and the identity is smooth. Choose
+ eta, then the scale error l vs L, then the planar error eps,
+ in that order, to make the total < eps. The result is a
+ smoothed Q, and a Hamiltonian flow tangent to V.
+ STEP 3 (outer faces). If b_{v,j}=0 the j-th motion does not
+ change b_v . p at all (it commutes). If b_{v,j}>0, a point on
+ the face b_v.p=c_v has p_j = L(y) its maximum, while the
+ construction keeps p_j <= (1-delta) l(y) with l(y)>=eta/2, so
+ b_v.p stays strictly below c_v near the face; the motion is
+ supported away from the face and preserves it as a set. This
+ gives the uniform outer-face control promised in Lemma 6.1.
+ QED (Lemma 6.1 and Lemma 6.9).
+
+WHAT WE HAVE. A single integral condition (mean positive) plus
+a positivity-away-from-center condition (H2) yields a strict
+pointwise gap (6.2), delivered by an honest Hamiltonian map that
+respects the slanted walls of the polytope. Part B converts that
+gap into disjoint layers.
+
+
+7. PART B: PACKING OVER A SURFACE WITH A HANDLE
----------------------------------------------------------------
-
-Now we actually place the balls. The setup is a base surface
-C^o of genus >= 1 (with finitely many points removed, at least
-one) times the toric fiber V from Section 5. A positive-genus
-surface has a "handle" -- an essential circle that cannot be
-shrunk to a point. That handle is what allows us to make
-disjoint layers.
-
-THE MODEL.
- We have a family of two-forms on Sigma x V, one for each
- compact bordered surface Sigma inside C^o. Each has the
- form
-
- Omega = omega_V + d Gamma,
-
- with Gamma a toric horizontal one-form (annihilates vectors
- tangent to V, coefficients depending smoothly on both base
- point and moment p). The base curvature
-
- d_C Gamma = kappa(p) dp_1 ^ dp_2
-
- is positive, and its integral over Sigma,
-
- A(p) = int over Sigma of kappa(p),
-
- is assumed to converge uniformly to a continuous positive
- limit H(p) as Sigma grows.
-
- This "surface form" is exactly what the geometric models of
- Section 7 provide.
-
-THE LEMMA (surface packing).
- Let r_1, ..., r_k > 0 be positive numbers, each simplex
- { p >= 0 : sum_j p_j <= r_i } strictly inside Delta and
- strictly away from every outer face. Define the leftover
- function
-
- P(p) = H(p) - sum_i (r_i - sum_j p_j)_+ .
-
- Suppose:
- (i) P is concave;
- (ii) P has positive average over Delta;
- (iii) P is positive on Delta minus tau Delta.
-
- Then for any capacities r_i' < r_i, some model (i.e. some
- sufficiently large Sigma) contains a compact packing of the
- closed balls of capacities r_i' in the interior of
- Sigma x V, with neighborhood embeddings.
-
- That is: the three hypotheses turn into an actual packing
- into the model.
-
-PROOF, IN FIVE STEPS.
-
- STEP 1: shrink to a finite model.
- Hamiltonian comparison (Section 5) provides K and q
- with (K-P)(qu) < K(u). The uniform gap is
-
- g = min_v [ K(q^{-1}v) - K(v) + H(p(v)) - C_r(p(v)) ]
-
- where C_r(p) is the cap sum. g > 0 by the lemma's
- hypotheses.
-
- Choose 0 < eps < g/8 (this is very small). Take Sigma
- so large that
- |A(p) - H(p)| < eps
- for all p in Delta (this is possible by uniform
- convergence of A to H). From now on, "A" means the
- finite base area, "H" the limiting area.
-
- We will use the comparison based on the LIMITING area H,
- not the finite A. This is why the lemma is stated in
- terms of H.
-
- STEP 2: place the base area on an annulus.
- Inside the handle choose an annular region A^+ with a
- positive coordinate s (going around the annulus) and an
- angular coordinate t in R/Z. A closed one-form
- beta on the base is supported in a transverse annulus
- B, and on A^+ looks like
-
- beta|A^+ = f(t) dt, f >= 0, integral f = 1.
-
- So beta has period exactly 1 around the essential
- circle. This is the "handle form"; its integral around
- the handle is 1.
-
- Now modify the base area. Pick a smooth smaller area
- kappa_sm(p) that (a) agrees with kappa(p) near the base
- boundary, (b) is independent of p on A^+, and (c) has
- small total integral:
-
- sup_p int_Sigma kappa_sm(p) < eps.
-
- Let
- H_*(p) = int_Sigma ( kappa(p) - kappa_sm(p) ),
- kappa_1(p) = kappa_sm(p) + H_*(p) chi_A,
-
- where chi_A is a bump on the annulus A with integral 1.
-
- Then kappa_1 > 0, |H_* - H| < 2 eps, kappa_1 - kappa has
- zero total integral. By a standard "relative area
- primitive" lemma, we can normalize the base one-form
- Gamma so that on the annulus
-
- d_C Gamma_1 = kappa_1, Gamma_1|A = gamma + s H_* beta,
-
- where d gamma = kappa_sm on the annulus. The total form
- is still symplectic.
-
- STEP 3: build the "layer path."
- Choose smooth positive toric functions h_i on Delta with
-
- h_i(p) > (r_i - sum_j p_j)_+,
- 0 < sum_i h_i - C_r < eps.
-
- (This makes h_i slightly larger than the "cap" function
- for the i-th ball.)
-
- Define
- J(v) = K(v) - H_*(p(v)),
- R_*(v) = K(q^{-1}v) - J(v) - sum_i h_i(p(v)).
-
- Then R_* >= g - 3 eps > 0 by the gap.
-
- The gap says: on each fiber v, the s-coordinate (going
- around the handle) can interpolate a certain coefficient
- from J(v) at the lower end of the handle to K(q^{-1}v)
- at the upper end, in steps of h_i(p(v)) with positive
- leftover R_*(v).
-
- Concretely, define a family of forms on A x V by
-
- Omega_lambda = d gamma + D_lambda^*(omega_V + d(G_lambda beta))
-
- where D_lambda is a Hamiltonian isotopy of the fiber
- (moving from identity to q as the s-coordinate runs
- from 0 to 1), and G_lambda is a scalar function on
- A x V determined by the interpolation. The suspension
- formula (standard in symplectic topology) shows that
- Omega_lambda is symplectic whenever the s-derivative
- of the scalar part is nonnegative.
-
- At the ends, the form matches the model from Step 2,
- so Omega_lambda extends globally.
-
- Refine G from "just interpolating" to "interpolating
- with marked layers": choose disjoint sub-intervals
- I_1, ..., I_k of the s-axis, and a residual tail I_*.
- Along I_i, increment G by h_i(p(v)) alone. Then the
- form on layer i is
-
- omega_V + d(G_f beta) with G_f = a_i(p) + rho_i(s) h_i(p).
-
- The "a_i(p)" part contributes a toric Hamiltonian to the
- fiber, and the "rho_i(s) h_i(p)" part contributes the
- linear growth in s.
-
- STEP 4: restore the original form via Moser.
- The modified form is symplectic and has the same
- boundary conditions on the base boundary. But we want
- it back on the original model. Use a Moser isotopy:
-
- - the family of forms has the same cohomology class
- at each stage,
- - the family preserves every outer face b_nu . p
- (because the construction uses toric functions and
- outer-face-commuting Hamiltonians),
- - the base boundary collar is fixed.
-
- The "Moser with outer faces" lemma guarantees an
- isotopy that undoes the deformation while keeping the
- base boundary collar fixed and (crucially) never
- pushing any trajectory across an outer face.
-
- So the final picture: a diffeomorphism psi from the
- "layer-modified" model back to the original model.
-
- STEP 5: embed one closed ball in each layer.
- In the i-th layer, the form is exactly
-
- omega_V(v') + dx ^ dT
-
- after a coordinate change, where:
- (s, T) are base coordinates,
- x = L(s,T) + rho_i(s) h_i(p) (a base function),
- v' = fiber coordinates with the p-parameters.
-
- The region of this product form that is available in
- the i-th layer is
- 0 < T < 1, 0 < x < h_i(p(v')).
-
- Now we want to fit a closed ball of capacity r_i' < r_i.
- A planar trick (a corollary of the nested-disk lemma):
- for any A > r_i' and delta > 0, a neighborhood of the
- closed disk of area A in the (x, T) plane embeds
- symplectically so that
-
- point of radial area w sits at
- x < w + delta, 0 < T < 1.
-
- Choose delta < r_i - r_i'. Then for every point of the
- ball,
- w + sum_j p_j(v') <= r_i' (by the ball's
- definition),
- x < w + delta < r_i - sum_j p_j(v') (subtract),
- hence x < h_i(p(v')) (by h_i > cap).
-
- So the ball fits strictly inside the layer with room to
- spare. The layer is compactly contained in the strip,
- and the strip is away from the outer faces. So the
- closed ball is inside the layer with compact closure,
- and the embedding extends to a neighborhood.
-
- Different layers have disjoint base intervals I_i, so
- their fiber images are automatically disjoint.
-
- Compose with psi^{-1} from Step 4. Done: we have a
- packing of the k closed balls inside the original model.
-
-WHAT WE GET.
- From the mean inequality of Section 5 and the handle of a
- positive-genus surface, we get a concrete packing in an
- explicit surface-plus-toric model. The packing stays away
- from the outer faces of Delta, so it survives any further
- deformation that respects those faces.
-
- Section 7 builds such models inside the symplectic ball.
-
-
-7. STEP C: GEOMETRIC MODELS AND TRANSFER
+Part A gave a strict pointwise gap in the fiber. Part B spends
+that gap: an essential loop ("handle") in a base surface lets a
+closed one-form with period 1 cut the model into disjoint slabs,
+one ball per slab.
+
+7.1 The model.
+ Let C^o be a closed connected oriented surface of genus >= 1
+ with at least one point removed, and let
+ Sigma_1 << Sigma_2 << ...
+ be an exhaustion by compact bordered subsurfaces. Suppose
+ that on a neighborhood of each Sigma_l x V (V the toric
+ domain of Section 2.6) there is a two-form
+ Omega_l = omega_V + d Gamma_l,
+ where Gamma_l is a toric horizontal one-form: it kills every
+ vector tangent to the fiber V, and its coefficients depend
+ smoothly on the base point and on p. Write d_C for the base
+ differential (p held fixed). Suppose
+ d_C Gamma_l = kappa_l(p) is a POSITIVE area form on
+ Sigma_l for every p, and
+ A_l(p) := int over Sigma_l of kappa_l(p)
+ converges to H(p) UNIFORMLY in p, H continuous > 0.
+ (The models for different l need not fit together.) The
+ identity that makes "positive curvature" mean "symplectic" is
+ (omega_V + d Gamma)^{m+1}
+ = (m+1) omega_V^m ^ d_C Gamma, (7.2)
+ for any toric horizontal Gamma on a surface; it holds because
+ the only other possible top-degree term would need two moment
+ one-forms squeezed into one slot. So d_C Gamma > 0 makes
+ Omega_l symplectic. (This also extends smoothly across the
+ fiber axes p_j = 0.)
+
+7.3 THE SURFACE PACKING LEMMA 7.4.
+ Let r_1,...,r_k > 0, with each full simplex
+ { p >= 0 : sum_j p_j <= r_i } lying strictly inside Delta and
+ disjoint from every outer face. Put
+ C_r(p) = sum_i ( r_i - sum_j p_j )_+ ,
+ P(p) = H(p) - C_r(p).
+ Suppose (i) P is concave; (ii) the average of P over Delta is
+ positive; (iii) P > 0 on Delta minus tau Delta for some
+ 0 < tau < 1. Then for any r_i' < r_i there is some l for
+ which Int(Sigma_l x V) contains a disjoint packing of the
+ closed balls B^{2m+2}(r_i'), each embedding defined on a
+ neighborhood of its ball.
+
+7.5 Step 1: pin the gap on a finite surface.
+ Lemma 6.1 gives a toric K and a Hamiltonian q with
+ (K-P)(q u) < K(u). Writing u = q^{-1} v,
+ K(q^{-1}v) - K(v) + H(p(v)) - C_r(p(v)) > 0
+ for every v in V. The minimum of the left side over the
+ compact V is some g > 0 (the uniform gap). Choose
+ 0 < eps < (1/8) min( g, min over Delta of H ).
+ Take l so large that sup over p of |A_l(p) - H(p)| < eps
+ (uniform convergence) and Sigma_l contains a torus with a
+ disk removed (genus >= 1 makes a handle). From now on H is
+ the limiting area; the finite A_l only approximates it.
+
+7.6 Step 2: move the base area onto a handle.
+ Inside the handle choose nested annuli A << A+ << Int(Sigma),
+ with positive coordinates (s,t) on A+, t in R/Z (t goes
+ around), and a transverse annulus B. There is a smooth
+ closed 1-form beta supported in B with
+ beta = f(t) dt on A+, f >= 0, int_0^1 f(t) dt = 1.
+ So beta has period exactly 1 around the essential loop: this
+ "handle form" is what will measure out the layers.
+ Choose a smooth nondecreasing S(s), 0 near the bottom of A
+ and 1 near the top, with S' supported inside A. Then
+ chi_A = S'(s) ds ^ beta
+ is a nonnegative 2-form, supported in A, of integral 1.
+ Next choose a positive area form kappa_sm(p) that agrees
+ with kappa on a thin boundary collar, is independent of p on
+ A+, and has tiny total area:
+ sup over p of int over Sigma of kappa_sm(p) < eps.
+ (Fix any reference area lambda, write kappa = a(y,p) lambda
+ with a bounded, blend to a small constant away from A+.)
+ Define
+ H_*(p) = int over Sigma of ( kappa(p) - kappa_sm(p) ),
+ kappa_1(p) = kappa_sm(p) + H_*(p) chi_A .
+ Then H_* > 0, |H_* - H| < 2 eps, kappa_1 > 0, and
+ kappa_1 - kappa has total integral zero and compact interior
+ support.
+
+ RELATIVE AREA PRIMITIVE LEMMA 7.5. Let eta_p be a smooth
+ family of 2-forms on a bordered surface, supported in a fixed
+ interior compact set, with int_Sigma eta_p = 0 for all p.
+ Then there is a smooth family of 1-forms alpha_p, supported
+ in a fixed interior set, with d_C alpha_p = eta_p.
+ Proof sketch. In one coordinate chart with eta = h dx ^ dy
+ and int h = 0, the explicit
+ alpha = A dy - rho(x) B(y) dx,
+ A(x,y)= int_{-inf}^x ( h(v,y) - rho(v) h_0(y) ) dv,
+ h_0(y)= int h(x,y) dx, B(y)= int_{-inf}^y h_0(v) dv,
+ gives d alpha = eta with compact support (rho a fixed bump of
+ integral 1). Patch finitely many charts with a partition of
+ unity and move the "common bump" part across overlaps; the
+ zero total integral kills the leftover common coefficient.
+ This is a fixed linear operator, so it is smooth in p.
+
+ Apply Lemma 7.5 to eta = kappa_1 - kappa, getting a toric
+ horizontal alpha with d_C alpha = kappa_1 - kappa. Set
+ Gamma_1 = Gamma_0 + alpha with Gamma_0 = Gamma_l, then
+ normalize on the annulus A. Choose a base one-form gamma on
+ A+ with d gamma = e lambda (e the constant value of kappa_sm
+ on A+); a primitive exists on a bipunctured annulus. The
+ form delta(p) = Gamma_1 - gamma - S(s) H_*(p) beta is
+ base-closed on A+; subtract its period c(p) times beta and an
+ exact correction d_C(zeta_A F), and one obtains Gamma_1
+ globally horizontal, toric, equal to Gamma_0 near the base
+ boundary, with
+ d_C Gamma_1 = kappa_1, Gamma_1 = gamma + S(s) H_*(p) beta on A.
+ Because (7.2) is linear in d_C Gamma and d_C Gamma_1 > 0, a
+ symplectic interpolation from Gamma_0 to Gamma_1 is immediate.
+
+7.7 Step 3: build the layer path (the exact deformation).
+ Choose smooth positive toric h_i on Delta with
+ h_i(p) > ( r_i - sum_j p_j )_+ and 0 < sum_i h_i - C_r < eps
+ (use a smoothed clamp (x + sqrt(x^2+eta_i^2))/2 to round the
+ corner of (.)_+ upward). Put
+ J(v) = K(v) - H_*(p(v)),
+ R_*(v) = K(q^{-1}v) - J(v) - sum_i h_i(p(v)).
+ Then R_* >= g - 3 eps > 0: on each fiber there is a positive
+ "residue" left over after the k bumps h_i are accounted for.
+ Let q_s := q_{S(s)} be the comparison isotopy, the identity
+ near the bottom of A and q near the top. Define a suspension
+ map D(s,u)=(s,q_s u). For a Hamiltonian q_s generated by Q_s
+ the standard suspension identity is
+ D^* omega_V = omega_V + d( (Q_s o q_s) ds ). (7.6)
+ (Differentiate in s; both slots agree.) Now form, on A x V,
+ Omega_lambda = d gamma + D^*( omega_V + d(G_lambda beta) )
+ gluing to omega_V + d Gamma_1 outside A. By (7.2) and (7.6),
+ Omega_lambda is symplectic as soon as d_s G_lambda >= 0; at
+ the ends G_lambda matches Gamma_1, so it extends globally.
+ Refine to put the bumps in STRICTLY ORDERED, disjoint
+ s-intervals I_1,...,I_k and a residue interval I_*: choose
+ nondecreasing step functions rho_i, rho_* and set
+ G_f(s,v) = J(v) + sum_i rho_i(s) h_i(p(v)) + rho_*(s) R_*(v).
+ Its s-derivative is >= 0; its bottom value is J and its top
+ value is (H_*+J) o q^{-1} = K o q^{-1}. On the i-th interval
+ it is a toric expression
+ G_f = a_i(p(v)) + rho_i(s) h_i(p(v)), a_i = J + sum_{j<i} h_j .
+ This is a genuine exact path of symplectic forms from the
+ model Omega_0 to a final form Omega_f. (This suspension "lift"
+ is the same device Schlenk uses in symplectic folding.)
+
+7.8 Step 4: undo the deformation with Moser.
+ Every variation primitive above is horizontal and is
+ supported away from the base boundary. Near each outer face
+ the comparison isotopy Poisson-commutes with the defining
+ moment mu_v = b_v . p (that was Lemma 6.1's boundary clause),
+ so all the functions used are invariant under the face's
+ Hamiltonian vector field. Thus
+ MOSER-WITH-OUTER-FACES LEMMA 7.7. Suppose Omega_t is a
+ smooth family of symplectic forms on a neighborhood of
+ Sigma x V with d_t Omega_t = d alpha_t, alpha_t horizontal
+ and vanishing in a fixed collar of d Sigma x V, and near
+ each outer face iota_{X_{mu_v}} Omega_t = -d mu_v. Then the
+ Moser isotopy preserves Sigma x V and its interior, is the
+ identity near the base boundary, and is defined on a
+ neighborhood of the compact region.
+ Proof. Solve iota_{Y_t} Omega_t = -alpha_t. Horizontality
+ gives alpha_t(X_{mu_v}) = 0, so d mu_v(Y_t)=0 near the face;
+ trajectories cannot cross any outer level. The Moser
+ computation d/dt(psi_t^* Omega_t) = psi_t^*( d alpha_t +
+ d iota_{Y_t} Omega_t ) = 0 finishes it.
+ Applying 7.7 yields a diffeomorphism psi, defined on a
+ neighborhood, with
+ psi^* Omega_f = Omega_0,
+ preserving Sigma x V and its interior. We will undo it with
+ psi^{-1} at the end.
+
+7.9 Step 5: one closed ball per layer.
+ Work in (s,t,v) coordinates on the i-th interval. Where
+ f > 0 the coordinate T with dT = beta runs over (0,1). Write
+ d gamma = ell(s,T) ds ^ dT, L(s,T) = int_{s_i^-}^s ell d sigma,
+ with s_i^- the bottom of I_i; then ell>0 and dL ^ dT = d gamma.
+ Set
+ x = L + G_f - a_i = L + rho_i(s) h_i(p(v)).
+ At fixed (T,v) this is strictly increasing in s (ell>0), so it
+ is a coordinate. In (x,T,v) the form becomes
+ omega_V(v) + dx ^ dT + d a_i ^ dT .
+ Set v' = phi_{a_i}^T v, the time-T flow of the toric
+ Hamiltonian a_i (its angles move, its moments are preserved).
+ This turns the last display into the clean product
+ omega_V(v') + dx ^ dT .
+ The available region in the i-th layer is therefore
+ 0 < T < 1, 0 < x < h_i(p(v')).
+ Now fit the ball B^{2m+2}(r_i') into the product. The planar
+ fact we need (a corollary of the nested-disk lemma 6.10): for
+ any target area A and any delta > 0, a neighborhood of the
+ closed planar disk of area A embeds symplectically into the
+ (x,T)-plane so that a point of radial area w lands at
+ 0 < T < 1, 0 < x < w + delta. (7.8)
+ Choose 0 < delta < r_i - r_i' and use (7.8) for the LAST
+ complex coordinate of B^{2m+2}(r_i'), keeping the first m
+ coordinates as v'. A point of the closed ball then has
+ w + sum_j p_j(v') <= r_i' (it is in the ball), so
+ x < w + delta < r_i - sum_j p_j(v') < h_i(p(v')) (two
+ uses above). Hence the whole closed ball lands compactly
+ inside the available region, away from the outer faces, with
+ uniform positive slack; so the embedding extends to a tubular
+ neighborhood of the ball. Different layers use disjoint base
+ intervals I_i, hence disjoint images. Undoing psi (Step 4)
+ returns the packing to the original model Omega_0. This
+ proves Lemma 7.4.
+
+WHAT WE HAVE. From "P has positive mean and is positive away
+from center" plus a handle, we get an actual disjoint packing of
+the requested balls in the abstract surface-times-fiber model,
+with honest symplectic embeddings and neighborhoods. Part C
+builds such models inside the real ball and carries the packing
+home.
+
+
+8. PART C: KAEHLER MODELS AND SYMPLECTIC TRANSFER
----------------------------------------------------------------
-
-Now we have to build surface-plus-fiber models inside the
-target ball, and to move packings between them. The four
-main tools live in geometry.tex.
-
-7.1 RELATIVE KÄHLER MOSER
-
- Classical Moser's method: given a smooth path of symplectic
- forms omega_t = omega_0 + d alpha_t with all omega_t
- symplectic and nondegenerate, integrate the vector fields
- X_t defined by
- iota_{X_t} omega_t = -alpha_t
- into an isotopy psi_t with psi_t^* omega_t = omega_0.
-
- On a compact complex manifold, this shows: two Kähler forms
- in the same cohomology class are related by an isotopy
- preserving the Kahler property.
-
- The paper's RELATIVE version: if you also want to
- preserve a list of disjoint smooth complex submanifolds
- S_1, ..., S_e as SETS, you can arrange that psi_t(S_j) = S_j.
- The trick: along each S_j, prescribe the first jet of a
- corrector function f_t so that the Moser field is tangent to
- S_j.
-
- This is the tool that lets us "leak" past forbidden
- divisors when we come back to the target ball. When we say
- "remove H_infty from PP^n," we mean "after applying a
- Moser isotopy that preserves both H_infty and the blowup
- divisor E."
-
-7.2 NORMAL-CONE TRANSFER
-
- Setup. Let T be a smooth projective variety, Z a smooth
- closed submanifold, L an ample rational polarization. Let
- T' = Bl_Z T, Y = PP_Z( 1 + N_{Z/T} ),
- where N_{Z/T} is the normal bundle of Z inside T, and
- PP_Z(1 + N_{Z/T}) is the projective bundle obtained by
- adding a trivial line to N_{Z/T}.
-
- Choose rational d > 0. Suppose two classes are ample:
-
- on T': pi^* L - d E_Z,
- on Y: L|_Z + d U.
-
- (Here E_Z is the exceptional divisor and U is the dual
- tautological class.)
-
- THE LEMMA.
- Let F subset T be a finite union of disjoint smooth
- complex submanifolds (a set we want to keep avoiding).
- Let omega_T be any Kähler form in class L, and omega_Y
- any Kähler form in class L|_Z + d U invariant under
- scalar rotation of the normal summand of Y.
-
- Then every compact subset of
- Y minus ( D_infty union pi_Y^{-1}(F cap Z) )
- has a neighborhood admitting a symplectic embedding
- into
- ( T minus F, omega_T ),
- with its given form omega_Y.
-
- Here D_infty = PP_Z(N_{Z/T}) is the "infinity
- section" of Y.
-
- Idea of the proof.
- Use the CLASSICAL construction of deformation to the
- normal cone:
-
- X = Bl_{Z x {0}} ( T x A^1 ) -> A^1.
-
- Its fiber at t != 0 is T; its fiber at t = 0 is the
- union of T' and Y, glued along D_infty.
-
- Put a rational polarization L - d [Y] on the total
- space X. It restricts to pi^* L - d E_Z on T' and
- to L|_Z + d U on Y. Ampleness on each component
- plus finiteness of the map
- T' u Y (disjoint) -> central fiber
- implies ampleness on the central fiber, which is an
- open condition; so the total polarization is ample
- near zero.
-
- For any compact K in Y away from D_infty and
- pi_Y^{-1}(F), the map Y -> T along the deformation
- (i.e. flowing from Y into a nearby smooth fiber T_t)
- is defined on a neighborhood of K. That flow is
- our symplectic embedding.
-
- In short: deformation to the normal cone is a "video"
- that starts at Y (at t = 0) and ends at T (at t -> 0 but
- nonzero). The construction carries symplectic structures
- along the video.
-
- Intuition for why this is useful. A projective manifold
- has a "zoom into Z" limit where it becomes the projective
- bundle Y. If Y is a nicer target (e.g. a toric model over
- a curve), we can do the packing there and zoom back. This
- is how the paper transfers packings from the "surface plus
- toric fiber" model into the actual ball.
-
-7.3 EXPLICIT NORMAL-BUNDLE FORM
-
- Suppose Z is a curve (complex dimension 1) and
- N_{Z/T} = A^b + A^b + ... + A^b (l summands),
- where A is an ample line bundle on Z with Chern curvature
- sigma > 0, and b > 0 is an integer. If bd < 1, the
- class sigma + d U on Y = PP_Z(1 + N) has an explicit
- Kahler representative of the shape
-
+Part C is the bridge. It builds, inside projective space, the
+"surface times toric fiber" models Part B needs, and it carries a
+packing done in such a model back into the real ball. We present
+five tools; Sections 9-11 assemble them.
+
+We fix one convention. Write PP^N for complex projective space
+and O(k) for its line bundles; a projective LINE measured by
+O(1) has area 1. "U" always means U = c_1(O(1)) of a projective
+bundle's tautological line, the "vertical size". All our models
+live on complex projective varieties and carry Kähler forms whose
+cohomology class is a positive rational combination of these
+classes.
+
+8.1 Tool 1: Moser's lemma, relative version.
+ The classical Moser trick: if a smooth path of Kähler forms
+ omega_t = omega_0 + d alpha_t stays Kähler, solve
+ iota_{X_t} omega_t = -alpha_t and integrate X_t to an isotopy
+ psi_t with psi_t^* omega_t = omega_0. So two Kähler forms in
+ the same cohomology class are isotopic.
+ We need to keep some complex submanifolds fixed as SETS.
+ RELATIVE KAEHLER MOSER LEMMA 8.1. Let X be compact
+ complex, omega_0, omega_1 cohomologous Kähler, and let S_1,...,
+ S_e be pairwise disjoint closed complex submanifolds. There
+ is an isotopy psi_t from the identity with
+ psi_1^* omega_1 = omega_0 and psi_t(S_j) = S_j .
+ If a compact Lie group acts holomorphically and the forms and
+ all S_j are invariant, the isotopy can be made equivariant,
+ and then preserves every connected piece of the fixed locus.
+ Proof. omega_t = (1-t)omega_0 + t omega_1 is Kähler for all t.
+ Pick alpha with d alpha = omega_1 - omega_0. Because S_j is
+ complex, T X restricted to S_j splits as T S_j plus a normal
+ bundle N_{j,t} = (T S_j)^{omega_t} (the omega_t-orthogonal
+ complement). Prescribe the first derivative of a corrector
+ f_t along S_j by
+ f_t = 0 and df_t = 0 on T S_j, df_t = -alpha on N_{j,t}.
+ This is a well-defined smooth family of covectors (the split
+ is a direct sum); realize it as a function in a fixed tube
+ around S_j and multiply by a cutoff equal to 1 near S_j.
+ Add up over j (the tubes are disjoint, compactness of the
+ parameter interval fixes the tubes). Set alpha_t = alpha+df_t.
+ Then iota_{X_t} omega_t = -alpha_t is tangent to each S_j,
+ because the right side kills N_{j,t}. Its flow therefore
+ preserves each S_j, and the standard computation
+ d/dt(psi_t^* omega_t) = psi_t^*( dd alpha_t + d iota_{X_t} omega_t )
+ = 0 gives psi_1^* omega_1 = omega_0. The equivariant claim:
+ average alpha over G; the normal splits are invariant; average
+ f_t; the field becomes invariant, hence equivariant, and its
+ flow fixes the fixed-point locus. Done.
+
+8.2 Tool 2: transport inside a projective degeneration.
+ A flat family f : X -> B over a smooth curve B (with a marked
+ point 0) has fibers that are projective varieties, all but
+ finitely many equal to a fixed one, and near 0 possibly a
+ singular or reducible central fiber X_0 whose components we
+ understand.
+ PROJECTIVE TRANSPORT LEMMA 8.2. Let L be a relatively
+ ample rational polarization. Let Y be a smooth projective
+ component of X_0. After shrinking B around 0:
+ (1) sections on Y extend, so a relative Kähler form can be
+ chosen whose restriction to Y is the Fubini-Study form
+ of Y in L; if a torus acts, the basis can be chosen by
+ weights so the restriction is torus-invariant.
+ (2) any compact K in the smooth ("submersion") locus of f
+ transports, by a genuine symplectic map, into a nearby
+ smooth fiber X_t, avoiding any closed subset K avoids.
+ (3) if a compact group acts and the total form is invariant,
+ this transport is equivariant; for a Hamiltonian circle
+ action it preserves the moment map up to an additive
+ constant on each connected transport domain (zero if
+ one point's moment matches).
+ Proof idea. Relative Serre vanishing R^1 f_*( I_Y L^N ) = 0
+ for large divisible N means every section of L^N on Y lifts to
+ a section over a neighborhood of 0. Lifting a basis of Y and
+ adding sections that vanish on Y gives a relative projective
+ embedding; its pullback Fubini-Study form (divided by N) is the
+ desired relative Kähler form Omega, equal to the Y-form on Y.
+ To move points: at each point of the submersion locus, lift a
+ base tangent vector to be Omega-orthogonal to the fiber. That
+ lift exists uniquely (the fiber restriction is symplectic).
+ Flow along it a short base path from 0 to t. The contraction
+ of the horizontal lift with Omega is zero on fiber vectors, so
+ the fiber forms are preserved: the flow is the symplectic
+ transport. Compact K flows for a uniform time. Invariance of
+ Omega makes the horizontal lift equivariant, hence the flow;
+ the moment difference mu_t circ Phi_t - mu_0 is d-closed, hence
+ constant on a connected domain.
+
+8.3 Tool 3: the normal-cone transfer.
+ This is the central geometric move: "zoom into a submanifold
+ Z, read off the projective normal bundle, do the packing there,
+ zoom back".
+ NORMAL-CONE TRANSFER PROPOSITION 8.3. Let T be smooth
+ projective, Z in T a smooth closed submanifold of positive
+ codimension (possibly disconnected), L ample rational. Set
+ T' = Bl_Z T, Y = PP_Z( 1 + N_{Z/T} ),
+ D_inf = PP_Z( N_{Z/T} ) (the "infinity section" of Y).
+ Here N_{Z/T} is the normal bundle of Z in T; "1 + " adds a
+ trivial line. Suppose for some rational d > 0 the two classes
+ pi^* L - d E_Z on T' (E_Z = exceptional divisor),
+ L|_Z + d U on Y
+ are ample. Let F in T be a finite union of disjoint smooth
+ complex submanifolds (a set to dodge), with a suitable
+ disjointness of its preimage. Let omega_T be any Kähler form
+ in class L and omega_Y any Kähler form in class L|_Z + dU
+ that is invariant under rotating the normal line.
+ Then every compact subset of
+ Y minus ( D_inf union pi^{-1}(F cap Z) )
+ has a NEIGHBORHOOD admitting a symplectic embedding into
+ ( T minus F , omega_T )
+ carrying the given form omega_Y. In particular a compact
+ packing done in Y transfers, with neighborhoods, into T.
+ Proof. Form the deformation to the normal cone,
+ X = Bl_{Z x {0}} ( T x A^1 ) -> A^1 .
+ Its fiber away from 0 is T. Its central fiber (fiber at 0)
+ is the reduced union of T' and Y, glued along D_inf. (This
+ is classical intersection theory: in local coordinates the
+ blow-up of the ideal (x_1,...,x_r,t) has a t-chart where
+ x_i = t v_i and Y is the exceptional at t=0, and x_1-chart
+ giving T'; one checks smoothness and flatness chart by chart.)
+ Put the rational polarization L - d[Y] on all of X. On the
+ two components it restricts to the two ample classes in the
+ statement. Ampleness on both components gives ampleness on
+ the (reduced) central fiber, because the natural map from the
+ disjoint union of components to the central fiber is finite
+ and surjective; ampleness is then open in t, so the
+ polarization is ample on X over a neighborhood of 0.
+ The nonzero fibers are T with class L, as desired.
+ Choose (Lemma 8.2) a relative Kähler form whose Y-restriction
+ is invariant; adjust omega_Y to it by the relative Moser
+ Lemma 8.1 (which preserves D_inf and the vertical forbidden
+ pieces). Now a compact K in Y off D_inf and off pi^{-1}(F)
+ sits in the submersion locus and its image in X is disjoint
+ from the inverse image of F; Lemma 8.2 transports a
+ neighborhood into a nearby T minus F with a Kähler form in L.
+ Finally Lemma 8.1 on T (preserving F) matches that form to
+ omega_T. All maps are defined on neighborhoods; shrink at
+ each of the finitely many compositions. Done.
+
+INTUITION. A projective T with a submanifold Z has a "zoom
+limit" where it looks like the projective bundle Y over Z. If Y is
+a nicer place to pack (a surface times toric fiber, thanks to Part
+B), we pack there, then run the zoom backwards into T. The
+forbidden set F (often a hyperplane, or curves) is what survives
+as "excluded" after the zoom.
+
+The remaining two tools (8.4, 8.5) give the explicit forms and
+identify the shell; they occupy the next sections of Part C.
+
+
+8.4 Tool 4: the explicit normal-bundle form.
+ This is where the abstract "surface times toric fiber" model
+ becomes concrete numbers.
+ BUNDLE-FORM LEMMA 8.4. Let A be an ample line bundle on a
+ smooth projective Z with a Hermitian metric whose Chern form
+ is sigma > 0 (a positive 2-form). Suppose the normal bundle is
+ N = A^b (+) A^b (+) ... (+) A^b (l summands),
+ b a positive integer. Let d > 0 with b d < 1. Then the
+ class sigma + d U on Y = PP_Z(1 + N) has an invariant
+ Kähler representative of the affine shape
omega_Y = sigma + d( sum_{j=1}^l p_j alpha_j ),
- p_j >= 0, sum_j p_j < d,
-
- where alpha_j is the angular connection on the j-th normal
- line (period one, curvature -b sigma), and p_j are the
- standard fiber moments.
-
- The (p_1, ..., p_l) coordinates are nonnegative and their
- sum is bounded by d. So the fiber over each point of Z is
- a polytope of ALLOWED moment values, which is exactly the
- "Delta" of Section 5's toric domain V. This is the
- precise sense in which Y is a "surface plus toric fiber"
- model.
-
- For Z = C a curve and Sigma subset C a bordered subsurface,
- the model restricted to Sigma x V has a toric horizontal
- primitive Gamma with base curvature
-
- kappa(p) = ( 1 - b sum_j p_j ) sigma.
-
- The integral of kappa over Sigma tends to (deg A) times
- ( 1 - b sum_j p_j ) as Sigma grows towards C. That is
- exactly the limiting base area H(p) used by the surface
- packing lemma (Section 6).
-
- So the normal-bundle model Y is precisely a surface-plus-
- fiber model, with H(p) = (deg A) (1 - b sum p_j). This is
- the reason the constructions work.
-
-7.4 BLOWUP COMPLEMENT IS A SHELL
-
- Let n >= 2, o a point of PP^n, H_infty a hyperplane not
- containing o. Let T = Bl_o PP^n, E the exceptional
- divisor, H the pullback of the hyperplane class of PP^n.
-
- For 0 < a < 1, the complement T minus (E union H_infty)
- with its standard Kähler form in class H - aE is
- symplectomorphic to the shell
-
- { z in C^n : a < pi sum_j |z_j|^2 < 1 }.
-
- Proof sketch. Use the "symplectic cut" construction
- (Lerman 1995): act on PP^n x C by a circle, reduce at a
- suitable level, and you get exactly the blowup T with its
- standard form. Removing E and H_infty leaves the shell.
- A direct model-check then matches the affine Fubini-Study
- coordinates to standard complex coordinates in C^n.
-
- This is the punchline: "blowup a point, remove two
- divisors, get the shell." The shell is precisely what
- Cases 2 and 3 need for the reserved-ball reduction.
-
-7.5 PUTTING IT TOGETHER
-
- A model for the target often looks like this:
-
- PP^n -> (blow up at o) T = Bl_o PP^n -> D subset T
- (a curve or
- a hyperplane)
- |
- v
- Y = PP_D(1 + N_{D/T})
-
- with the class L|_D + d U on Y. That class is ample by
- choosing d small.
-
- The surface-packing lemma (Section 6) works inside Y when
- we can identify Y's base D as a punctured curve of positive
- genus. So:
-
- - pack into Y,
- - transfer back to T via normal-cone transfer,
- - identify T minus (E u H_infty) with a shell in C^n
- via the blowup-complement lemma,
- - add the reserved central ball if applicable.
-
- That is the whole pipeline. Sections 8 flesh it out for
- the three cases.
-
-
-8. THE THREE CASES IN DETAIL
+ p_j >= 0, sum_j p_j < d, (8.5)
+ where alpha_j is the period-1 angular connection of the j-th
+ normal line (curvature -b sigma) and p_j are the standard
+ fiber moments. The term p_j alpha_j is smooth across the
+ axis z_j = 0. If Z is a CURVE and Sigma in Z is a compact
+ bordered subsurface, the truncation of this model to any
+ compact moment polytope strictly inside sum p_j < d admits
+ a toric horizontal primitive (as in Part B), with base
+ curvature
+ kappa(p) = ( 1 - b sum_j p_j ) sigma . (8.6)
+ For bordered exhaustions of C minus finitely many points the
+ integrated base areas converge uniformly (on that polytope) to
+ H(p) = (deg A) ( 1 - b sum_j p_j ) . (8.7)
+ Proof. In the affine chart N in PP(1+N), with normal
+ coordinates w_1,...,w_l and their Hermitian norms ||.||, set
+ omega_Y = sigma + dd^c[ d log( 1 + sum_j ||w_j||^2 ) ].
+ (dd^c = i d d-bar up to a constant; it is a real (1,1) form.)
+ The bracket is d times the local weight of the dual
+ tautological metric, so its curvature represents dU globally.
+ The normal-circle moments come out
+ p_j = d ||w_j||^2 / ( 1 + sum_l ||w_l||^2 ),
+ and using radii sqrt(p_j/pi) identifies each affine fiber with
+ the open ball sum_j p_j < d. Differentiating the potential
+ and writing the metric connection gives (8.5); in a unitary
+ frame alpha_j = d theta_j + A_j and p_j d theta_j is the
+ standard smooth angular part, so all is smooth at the axes.
+ To see positivity, at any base point pick a holomorphic frame
+ whose metric has zero first derivative; the form then splits
+ into a positive VERTICAL Fubini-Study piece plus the
+ HORIZONTAL piece (1 - b sum_j p_j) sigma. This still holds
+ on the projective infinity chart, where sum_j p_j = d, and
+ there the horizontal factor is at least 1 - b d > 0. So
+ omega_Y is Kähler. For rational d a divisible multiple is the
+ curvature of a positive Hermitian line bundle, hence ample by
+ Kodaira's theorem.
+ For Z a curve: every complex line bundle on a bordered
+ surface is smoothly trivial (the obstruction lives in
+ H^2(Sigma) = 0), so pick unitary frames on a slightly larger
+ bordered neighborhood, giving alpha_j = d theta_j + A_j, and a
+ base primitive gamma of sigma. Then
+ omega_Y = sum_j d p_j ^ d theta_j + d Gamma_0,
+ Gamma_0 = gamma + sum_j p_j A_j,
+ which is toric horizontal with base differential (8.6).
+ Integrating over Sigma, and using int_Sigma sigma -> int_C
+ sigma = deg A (uniform because the coefficient is continuous
+ and bounded on the fixed compact polytope), gives (8.7).
+ Done.
+
+ REMARK. Formula (8.7) is EXACTLY the "limiting area H(p)"
+ demanded by the surface packing Lemma 7.4: H is affine in
+ p = sum p_j, positive as long as sum p_j < 1/b, and its
+ "slope" -b controls how fast the base area shrinks as moments
+ grow. The negative caps (r_i - sum p_j)_+ are then subtracted
+ from this H to form P. This is the whole reason the proof
+ works: the normal-bundle model IS a surface-times-toric-fiber
+ model with an explicitly known H.
+
+8.5 Tool 5: the target ball and its exterior shell.
+ Two identities turn projective data back into honest balls in
+ C^n.
+
+ 8.5.1 The unit ball is PP^n minus a hyperplane.
+ In affine coordinates w in C^n the Fubini-Study moments and
+ phases are
+ p_j = |w_j|^2 / (1 + sum_l |w_l|^2), theta_j = arg(w_j)/2pi,
+ the form is sum_j d p_j ^ d theta_j, and the map
+ w |-> w / sqrt( pi(1 + sum_l |w_l|^2) ) (8.8)
+ is a symplectomorphism from PP^n minus the hyperplane
+ {z_0 = 0} onto the open unit ball { z : pi sum_j |z_j|^2 < 1 }.
+ (Check the form identity where all w_j are nonzero, then
+ extend smoothly across the axes.)
+
+ 8.5.2 The shell is a blow-up minus two divisors.
+ SHELL LEMMA 8.5. Let n >= 2, o a point of PP^n, H_inf a
+ hyperplane not through o. Let T = Bl_o PP^n, E the
+ exceptional divisor, H = pi^* c_1(O_{PP^n}(1)). For 0 < a < 1
+ there is a Kähler form omega_a in class H - aE such that
+ ( T minus (E union H_inf) , omega_a )
+ is symplectomorphic to
+ ( { z in C^n : a < pi sum_j |z_j|^2 < 1 } , omega_0 ),
+ and the same holds for EVERY Kähler form in that class.
+ Also, for rational A > B > 0, the class AH - BE is ample.
+ Proof. Use the symplectic cut (Lerman). Put homogeneous
+ coordinates o = [1:0:...:0], H_inf = {z_0 = 0}. On
+ PP^n x C let the circle act by
+ ([z_0:z'], v) |-> ([z_0 : e^{2 pi i t} z'], e^{-2 pi i t} v),
+ with Hamiltonian, for unit Fubini-Study on PP^n and the
+ standard form on C,
+ F = mu - pi|v|^2, mu = |z'|^2/(|z_0|^2 + |z'|^2).
+ The level F = a (with 0<a<1) is compact and regular and the
+ circle acts freely on it (if v != 0 it is free in v; if v = 0
+ then mu = a in (0,1) where the projective scalar action is
+ free). So the symplectic reduction is a smooth compact Kähler
+ manifold: the "cut" of PP^n at level a. Its complex model is
+ exactly T: the holomorphic map ([z_0:z'],v) -> ([z_0:vz'],[z'])
+ has values in the graph resolution of projection from o (which
+ is Bl_o PP^n) and its fibers are the free complex scalar
+ orbits; each such orbit meets the level circle in exactly one
+ point, so the quotient is the blow-up. On the open
+ mu > a part, choose v = sqrt((mu-a)/pi) > 0; the extra
+ complex-line form vanishes on that real slice, so the
+ Fubini-Study form is unchanged and (by 8.5.1) the retained open
+ region is precisely the shell of the statement. Reduction of
+ the level mu = a (v = 0) gives E = PP^{n-1} of line area a,
+ matching E; and the proper transform of H_inf is retained with
+ unit line area. These two restrictions fix the class H - aE.
+ Any other Kähler form in H - aE is related to omega_a by the
+ relative Moser Lemma 8.1, which preserves the disjoint
+ divisors E, H_inf and hence their complement.
+ Ampleness of AH - BE: the graph model is a closed subvariety
+ of PP^n x PP^{n-1}; the two hyperplane classes pull back to
+ H and to H - E (projection from o is by hyperplanes through
+ o). Restricting a positive rational product polarization is
+ ample, and
+ AH - BE = (A-B) H + B (H - E)
+ exhibits AH - BE as a sum of a positive multiple of H (ample)
+ and a positive multiple of H - E (ample). Done.
+
+WHAT PART C SUPPLIES. Concrete models: blow up a point, look at
+the projective normal bundle over a hyperplane or over a curve,
+and Lemma 8.4 gives the model an explicit H(p) = (deg A)(1 - b
+sum p_j); Lemma 8.3 moves a packing from that model into PP^n, and
+Lemma 8.5 plus 8.5.1 identify the result as the open unit ball, or
+as the exterior shell of a central ball. Sections 9-11 do this,
+case by case, and check the surface-packing hypotheses (concavity,
+positive mean, positivity away from center) in each case.
+
+
+9. CASE 1: EVERY CAPACITY BELOW 1/2
----------------------------------------------------------------
-
-Normalize the target capacity to 1. In every case, we produce
-an ample model with base a punctured positive-genus curve, whose
-limiting base area H(p) has all the required properties.
-
-8.1 CASE 1: ALL CAPACITIES BELOW 1/2
-
- Take PP^n. Choose a hyperplane H_infty (a generic one). Pick
- n-1 quadrics Q_1, ..., Q_{n-1} in general position. Their
- intersection
-
- C = Q_1 cap Q_2 cap ... cap Q_{n-1} subset PP^n
-
- is a smooth connected curve of degree 2^{n-1} (Bertini's
- theorem ensures this for a generic choice of quadrics). Its
- genus is
-
- g(C) = 1 + (n-3) * 2^{n-2},
-
- which is 1 for n=3, 5 for n=4, 17 for n=5, and so on. In any
- case, POSITIVE for every n >= 3.
-
- Its NORMAL bundle in PP^n is ( O_C(2) )^{oplus (n-1)}: one
- degree-2 line bundle per quadric. Each quadric gives a
- section of O(1) that vanishes precisely on C, so the normal
- direction of C inside Q_1 cap ... cap Q_j has class the
- corresponding O(2).
-
- Deform to normal cone. T = PP^n, Z = C, L = O(1). The
- resulting model is
-
- Y = PP_C( 1 + O_C(2)^{oplus (n-1)} ),
-
- with class L|_C + d U. Ampleness of the two components
- (Bl_C PP^n with pi^*L - d E_C, and Y with L|_C + d U) holds
- for 0 < d < 1/2, which is standard (adjunction in
- PP^n).
-
- The base area (from the explicit normal-bundle form, Section
- 7.3) is
-
- H_0(p) = 2^m ( 1 - 2 sum_{j=1}^m p_j ), m = n-1.
-
- (The 2^m is deg O_C(1) = 2^{n-1} = 2^m, the degree of C as
- a curve in PP^n. The factor (1 - 2 sum p_j) is 1 - b sum p_j
- with b = 2, one per summand of the normal bundle O_C(2).
- Also m = n-1 is the number of normal summands.)
-
- The truncation is at sum p_j <= h for small h. Its integral
- is
-
- J_m(h) = int_{sum p_j <= h} 2^m ( 1 - 2 t ) dp
- = (2^m h^m / m!) ( 1 - 2mh / (m+1) )
-
- which tends to 1 / n! as h -> 1/2.
-
- And 1 / n! is EXACTLY the volume of the unit ball B^{2n}(1).
- So as h approaches 1/2, the model captures ALL of the volume.
-
- Fixing parameters. Given requested capacities R_1, ..., R_k
- all < 1/2 with sum R_i^n < 1, choose rationals
-
- r_* < h < d < 1/2,
-
- where r_* = max r_i with each r_i chosen in
- (R_i, 1/2), and J_m(h) > sum r_i^n / n!.
-
- These exist because J_m(h) -> 1/n! as h -> 1/2, and the
- volume inequality is strict.
-
- Now define P(p) = H_0(p) - sum_i (r_i - t)_+, t = sum p_j.
- Check:
- concavity: H_0 is affine in t, each cap (r_i - t)_+
- is convex, so P is concave.
- mean positive: int P dp = J_m(h) - sum r_i^n/n! > 0.
- boundary: taking tau with r_*/h < tau < 1, we
- have that for p outside tau Delta,
- all caps vanish, and H_0(p) >= positive.
- So all hypotheses of the surface packing lemma hold.
-
- Apply surface packing lemma (Section 6) with parameters
- r_i' < r_i. Get a compact packing of the closed balls of
- capacities r_i' inside one model Y.
-
- The packing lives away from D_infty (the infinity section
- of Y) and away from pi_Y^{-1}(C cap H_infty). Therefore
- it transfers back into PP^n minus H_infty via normal-cone
- transfer.
-
- PP^n minus H_infty is the open ball B^{2n}(1) via
- affine Fubini-Study coordinates (the map
- z -> z / sqrt(pi (1 + sum |z|^2))
- is a symplectomorphism). So the packing lands in the
- desired target.
-
- Neighborhood statement: all constructions are compact and
- defined on neighborhoods; shrink at each composition. Since
- R_i < r_i' < r_i, we have the required slack.
-
-8.2 CASE 2: ONE LARGE CAPACITY, DIMENSION n >= 4
-
- Let A >= 1/2 be the large capacity. Choose rationals
-
- 1/2 < a < 1, A < a, s = 1 - a.
-
- The reserved central ball has capacity A < a. The remaining
- balls (R_1, ..., R_k) must fit into the shell
- { a < pi sum |z_j|^2 < 1 }.
-
- For each i, choose r_i > R_i so that r_i < s and
- sum r_i^n < 1 - a^n.
-
- Construct a "two-stage" model.
-
- Stage 1: PP^n -> hyperplane D = PP^{n-1} -> normal cone.
- Let T = Bl_o PP^n, L = H - aE. Choose a hyperplane D
- not containing o. Deform to normal cone of D. Component
- class on Bl_D T is L - d_1 D, ample for d_1 < s (this
- is standard for blowups at a point of projective space).
- Component class on Y_1 = PP_D(1 + O_D(1)) is L|_D +
- d_1 U_1, ample for d_1 < 1 - a = s. Pick rationals
- r_* < h_1 < d_1 < s.
-
- The base area function is
- H_0^(1)(u) = 1 - u, u = first moment on D,
- i.e. deg(O_D(1)) = 1.
-
- Stage 2. Inside D = PP^{n-1}, run Case 1 with the
- appropriate dimension (n-1 instead of n, so n-1 >= 3
- requires n >= 4). The curve
-
- C = intersection of (n-2) generic quadrics in D
-
- is smooth of genus g(C) = 1 + (n-4) * 2^{n-3}, which
- is 1 for n=4, 5 for n=5, 17 for n=6, and so on. Positive
- for every n >= 4.
-
- The stage-2 model is Y_2 = PP_C(1 + O_C(2)^{oplus
- (n-2)}), with base area
- H_0^(2)(v) = 2^{n-2} ( 1 - 2 sum_{j=2}^{n-1} v_j ),
- v = (v_2, ..., v_{n-1}) the second-stage moments.
-
- The truncation parameters are r_* < h_2 < d_2 < 1/2.
-
- Combining the two stages. The stage-2 model pulls back
- into the stage-1 fiber (Stage 1 gives a ball for the first
- normal direction over D, then Stage 2 gives another toric
- ball inside D). After the "moment change" coordinate
- (z_1, z_2, ..., z_{n-1}) with u = pi |z_1|^2,
- v_j = pi |z_j|^2 (j >= 2), define
-
- zeta_1 = z_1, zeta_j = sqrt(1 - u) z_j (j >= 2),
-
- with new moments
-
- p_1 = u, p_j = (1 - u) v_j (j >= 2).
-
- This is a smooth diffeomorphism on u < 1 including all axes.
- Set w = p_2 + ... + p_{n-1}. The combined polytope is
-
- Delta(h_1, h_2) = { p >= 0 : p_1 <= h_1,
- w + h_2 p_1 <= h_2 }.
-
- Combined base area function (verified in Section 5 of
- applications.tex):
-
- H_0(p) = 2^{n-2} ( 1 - p_1 - 2w ).
-
- Its coefficient (1 - p_1 - 2 w) stays positive on Delta
- because Delta has p_1 <= h_1 < 1 and w + h_2 p_1 <= h_2,
- so w <= h_2 (1 - p_1) and 1 - p_1 - 2w
- >= (1 - p_1)(1 - 2 h_2) >= (1 - h_1)(1 - 2 h_2) > 0
- when h_1 < 1 and h_2 < 1/2.
-
- Volume. The integral of H_0 over Delta(h_1, h_2) is
-
- B_q(h_2) * ( 1 - (1 - h_1)^n ) / n,
-
- where
- B_q(h_2) = 2^q ( h_2^q / q! - 2q h_2^{q+1} / (q+1)! )
- -> 1 / (q+1)! as h_2 -> 1/2,
-
- q = n - 2. So (B_q vol) -> ( 1 - a^n ) / n! as
- h_1 -> s and h_2 -> 1/2.
-
- And (1 - a^n) / n! is EXACTLY the volume of the shell
- { a < pi sum |z_j|^2 < 1 }.
-
- Same concavity / mean / outer-region checks as Case 1. The
- cap functions are (r_i - p_1 - w)_+ and the "+" is still
- concave. Choose truncations so that int P dp > 0, and
- r_*/h_1, r_*/h_2 are both < tau < 1. Then P is positive
- on Delta minus tau Delta.
-
- Apply surface packing lemma to get a compact packing inside
- Y_2. Then transfer:
-
- Y_2 -> (embed via phi into) D minus (C cap H_infty)
- -> (transfer to) Y_1 minus (its infinity + bad set)
- -> (transfer to) T minus (E union H_infty)
- -> (identify) shell in C^n.
-
- Add the reserved central ball of capacity A < a at the
- center. All disjoint, all with compact closure, all with
- neighborhood embeddings.
-
-8.3 CASE 3: ONE LARGE CAPACITY, DIMENSION n = 3
-
- Same setup as Case 2, but n = 3 so the plane inside PP^3
- is PP^2, and a conic in PP^2 has genus 0. No handle.
-
- Replace conic by a plane CUBIC curve C through two
- distinguishable points q_1, q_2. Such C has genus 1.
- Moreover, C minus { q_1, q_2 } still has (non-compact)
- genus 1. So the surface packing lemma still works, with
- C^o = C minus { two points and possibly extra points }.
-
- Extra wrinkle: we need to cut the moment at p_1 = c
- for c < s, to allow Stage 1 and Stage 2 to fit together.
- This is a KÄHLER CUT.
-
- Setup for the cut.
- V_hat = Bl_{(q_1,0), (q_2,0)} PP_{PP^2}(1 + O_{PP^2}(3)).
- This is a projective bundle over PP^2 with a line
- bundle O(3), blown up at two points.
-
- The standard circle rotating the O(3) summand lifts to
- V_hat. Its Hamiltonian is p_1 (the first moment).
- Normalize p_1 = 0 on the "bottom" (the strict transform
- of the zero section).
-
- The fixed loci of the circle are at levels:
- 0, s, d_1,
-
- where s = 1 - a (mass of the exceptional curves) and
- d_1 the ordinary fiber mass. Both are positive for
- 0 < c < s.
-
- The cut.
- Take the Kähler reduction of V_hat x C by the
- diagonal circle at level
- p_1 + pi |v|^2 = c.
-
- Because c < s, every point of the level lies over the
- attracting basin of the bottom section. The complex
- quotient is a projective bundle
-
- Y = PP_D( 1 + N_{D/V_hat} ), D = Bl_{q_1,q_2} PP^2,
-
- with class L|_D + c U_c. This Y is a projective bundle
- over D (a plane surface blown up at two points) with
- normal bundle
-
- N_{D/T} = O_D(3M - e_1 - e_2),
-
- in classes M = pullback of plane hyperplane class,
- e_i = exceptional divisors over q_1, q_2.
-
- Ample classes: L = 3b M - s (e_1 + e_2) with
- b = (2 - a)/3 = (1 + s)/3, s = 1 - a, and L is the
- class on D.
-
- Now Stage 1 is a quadric-surface normal model (different
- from Case 2's hyperplane):
- D = Bl_{q_1,q_2} PP^2,
- Y_1 = PP_D(1 + O_D(3M - e_1 - e_2)).
- Stage 2 is a cubic curve model living in D:
- C = strict transform in D of a smooth plane cubic
- through q_1, q_2.
- Y_2 = PP_C(1 + O_C(3)) is a toric model over C.
- deg_C O(3) = 9.
-
- Base area function (Section 4 of sixdim.tex).
- The full projective system on the cubic has the
- curve-vanishing weight computed as follows. Two
- marked points, each contributing one logarithmic
- mass (s - p_1 - p_2)_+ to the base area. And
- deg O_C(3) = 9, so the base area is
-
- H_0(p) = 9 ( b - p_1 - p_2 ) - 2 ( s - p_1 - p_2 )_+.
-
- Here b = (1 + s) / 3 as above.
-
- The 9 b is the constant coefficient at zero moment:
- it comes from deg O_C(3) = 9 times the parameter b.
- The -9 p_1 - 9 p_2 is the base area loss per unit
- moment. The -2(s - t)_+ is the loss due to the two
- marked-point atoms, each missing mass (s - t)_+.
-
- Volume.
- Integrate over Delta_infty = { 0 <= p_1 <= s,
- 0 <= p_1 + p_2 <= b }:
-
- int H_0(p) dp = (1 - a^3) / 6,
-
- which is EXACTLY the volume of the shell
- { a < pi sum |z_j|^2 < 1 } in complex dimension 3.
-
- (The computation is direct integration in one variable.)
-
- Concavity of H_0.
- As a function of t = p_1 + p_2, H_0 is piecewise affine
- with slopes -7 for t < s and -9 for t > s. Concave.
-
- Apply surface packing lemma with truncated polytope
-
- Delta = { p >= 0 : p_1 <= h_1, p_1 + p_2 <= h_2 },
-
- for h_1 near s and h_2 near b. The cap supports are
- inside Delta; there is enough volume for all balls.
-
- Transfer chain:
-
- cubic model Y_2 ->
- moment-preserving parallel transport into V_hat
- (p_1 <= c preserved exactly) ->
- lower Kähler cut at p_1 = c ->
- Y_1 = PP_D(1 + O_D(3M - e_1 - e_2)) ->
- normal-cone transfer into T = Bl_o PP^3 minus
- (E union H_infty) ->
- identified with shell in C^3 ->
- add reserved central ball of capacity A < a.
-
- All disjoint, all compact, all with neighborhood embeddings.
-
-8.4 WHY THESE THREE COVER EVERYTHING
-
- Given R_1, ..., R_k with sum R_i^n < 1 and all pairs
- R_i + R_j < 1:
-
- - if k = 1, put it by inclusion.
- - if all R_i < 1/2, use Case 1.
- - otherwise exactly one is >= 1/2 (uniqueness from
- the two-ball inequality). Call it A.
- - if n >= 4, use Case 2.
- - if n = 3, use Case 3.
-
- In every case, the assumed hypotheses give exactly the
- properties needed (concavity, positive mean, boundary gap).
- So the sufficiency direction of the theorem is proven.
-
-
-9. WHY DIMENSION FOUR IS DIFFERENT
+Here all requested capacities are < 1/2 and their n-th powers sum
+to < 1. We pack them into the unit ball using a curve that is a
+complete intersection of quadrics, whose normal-bundle model
+captures essentially all the volume.
+
+9.1 Two integral formulas we use.
+ For p = (p_1,...,p_m) >= 0 put t = p_1+...+p_m.
+ (a) int_{sum p <= h} f(t) dp = (1/(m-1)!) int_0^h f(t) t^{m-1} dt,
+ because {sum p <= h} has (m-1)-dimensional cross-sections
+ of volume (h-t)^{m-1}/(m-1)!; differentiating in h gives
+ the stated formula.
+ (b) int_{R^m_{>=0}} (r - t)_+ dp = r^{m+1}/(m+1)!,
+ by taking f(t) = r - t on [0,r] in (a) and using
+ int_0^r (r-t)t^{m-1} dt = r^{m+1}/(m(m+1)) / ((m-1)!).
+ These are the "layer cake" of the unit simplex; the second one
+ is exactly the volume of a ball of capacity r, which is why
+ caps and balls match.
+
+9.2 The quadrics complete intersection.
+ LEMMA 9.3 (curves cut by quadrics). Let N >= 3 and let
+ F in PP^N be a hyperplane. There is a smooth connected curve
+ C, the complete intersection of N-1 quadrics, meeting F
+ transversally. With A = O(1)|_C,
+ deg A = 2^{N-1}, N_{C/PP^N} = (A^2)^{oplus (N-1)},
+ g(C) = 1 + (N-3) 2^{N-2}. (9.4)
+ Moreover, with H the hyperplane pullback and E_C the
+ exceptional divisor of Bl_C PP^N, the class H - d E_C is ample
+ for every rational 0 < d < 1/2, and the class A + d U is ample
+ on PP_C( 1 + (A^2)^{oplus(N-1)} ) for the same d.
+ Proof. Bertini's theorem, applied to the (basepoint-free on
+ each smooth intermediate step, plus on F) quadratic Veronese
+ system, gives N-1 quadrics whose common intersection is smooth
+ and meets F transversally. The Koszul resolution of O_C has
+ terms in negative twists O(-2j), 1 <= j <= N-1; the vanishings
+ H^i(PP^N, O(t)) = 0 (0<i<N) and the absence of sections of
+ negative twists give H^0(O_C) = C, so C is connected. The
+ intersection class is (2H)^{N-1}, so deg A = 2^{N-1}. The
+ defining equations form a regular sequence, so the conormal
+ bundle is (A^{-2})^{oplus(N-1)}; dualizing gives the normal
+ bundle. Adjunction K_C = K_{PP^N}|_C (x) det N_{C/PP^N}
+ = A^{N-3}, giving 2g(C)-2 = (N-3)2^{N-1}, i.e. (9.4).
+ Ample class H - dE_C: the quadric equations generate the ideal
+ I_C(2), so the Rees algebra (+)_j I_C^j(2j) is a quotient of
+ the polynomial algebra; its relative Proj is Bl_C PP^N, and
+ that quotient map is a closed embedding
+ Bl_C PP^N -> PP^N x PP^{N-2}.
+ The two hyperplane classes pull back to H and to 2H - E_C. For
+ 0<d<1/2,
+ H - d E_C = (1 - 2d) H + d (2H - E_C)
+ is the restriction of a positive rational product polarization,
+ hence ample (a projective embedding after clearing denominators).
+ The class A + dU: Lemma 8.4 (with b = 2, l = N-1) gives it an
+ explicit Kähler representative whose horizontal coefficient is
+ 1 - 2 sum_j p_j >= 1 - 2d > 0 everywhere, including infinity;
+ a rational class with a Kähler representative is ample by
+ Kodaira. Done.
+
+9.3 The limiting base area.
+ Now specialize N = n, m = n-1, and apply Lemma 8.4 to the
+ normal bundle (A^2)^{oplus m}: the exponent is b = 2 and
+ deg A = 2^{n-1} = 2^m. Formula (8.7) gives the limiting base
+ area of the normal-bundle model Y = PP_C(1 + (A^2)^{oplus m}):
+ H_0(p) = 2^m ( 1 - 2 sum_{j=1}^m p_j ). (9.5)
+ Its truncation to Delta = { p >= 0 : sum_j p_j <= h }, for
+ 0 < h < 1/2, integrates to
+ J_m(h) = int_{sum p <= h} 2^m (1 - 2t) dp
+ = (2^m h^m / m!) ( 1 - 2mh/(m+1) ), (9.6)
+ by 9.1(a). As h -> 1/2, J_m(h) -> (2^m (1/2)^m/m!)(1 -
+ 2m(1/2)/(m+1)) = (1/m!)(1/(m+1)) = 1/(m+1)! = 1/n!, which is
+ EXACTLY the volume of the unit ball B^{2n}(1). So as h
+ approaches 1/2 the model captures all of the target's volume.
+
+9.4 Fixing the parameters.
+ Let rho_1,...,rho_k be the requested capacities, all < 1/2,
+ with sum rho_i^n < 1. Increase them slightly to RATIONAL
+ r_i > rho_i with r_* := max r_i < 1/2 and sum r_i^n < 1 (open
+ conditions, so rationals are dense enough). By 9.6 and the
+ strict volume inequality choose RATIONAL parameters
+ r_* < h < d < 1/2, with J_m(h) > (1/n!) sum_i r_i^n . (9.7)
+ (First push h close to 1/2 to make J_m(h) exceed the sum, then
+ insert d strictly between h and 1/2.)
+
+9.5 The surface-packing check.
+ Define, on Delta = {sum_j p_j <= h},
+ P(p) = H_0(p) - sum_i ( r_i - sum_j p_j )_+ .
+ We verify Lemma 7.4's three hypotheses.
+ CONCAVE: H_0 is affine in t = sum p_j; each cap (r_i - t)_+
+ is concave (a minimum of two affine functions); a sum of
+ concave functions is concave. Also H_0 > 0 on Delta because
+ t <= h < 1/2 implies 1 - 2t > 0.
+ MEAN POSITIVE: by 9.1(b) and 9.7,
+ int_Delta P dp = J_m(h) - sum_i r_i^{n}/n! > 0.
+ POSITIVE AWAY FROM CENTER: choose tau with r_* / h < tau < 1.
+ If p is in Delta but NOT in tau Delta, then t = sum p_j > tau h
+ > r_*, so every cap (r_i - t)_+ vanishes (since r_i <= r_*
+ < t); hence P = H_0 >= 2^m (1 - 2h) > 0 there. So P > 0 on
+ Delta minus tau Delta, with a uniform lower bound.
+ The auxiliary simplices {sum p_j <= r_i} lie strictly inside
+ Delta (r_i < h) and away from the outer face t = h. All
+ hypotheses of Lemma 7.4 hold.
+
+9.6 Producing and transporting the packing.
+ Take any r_i' with rho_i < r_i' < r_i. Lemma 7.4 packs
+ disjoint closed balls B^{2m+2}(r_i') = B^{2n}(r_i') into
+ Int(Sigma_l x V) for the normal-bundle model, with neighborhood
+ embeddings. (This model sits inside Y = PP_C(1+(A^2)^{oplus
+ m}); it is truncated at t <= h < d, so it lies in the affine
+ part, away from the infinity section D_inf and away from the
+ fibers over the finitely many points C cap H_inf.)
+ Apply the normal-cone transfer Proposition 8.3 to the pair
+ (T, Z) = (PP^n, C), forbidden set F = H_inf. Its hypotheses:
+ the two classes H - d E_C and A + dU are ample by Lemma 9.3
+ since d < 1/2. So the compact packing transfers, with
+ neighborhood embeddings and using the prescribed Fubini-Study
+ form, into
+ PP^n minus H_inf .
+ (Avoiding D_inf and the vertical fibers over C cap H_inf is
+ exactly what keeps the image inside PP^n minus H_inf and off
+ its boundary.) Finally, by 8.5.1 the map (8.8) identifies
+ PP^n minus H_inf symplectically with the open unit ball. Since
+ rho_i < r_i' < r_i < h < 1/2 at every step there is slack, so
+ each closed ball lands with a compact neighborhood inside the
+ target. This packs the rho_i (equivalently the requested
+ R_i) into Int B^{2n}(1) and proves Case 1.
+
+REMARK ON "most of the volume". The model at level h<1/2 captures
+J_m(h) of the 1/n! total, and J_m(h) -> 1/n! as h -> 1/2. That is
+the precise sense in which a thin normal neighborhood of a
+degree-2^{n-1} curve can hold almost the entire volume of
+projective space. The reason: the curve's normal bundle is
+O(2)^{oplus(n-1)}, so its fiber polytope {sum p_j < d} of size
+d up to nearly 1/2 fills almost the whole moment simplex.
+
+
+10. CASE 2: ONE LARGE CAPACITY, n >= 4
----------------------------------------------------------------
-
-The theorem is for n >= 3, i.e. 6 real dimensions and up. What
-about n = 1 (2 real dimensions) and n = 2 (4 real dimensions)?
-
-n = 1 (ordinary plane, real dimension 2):
- Symplectic = area preserving. Packing is purely a matter
- of area; the criterion is just sum R_i < R. No pairwise
- condition needed. Nothing interesting.
-
-n = 2 (real dimension 4):
- This is the richest case. Gromov proved the two-ball
- obstruction in 1985. McDuff and Polterovich (1994) then
- showed that volume + pairwise is NOT sufficient: there
- are infinitely many additional obstructions, coming from
- exceptional classes in blowups of CP^2 and their
- J-holomorphic curves. These are families of "rigid rulers"
- that further restrict packing configurations.
-
- Biran (1997, 1999) established "packing stability" in this
- dimension: on a closed symplectic four-manifold with rational
- symplectic class, any sufficiently large number of EQUAL
- balls can fill an arbitrarily large fraction of the volume.
- Buse-Hind (2011, 2013) extended equal-ball stability to
- higher-dimensional balls and to all closed rational
- symplectic manifolds. Buse-Hind-Opshtein (2016) proved
- strong stability for unequal balls, provided all capacities
- are sufficiently small.
-
- But the sharp criterion (exactly which configurations fit)
- remains an open, and much harder, question in dimension 4.
-
-n >= 3 (6 real dimensions and up):
- The extra dimensions give "floppiness." The paper proves
- that volume + two-ball IS sufficient. All other
- obstructions disappear. So the answer is exactly:
-
- pack iff (sum R_i^n) < R^n AND (all R_i+R_j < R).
-
-This is a strange but common phenomenon in high-dimensional
-geometry: things that are rigid in low dimensions become
-flexible once there is enough room.
-
-
-10. WHAT THE PROOF DOES AND DOES NOT GIVE
+Now exactly one capacity, call it A, is >= 1/2 (uniqueness is
+Section 4.2). The remaining capacities are the rho_i, and by
+hypothesis
+ A^n + sum_i rho_i^n < 1, A + rho_i < 1 for all i.
+The idea: put a slightly larger ball of capacity a > A at the
+center, and pack the rho_i into the shell { a < pi sum |z_j|^2 < 1}
+of the central ball. This is a two-stage construction: first a
+HYPERPLANE inside T = Bl_o PP^n, then a QUADRICS CURVE inside that
+hyperplane.
+
+10.1 Choosing the parameters.
+ The inequalities are strict and finite in number, so we can
+ perturb up to RATIONAL numbers
+ A < a < 1, 1/2 < a, rho_i < r_i < s := 1 - a,
+ sum_i r_i^n < 1 - a^n. (10.1)
+ Note 0 < s < 1/2. Let q = n - 2 (so q >= 2 since n >= 4) and
+ r_* = max_i r_i. Later we will have parameters
+ 0 < r_* < h_1 < d_1 < s, 0 < r_* < h_2 < d_2 < 1/2. (10.2)
+
+10.2 The shell model for the central ball.
+ By the shell lemma 8.5, the complement of (E union H_inf) in
+ T = Bl_o PP^n, with a Kähler form omega_a in the class
+ H - aE, is symplectomorphic to the shell {a < pi sum|z_j|^2<1}.
+ So if we pack the small balls into a compact piece of this
+ shell we can add the central ball of capacity A < a. We now
+ build a surface-times-toric-fiber model of that shell, inside T,
+ carrying class H - aE.
+
+10.3 Stage 1: a hyperplane inside T.
+ E and H_inf are two disjoint smooth divisors in T (E is the
+ blow-up divisor, H_inf the proper transform of a hyperplane
+ through nothing special). Pick a hyperplane D0 in PP^n that
+ does not pass through o and differs from H_inf; its strict
+ transform D = D0 in T is isomorphic to PP^{n-1}, lies in the
+ class [D] = H (it is disjoint from E), and meets H_inf in a
+ hyperplane F of D. The normal line of D in T is O_D(1), and
+ L|_D = O_D(1).
+ Deform to the normal cone of D with a rational parameter d_1.
+ Component classes are
+ (i) on the blow-up Bl_D T: (1 - d_1) H - a E ,
+ (ii) on Y_1 = PP_D(1 + O_D(1)): O_D(1) + d_1 U_1 .
+ Class (i) is (1 - d_1) H - a E, and on the blow-up of PP^n at
+ one point x H - y E is ample for x > y > 0 (Lemma 8.5); here
+ x = 1 - d_1 > a = y because d_1 < s = 1 - a. So (i) is ample.
+ Class (ii): use Lemma 8.4 with A = O_D(1) (b = 1, l = 1,
+ deg A = 1); with d_1 < s < 1 it is Kähler of the affine shape
+ omega_D + d(u alpha_1), dd alpha_1 = -omega_D, 0 <= u < d_1,
+ (omega_D = unit Fubini-Study on D) with base area
+ H_0^(1)(u) = 1 - u . (10.3)
+ The forbidden preimage in Y_1 lies over F (= D cap H_inf);
+ the preimage of E is empty.
+
+10.4 Stage 2: a quadrics curve inside D = PP^{n-1}.
+ Apply the quadrics Lemma 9.3 in D with N = n-1 (needs
+ n-1 >= 3, i.e. n >= 4) and forbidden hyperplane F. Get a
+ smooth curve C, the complete intersection of n-2 quadrics,
+ meeting F transversally, with
+ deg C = 2^{n-2} = 2^q, N_{C/D} = (O_C(2))^{oplus q},
+ g(C) = 1 + (n-4) 2^{n-3} >= 1 .
+ Deform to the normal cone of C in D with parameter d_2. By
+ Lemma 9.3 both component classes are ample when d_2 < 1/2.
+ The normal-bundle component Y_2 = PP_C(1 + (O_C(2))^{oplus q})
+ has, by Lemma 8.4 with sigma = a positive curvature form of
+ O_C(1) of total integral 2^q (b = 2), the affine model
+ omega_X = sigma + d( sum_{j=2}^{n-1} v_j alpha_j ),
+ dd alpha_j = -2 sigma, sum_{j=2}^{n-1} v_j < d_2, (10.4)
+ with base area
+ H_0^(2)(v) = 2^q ( 1 - 2 sum_{j=2}^{n-1} v_j ). (10.5)
+
+10.5 Assembling the two stages.
+ The whole point: the stage-2 model lives over the curve C,
+ which sits inside D, which sits inside the first normal line in
+ T. Explicitly, exhaust C minus F by bordered surfaces Sigma;
+ for a large one, the truncation sum v_j <= h_2 (< d_2) is a
+ compact region in Y_2 away from the infinity section and away
+ from the finitely many forbidden fibers over C cap F. The
+ normal-cone transfer Proposition 8.3, applied to (D, C) with
+ forbidden set F, embeds a neighborhood X of that region
+ symplectically into (D minus F, omega_D). Call it phi : X -> D.
+ (All line bundles on bordered curve pieces are smoothly framed,
+ so no obstruction; phi need only be symplectic.)
+ Now pull back the first normal line of (10.3) along phi. The
+ disk bundle X over Sigma has H^2(X;Z) = H^2(Sigma;Z) = 0, so the
+ pulled-back normal line and the trivial line with the curve's
+ own O_C(1)-connection alpha_{1,C} are equivalent; the
+ difference of their connections is a closed one-form xi on X.
+ Writing eta = i^*xi on the curve (i the zero section) and
+ contracting the disk fibers gives xi = pi^*eta + df for a
+ function f; a gauge change removes df. Hence the pulled-back
+ first connection reads exactly
+ alpha_{1,C} - sum_{j=2}^{n-1} v_j alpha_j + eta ,
+ with eta generally NOT exact (its periods are kept). These are
+ smooth unitary operations, so they fix the first moment u.
+ The lifted (10.3) therefore becomes, on X,
+ sigma + d( u(alpha_{1,C}+eta) + (1-u) sum_{j=2}^{n-1} v_j alpha_j ). (10.6)
+
+10.6 The moment change and the combined model.
+ Use disk coordinates z_1,...,z_{n-1} on the fibers of X with
+ u = pi|z_1|^2, v_j = pi|z_j|^2. The smooth change of variables
+ zeta_1 = z_1, zeta_j = sqrt(1 - u) z_j (2 <= j <= n-1) (10.7)
+ is a diffeomorphism on u < 1 with a smooth inverse there,
+ including all coordinate axes. Its moments are
+ p_1 = u, p_j = (1 - u) v_j (2 <= j <= n-1),
+ with the same angles. Put w = p_2 + ... + p_{n-1}. The
+ truncations u <= h_1 and sum_{j>=2} v_j <= h_2 become
+ Delta = Delta(h_1,h_2) = { p >= 0 : p_1 <= h_1, w + h_2 p_1 <= h_2 }. (10.8)
+ In the new variables the horizontal coefficient of (10.6) is
+ 1 - u - 2w = 1 - p_1 - 2w ,
+ and using w + h_2 p_1 <= h_2 (so w <= h_2(1-p_1)) together with
+ p_1 <= h_1 < 1, it stays positive on Delta:
+ 1 - p_1 - 2w >= (1 - p_1)(1 - 2 h_2) >= (1 - h_1)(1 - 2 h_2) > 0. (10.9)
+ So the combined model is a genuine surface (Sigma) times toric
+ fiber (over Delta), and its limiting base area is
+ H_0(p) = 2^q ( 1 - p_1 - 2w ). (10.10)
+
+10.7 Enough volume.
+ Integrate H_0 over Delta. For fixed p_1 = u, the q = n-2
+ remaining moments range over the simplex {w >= 0 : w <=
+ h_2(1-u)}; using 9.1(a) and the substitution s = (1-u)sigma,
+ int_{w <= h_2(1-u)} 2^q (1 - u - 2w) dw
+ = B_q(h_2) (1 - u)^{q+1},
+ B_q(h_2) = 2^q ( h_2^q/q! - 2q h_2^{q+1}/(q+1)! )
+ -> 1/(q+1)! as h_2 -> 1/2.
+ Then integrating p_1 from 0 to h_1,
+ int_Delta H_0 dp
+ = B_q(h_2) * (1 - (1 - h_1)^n)/n
+ -> (1 - a^n)/n! as h_1 -> s = 1-a, h_2 -> 1/2. (10.11)
+ And (1 - a^n)/n! is EXACTLY the volume of the shell
+ { a < pi sum |z_j|^2 < 1 } (the target minus the central ball
+ of capacity a). So with h_1 close to s and h_2 close to 1/2 the
+ model captures all the shell volume.
+ Fix these parameters rational, by (10.1), so close to the limits
+ that BOTH
+ r_* < h_1 < d_1 < s, r_* < h_2 < d_2 < 1/2, and
+ int_Delta H_0 dp > (1/n!) sum_i r_i^n (10.12)
+ hold simultaneously (the last is an open condition near the
+ limits; d_1,d_2 then fit between the h's and their limits).
+
+10.8 The surface-packing check.
+ Define on Delta
+ P(p) = H_0(p) - sum_i ( r_i - p_1 - w )_+ , w = sum_{j>=2} p_j.
+ CONCAVE: H_0 is affine in (p_1, w); each cap (r_i - p_1 - w)_+
+ is concave. So P is concave.
+ SIMPLICES INSIDE: if p_1 + w <= r_i then p_1 <= r_i < h_1 and
+ w + h_2 p_1 <= (p_1 + w) <= r_i < h_2, so each auxiliary simplex
+ { p_1 + w <= r_i } lies strictly inside Delta and away from both
+ outer faces u = h_1 and w + h_2 u = h_2.
+ MEAN POSITIVE: by 9.1(b) and (10.12),
+ int_Delta P dp = int_Delta H_0 dp - sum_i r_i^{n}/n! > 0
+ (each cap integrates to a full ball of capacity r_i).
+ AWAY FROM CENTER: choose tau with max{ r_*/h_1, r_*/h_2 } <
+ tau < 1. If p is not in tau Delta, either p_1 > tau h_1 > r_*
+ or w + h_2 p_1 > tau h_2 > r_* > p_1 + w; in either case every
+ cap vanishes and P = H_0 >= 2^q (1 - h_1)(1 - 2 h_2) > 0 (no
+ worse than (10.9)). So P > 0 on Delta minus tau Delta with a
+ uniform bound. All hypotheses of Lemma 7.4 hold.
+
+10.9 Producing and transporting the packing.
+ Pick r_i' with rho_i < r_i' < r_i. Lemma 7.4 packs the disjoint
+ closed balls B^{2m+2}(r_i') (here 2m+2 = 2(n-1)+2 = 2n) into one
+ large model of the form (10.6), with neighborhood embeddings,
+ with all moments below p_1 <= h_1 < d_1 and total sum below
+ h_2 < d_2. The chain of transfers, each with neighborhood:
+ (a) The model lies in the stage-2 part, over (D minus F), by
+ the lifting 10.5. The first moment u = p_1 < h_1 < d_1
+ keeps it inside the affine part of Y_1 (below the first
+ normal infinity) over D minus F.
+ (b) Normal-cone transfer Proposition 8.3 with (D, C), forbidden
+ F, moves it into (D minus F) inside D = PP^{n-1}, i.e.
+ puts the packing in the first-stage fiber over D minus F.
+ (This is the phi of 10.5 read backwards.)
+ (c) Now the packing sits in Y_1 = PP_D(1+O_D(1)), with
+ u < h_1 < d_1 < s, off infinity and off the preimage of F.
+ Apply the first normal-cone transfer Proposition 8.3 with
+ (T, D), forbidden F union (points over D cap H_inf). Its
+ component classes are ample: (i) (1-d_1)H - aE (shown
+ ample above) and (ii) O_D(1)+d_1 U_1. So the packing moves
+ into T minus (the forbidden set), inside T minus (E union
+ H_inf).
+ (d) By the shell lemma 8.5, T minus (E union H_inf) with the
+ Kähler form in class H - aE is symplectomorphic to
+ { a < pi sum |z_j|^2 < 1 }.
+ So the rho-balls land in the shell with compact closure. Add
+ the central closed ball of capacity A < a in the middle of the
+ target ball. It is disjoint from the shell packing. Every
+ capacity had slack A < a and rho_i < r_i' < r_i < s, and every
+ transfer was defined on a neighborhood of a compact set, so the
+ closed balls land inside the open unit ball with neighborhood
+ embeddings. This proves Case 2.
+
+WHY TWO STAGES. The hyperplane step converts the outer shell
+{ a < sum < 1 } into "one normal line over D". Since D is just
+PP^{n-1}, and n-1 >= 3, we can re-run the Case-1 quadrics
+construction INSIDE D to get a base curve of positive genus
+g = 1+(n-4)2^{n-3}, exactly as before. When n = 3 there is no
+"n-1 >= 3" to spend: the plane inside PP^3 is PP^2, and a conic
+has genus 0. That is the sole reason Case 3 (Section 11) needs a
+different base curve.
+
+
+11. CASE 3: ONE LARGE CAPACITY, n = 3
----------------------------------------------------------------
-
-The proof is a pure existence proof. Specifically:
-
- It tells you a packing EXISTS.
- It does NOT tell you an algorithm for constructing one.
- It does NOT tell you the shapes of the packed balls.
- It does NOT give numerical effective bounds on how small
- the neighborhoods around the closed balls can be.
-
-That last point is why every "embedding" here means "symplectic
-embedding defined on an open neighborhood of the closed source
-ball." This is the standard convention, but it's worth
-being explicit about.
-
-What the proof does open up:
-
- - The surface-plus-toric-fiber machine (Sections 5-6) is
- quite general. Similar constructions (symplectic folding,
- Moser-type normalizations, positive-genus bases) appear
- across symplectic topology.
- - The interplay between packing and the algebraic structure
- of the Hamiltonian diffeomorphism group (Edtmair 2025)
- was a motivation. The paper's argument doesn't use those
- results, but the connection is real.
- - The normal-cone transfer with a lower Kähler cut
- (Section 7.2 / 8.3) is a reusable device.
-
-What the proof does NOT settle:
-
- - Dimension four. As discussed, the criterion here is
- different (richer set of obstructions) and is still an
- active research area.
- - Stabilized packing: packings after crossing with a fixed
- symplectic surface. Siegel and Yao's original paper
- studied these too; they are not resolved by the present
- theorem.
- - Complex dimension 1 and 2 - trivially resolved and
- richer-respectively, as noted above.
-
-
-11. GLOSSARY
+This is the one six-real-dimensional case that Case 2 cannot
+reach. By Section 4.2 exactly one capacity A is >= 1/2; the rest
+are the rho_i, with
+ A^3 + sum_i rho_i^3 < 1, A + rho_i < 1.
+We again reserve a central ball of capacity a > A and pack the
+rho_i into the shell { a < pi sum_{j=1}^3 |z_j|^2 < 1 }. The
+difficulty (Section 10's "WHY TWO STAGES"): the hyperplane inside
+PP^3 is PP^2, and a conic in PP^2 has genus 0, no handle. We
+replace the conic by a plane CUBIC through two marked points,
+which has genus 1, and we splice the pieces with a KAHLER CUT.
+
+11.1 Parameters.
+ Choose rational
+ 1/2 < a < 1, s := 1 - a (so 0 < s < 1/2),
+ b := (2 - a)/3 = (1 + s)/3 . (11.2)
+ Then 0 < s < b < a < 1 (check b < a and s < b are both easy
+ from s < 1/2). Choose the rho_i's padded up to rationals
+ A < a, rho_i < r_i < s, sum_i r_i^3 < 1 - a^3 . (11.3)
+ The target is the shell of Lemma 8.5:
+ { z in C^3 : a < pi sum_{j=1}^3 |z_j|^2 < 1 }.
+
+11.2 A quadric normal model of the shell.
+ Let T = Bl_o PP^3, o not in H_inf, and L = H - aE. By Lemma
+ 8.5, T minus (E union H_inf) in class H - aE is the shell.
+ Choose a smooth quadric surface through o, general with respect
+ to H_inf, and let D be its strict transform in T.
+ LEMMA 11.4 (the quadric model). D is isomorphic to the
+ plane blown up at two points q_1, q_2:
+ D = Bl_{q_1,q_2} PP^2.
+ Let M be the pullback of the plane hyperplane class and e_i the
+ exceptional classes over q_i. Then
+ H|_D = 2M - e_1 - e_2, E|_D = M - e_1 - e_2, (11.5)
+ hence (the normal class of D in T is [D]|_D = (2H - E)|_D, and)
+ N_{D/T} = O_D( 3M - e_1 - e_2 ), L|_D = 3b M - s(e_1+e_2). (11.6)
+ For every rational 0 < c < s the two normal-cone component
+ classes
+ (1 - 2c) H - (a - c) E on the blow-up Bl_D T, (11.7)
+ 3b M - s(e_1 + e_2) + c U on Y = PP_D(1 + O_D(3M - e_1 - e_2)), (11.8)
+ are ample. Also D cap E and D cap H_inf are two disjoint smooth
+ plane curves (a LINE and a smooth CONIC), both through q_1, q_2.
+ Proof. Projection from o restricted to the quadric becomes a
+ morphism after blowing up o; it contracts the strict transforms
+ of the two ruling lines through o (each has self-intersection
+ -1). In coordinates the quadric is x_0 x_1 = x_2 x_3 with
+ o = [1:0:0:0]; projection to [x_1:x_2:x_3] has the quadratic
+ inverse [u:v:w] -> [vw:u^2:uv:uw], whose resolution is the
+ two-point blow-up at [0:1:0] and [0:0:1]. This gives (11.5),
+ and [D] = 2H - E gives (11.6). Class (11.7) is ample because
+ a - c > 0 and (1-2c)-(a-c) = s - c > 0 (Lemma 8.5 criterion).
+ Class (11.8) is ample by the scalar-invariant-cycle argument:
+ subtracting a small ample class leaves a class that is
+ nonnegative on every invariant curve (the two fixed sections
+ and the fibers, all with positive sizes), hence nef, hence
+ ample after adding the small class back. The curves D cap E
+ and D cap H_inf are the plane line and plane conic through the
+ two marks; for a general quadric the conic is smooth and the
+ line and conic meet only at q_1, q_2 with distinct tangents.
+ Done.
+
+11.3 The lower Kaehler cut.
+ There is an auxiliary threefold whose moment sublevel IS the
+ Y of (11.8). Put
+ V_0 = PP_{PP^2}( 1 + O_{PP^2}(3) ), (a PP^2-bundle of
+ projective LINES), and
+ V_hat = Bl_{(q_1,0),(q_2,0)} V_0, (blow up the two points
+ (q_i, the zero in the fiber)).
+ Write U_1 for the pullback of the tautological class of V_0 and
+ Ehat_i for the point-exceptional divisors. A circle turns the
+ O(3) summand; it lifts to V_hat.
+ LEMMA 11.9 (the cut). Fix s < d_1 < b and suppose V_hat
+ carries a circle-invariant Kähler form in the class
+ A_class = 3b M + d_1 U_1 - s(Ehat_1 + Ehat_2). (11.10)
+ Normalize its first moment p_1 to be 0 on the strict bottom
+ section D = Bl_{q_1,q_2} PP^2. For every rational 0 < c < s,
+ the symplectic (Kähler) cut retaining p_1 <= c is the complex
+ projective bundle Y of (11.8), with an invariant Kähler form in
+ the class 3bM - s(e_1+e_2) + c U. The retained open region
+ { p_1 < c } sits in V_hat and is symplectomorphic to the affine
+ part of Y, in a way that preserves the projection to D on the
+ attracting basin of the bottom section (so it preserves
+ avoidance of the preimages of the plane line-and-conic).
+ Proof. The circle is Hamiltonian (H^1(V_hat;R) = 0: the
+ PP^2-bundle has vanishing H^1 and point blow-ups preserve it).
+ At either blown-up point the tangent weights are 0,0,1 (two
+ plane directions, one vertical), so the fixed loci on the
+ exceptional PP^2 are the projectivized base directions (a line
+ in the bottom section) and the pure vertical point; a line
+ joining them has area s and weight 1, giving moment gap s. An
+ ordinary projective fiber has area d_1 and weight one at the
+ bottom. So the fixed levels are exactly
+ 0, s, d_1 . (11.11)
+ The attracting basin of the bottom section under complex scalar
+ contraction is the total space of the normal line
+ N' = O_D(3M - e_1 - e_2)
+ (checked in local charts at a marked point: y_2 = y_1 v,
+ z = y_1 w, contraction w -> lambda w; the other chart w' = w/v
+ is a fiber coordinate of pi^*O(3)(x)O_D(-e_1-e_2)). It is an
+ attracting BASIN, not a moment sublevel, but it contains
+ {p_1 < s}: contraction strictly lowers the moment on every
+ nonconstant orbit, and a point of moment < s must contract to
+ the bottom (levels 11.11).
+ Now form the Kähler reduction of V_hat x C by the diagonal
+ circle at level
+ p_1 + pi|v|^2 = c . (11.12)
+ Every point of the level lies over the basin (its first moment
+ is at most c < s). On any nonzero pair (w,v) in N' + 1 the
+ complex scalar orbit meets (11.12) exactly once (total moment
+ rises from 0 at contraction to beyond c at expansion), and the
+ circle acts freely there (fiber weights one). The holomorphic
+ quotient of the nonzero pairs is
+ PP_D( N' + 1 ) = Y ,
+ which is exactly (11.8); its Kähler structure is the quotient
+ of the invariant form. The reduction class restricts to
+ 3bM - s(e_1+e_2) on the bottom and has projective fiber area c,
+ determining 3bM - s(e_1+e_2) + cU. A point of {p_1 < c} maps
+ to [x, sqrt((c - p_1(x))/pi)] symplectically (the added real
+ coordinate contributes no form), preserving the base point in
+ D. Done.
+
+
+11.4 The cubic degeneration, and where the cut's Kähler form
+ comes from.
+ Lemma 11.9 assumed V_hat has a circle-invariant Kähler form in
+ the class (11.10). That form is produced by deforming V_hat
+ itself along a family whose special fibre contains a model over
+ a plane CUBIC. Choose a smooth plane cubic C through q_1, q_2,
+ general enough that: it meets each of the line l and conic Q in
+ finitely many points, and it is smooth at q_1, q_2. Such
+ cubics exist (the cubics through q_1, q_2 form a linear system,
+ and avoiding the finitely many bad conditions is open and
+ nonempty). With M = O_{PP^2}(1)|_C, this C is an elliptic curve
+ of degree deg_C M = 3.
+
+ Build the family
+ Z = Bl_{C x {0}} ( PP^2 x A^1 ) -> A^1 .
+ Its fibre away from t=0 is PP^2; its central fibre is the
+ reduced union of the strict plane P and
+ Z_0 = PP_C( 1 + O_C(3) ) .
+ Write U_2 for the dual tautological class on Z_0 and put
+ N = O_Z( 3M - [Z] ).
+ Since [Z] restricts to -U_2 on Z_0 and to 3M on P,
+ N|_{Z_0} = O_Z(3M + U_2), N|_P = trivial . (11.14)
+ Now take the projective bundle PP_Z(1 + N) and blow it up
+ along the two parameter-sections arising from q_1, q_2 in the
+ zero section; call the result f : X -> A^1. These two sections
+ are disjoint, meet the central fibre in its smooth locus, and
+ every NONZERO fibre of f is exactly V_hat. The central fibre
+ has two smooth components
+ W = Bl_{(q_1,0,0),(q_2,0,0)} PP_{Z_0}( 1 + O_Z(3M+U_2) ),
+ W' = P x PP^1 ,
+ glued along the first projective bundle over the infinity
+ section of Z_0. The family is flat (smooth total space with
+ local rings torsion-free over the parameter's local ring).
+ Choose rational parameters s < d_1 < d_2 < b and the relative
+ rational class
+ A_class = 3b M - d_2[Z] + d_1 U_1 - s(E_1 + E_2), (11.15)
+ where U_1 is the dual tautological class of the first bundle and
+ E_1,E_2 are the family blow-up divisors. Its restrictions are
+ A_class|_W = 3b M + d_2 U_2 + d_1 U_1 - s(E_1+E_2),
+ A_class|_{W'} = 3(b - d_2) M + d_1 U_1 . (11.16)
+
+ CLAIM 11.17 (both component classes are ample).
+ W': the class 3(b-d_2)M + d_1 U_1 is a positive product class
+ (b-d_2 > 0 and d_1 > 0), so ample. W: use the two scalar
+ factors, the second lifted to the first bundle by its
+ linearization on U_2. An ordinary surface fibre over C is
+ PP_{PP^1}(1 + O_{PP^1}(1)); its toric diagram is
+ Q = { p_1, p_2 >= 0 : p_1 <= d_1, p_1 + p_2 <= d_2 },
+ with vertices (0,0), (d_1,0), (d_1, d_2-d_1), (0,d_2) and
+ boundary curve areas d_1, d_2-d_1, d_1, d_2, all positive.
+ Over either marked point the fibre adds the exceptional plane
+ of size s; the strict surface fibre is then the corner (0,0)
+ cut by p_1 + p_2 >= s, with five positive edge lengths
+ s, d_1-s, d_2-d_1, d_1, d_2-s (all positive by 11.15).
+ A horizontal invariant irreducible curve must be a strict
+ vertex section (intersecting a generic fibre over C in a
+ torus-fixed finite set); the four vertex sections have classes
+ 9b - 2s, 9(b-d_1), 9(b-d_2), 9(b-d_2) against A_class|_W, all
+ positive (the first is 3+s because 9b-2s = 9(1+s)/3-2s =
+ 3+3s-2s = 3+s). Only the bottom vertex section passes through
+ the two blow-up centres, losing 2s. Now fix any ample class B
+ on W; for small epsilon > 0 every listed curve has
+ nonnegative degree against A_class|_W - epsilon B. Every curve
+ on W is numerically equivalent to a torus-invariant effective
+ cycle (the classical invariant-cycle theorem; the torus is
+ connected and solvable, finitely many orbits not required),
+ so A_class|_W - epsilon B is nef; adding the small ample class B
+ gives ample. Ampleness on the reduced central fibre is
+ ampleness on each component, and relative ampleness is open, so
+ A_class is relatively ample near 0. Done.
+
+ Finally, the first circle acts on the WHOLE family X and
+ preserves the blow-up centres. Averaging any relative Kähler
+ form over this circle keeps it closed, Kähler on the fibres, and
+ invariance-preserving. The nonzero fibres V_hat therefore
+ acquire a circle-invariant Kähler form in the class (11.10), as
+ Lemma 11.9 required. This closes the loop: the cut of 11.3 is
+ fed by the cubic degeneration of 11.4.
+
+
+11.5 The toric model over the cubic, and the base area.
+ We must compute, for the cubic model, the limiting horizontal
+ area H_0 as a function of the two fiber moments. The idea is
+ that the two marked points q_1, q_2 force certain coefficient
+ sections to vanish, and that vanishing is what shrinks the base
+ area.
+
+ The weight decomposition.
+ On the affine chart of Z_0 = PP_C(1 + O_C(3)) the two affine
+ line coordinates w_1, w_2 both have transition bundle O_C(3).
+ WEIGHTS LEMMA 11.18. For the full projective system (of a
+ large divisible power N times A_class) the weight (k_1,k_2)
+ runs over 0 <= k_1 <= N d_1, 0 <= k_2 <= N d_2 - k_1, and
+ its coefficient space on C is
+ H^0( C, O_C((3bN - 3k_1 - 3k_2) M)(-m_k q_1 - m_k q_2) ),
+ m_k = ( sN - k_1 - k_2 )_+ . (11.18)
+ These systems are basepoint-free for large divisible N.
+ Proof. Expand first in the U_1-coordinate: a term of degree k_1
+ leaves coefficient class (3bN-3k_1)M + (Nd_2-k_1)U_2. Expand
+ in the U_2-coordinate: this gives the stated range and the
+ coefficient line. Sections on the blow-up are exactly sections
+ before the blow-up vanishing to total order >= sN at each center;
+ in local coordinates (y, w_1, w_2) the coefficient of
+ w_1^{k_1} w_2^{k_2} must vanish in y to order >= m_k. Writing
+ t = (k_1+k_2)/N, the degree of that line, divided by N, is
+ 9(b - t) - 2(s - t)_+ , (11.19)
+ which is decreasing in t on 0 <= t <= d_2 and at least
+ 9(b - d_2) > 0; the degrees therefore grow uniformly in the
+ weights as N grows. On a genus-one curve every line of degree
+ >= 2 is basepoint-free (evaluate at a point; H^1 vanishes after
+ subtracting it). Done.
+
+ The reduced potential and the base form.
+ TORIC-MODEL LEMMA 11.20. Let Delta be a compact downward
+ polytope strictly below the two upper boundaries of Q (the
+ diagram (11.17)). Over every compact bordered subsurface
+ Sigma of C^o = C minus {q_1, q_2}, the affine moment region
+ has smooth product disk coordinates (including the lower axes)
+ in which the form is
+ Omega = omega_V + d Gamma, d_C Gamma = kappa(p) > 0,
+ with Gamma horizontal and toric, and for a bordered exhaustion
+ of C^o the integrated areas converge UNIFORMLY on Delta to
+ H_0(p) = 9(b - p_1 - p_2) - 2( s - p_1 - p_2 )_+ . (11.21)
+ The same limit holds after deleting finitely many further
+ points.
+ Proof (the parts that matter).
+ In a local frame of M|_C the projective potential is
+ B(y, rho) = (1/N) log( sum_k g_k(y) e^{k . rho} ),
+ rho_j = log|w_j|^2 ,
+ where g_k is the sum of squared lengths of a basis of the
+ coefficient space (11.18). The moments are p = d_rho B; their
+ gradient is a diffeomorphism onto the interior of Q and is
+ smooth at every lower face (the coefficient system contains
+ weights with k_j = 1, so the Jacobian is block-triangular with
+ invertible diagonal blocks there). Taking the partial Legendre
+ transform G(y,p) = inf_rho(B - p.rho) gives, in moment and
+ phase coordinates, the form sum d p_j ^ d theta_j +
+ d(d_y^c G), whose base part kappa(p) = d_y d_y^c G is positive
+ (a Schur-complement of a positive Hermitian Hessian). If the
+ frame changes by e' = h e with lambda = log|h|^2, the potential
+ transforms as G' = G - 3(b - p_1 - p_2) lambda; since
+ d^c lambda = d(arg h /2pi), the primitive changes by an exact
+ term. Hence the local base forms have the transition law of
+ the real class
+ 3(b - p_1 - p_2) c_1(M|_C) . (11.22)
+ On a bordered Sigma (where H^2 = 0) we can pick a global frame,
+ and a radial-homotopy argument produces Gamma with
+ d Gamma = Omega - omega_V, proving the smooth model.
+ The base AREA is the spatial integral of kappa. Near a marked
+ point y=0, sN - k_1 - k_2 > 0 forces the coefficient system to
+ vanish to order m_k, so g_k(y) = |y|^{2m_k} a_k(y) with a_k > 0
+ smooth, and the potential differs by a bounded amount from
+ max over weights of (k/N . rho + lambda . (s - t)_+), lambda =
+ log|y|^2. Taking the Legendre transform gives
+ G(y,p) = ( s - p_1 - p_2 )_+ log|y|^2 + O(1) . (11.23)
+ So G has, at each marked point, an ATOM of the current d d^c G
+ of mass (s - p_1 - p_2)_+ (subtracting the log term leaves a
+ bounded subharmonic function, which has no atom). By (11.22)
+ the TOTAL mass of the class is int_C 3(b-t) c_1(M|_C) =
+ 3(b-t) deg_C M = 9(b-t). Removing the two atoms (one per
+ marked point) leaves the smooth base area
+ H_0 = 9(b - t) - 2(s - t)_+ , t = p_1 + p_2 ,
+ which is (11.21). Uniform convergence of the bordered
+ integrals to H_0 is Dini's theorem (continuous positive forms
+ increasing to a continuous limit on a compact polytope). Done.
+
+11.6 The volume of the model equals the shell volume.
+ The limiting moment domain is
+ Delta_inf = { p_1, p_2 >= 0 : p_1 <= s, p_1 + p_2 <= b }.
+ Because b > s, direct integration of (11.21), with u = p_1,
+ gives
+ int_{Delta_inf} H_0 dp
+ = int_0^s ( (9/2)(b-u)^2 - (s-u)^2 ) du
+ = (3/2)( b^3 - (b-s)^3 ) - s^3/3
+ = s/2 - s^2/2 + s^3/6 .
+ Substituting s = 1 - a, a one-line check gives
+ s/2 - s^2/2 + s^3/6 = (1 - a^3)/6 . (11.24)
+ And (1 - a^3)/6 is EXACTLY the volume of the shell
+ { a < pi sum_{j=1}^3 |z_j|^2 < 1 } = B^{6}(1) minus B^6(a).
+ Meanwhile the full cap of size r_i (< s) integrates to a ball:
+ int_{p_1,p_2>=0} (r_i - p_1 - p_2)_+ dp = r_i^3/6 ,
+ which is the volume of B^6(r_i). So at the limiting domain the
+ model carries exactly the shell volume, and each requested ball
+ is exactly a cap of the H_0 profile.
+
+ CONCAVITY of H_0. As a function of t = p_1 + p_2, H_0 is
+ piecewise affine with slopes -7 (for t < s: derivative of
+ 9(b-t) - 2(s-t) is -9 + 2 = -7) and -9 (for t > s). The slope
+ drops at t = s, so H_0 is CONCAVE. This is the crucial sign:
+ the proof's engine (Part A) needs P concave, and the base area
+ of a cubic with two marks is concave precisely because the two
+ atoms have positive mass.
+
+
+11.7 Packing and the transfer chain for Case 3.
+ Fix rational parameters
+ max_i r_i < h_1 < c < s < d_1 < d_2 < b, s < h_2 < d_2, (11.25)
+ so close to h_1 = s and h_2 = b that on
+ Delta = { p_1, p_2 >= 0 : p_1 <= h_1, p_1 + p_2 <= h_2 }
+ the function
+ P(p) = H_0(p) - sum_i ( r_i - p_1 - p_2 )_+
+ has positive integral (hence positive mean). This is possible
+ because of (11.24), the convergence of Delta to Delta_inf, and
+ the fact that the caps already fit inside every sufficiently
+ close truncation.
+
+ Surface-packing check (Lemma 7.4).
+ H_0 > 0 on Delta (it is decreasing in t and t <= h_2 < b). H_0
+ is concave (slopes -7 then -9); each negative cap is concave, so
+ P is concave. The auxiliary simplices {p_1 + p_2 <= r_i} lie
+ strictly inside Delta and away from both outer faces. Choose
+ tau < 1 with tau h_1 > max_i r_i and tau h_2 > max_i r_i. If
+ p is not in tau Delta, either p_1 > tau h_1 or p_1 + p_2 >
+ tau h_2, so all caps vanish and P = H_0 has a uniform positive
+ lower bound. So all three hypotheses of Lemma 7.4 hold.
+
+ Produce the packing.
+ Construct the cubic family of 11.4 with the chosen d_1, d_2.
+ Remove from its base curve C the two marked points AND all
+ intersection points with l union Q (finitely many). The
+ remaining curve still has genus one and admits connected
+ bordered exhaustions of genus one. Lemma 11.20 supplies the
+ toric models there, with uniform limiting area H_0. Lemma 7.4
+ then packs the closed balls of capacities r_i' (with rho_i <
+ r_i' < r_i) disjointly in one such model, with neighborhood
+ embeddings. The whole chosen model has p_1 <= h_1 < c, avoids
+ the double locus (p_1+p_2 = d_2) and the two blow-ups, and
+ avoids the inverse images of l union Q under the map to the
+ plane.
+
+ MOMENT-PRESERVING TRANSPORT LEMMA 11.26. Let Sigma be a
+ compact bordered surface in the unmarked base, K the compact
+ moment region over it for a polytope strictly below the two
+ upper faces of Q. For sufficiently small nonzero parameter,
+ symplectic parallel transport from the central component W to a
+ nearby fibre V_hat is defined on a neighborhood of K and
+ preserves p_1 EXACTLY, when both moments are normalized to
+ vanish on their bottom sections. If K avoids the inverse image
+ of a closed set, its image does too.
+ Proof. K avoids the central double locus and the two blow-ups,
+ so it lies in the submersion locus. Parallel transport for the
+ averaged total form exists for short time near this compact set
+ (Lemma 8.2). The connection is invariant under the first
+ circle, so the transport Phi is symplectic and equivariant; hence
+ d(p_1^{V_hat} circ Phi) = d p_1^{W}. The difference is thus
+ constant on the connected region. The region includes points
+ with first affine coordinate zero, which are fixed on the first
+ bottom section; a short equivariant transport keeps them there,
+ so both normalized moments are zero and the constant is zero.
+ The avoidance is preserved because the excluded set is closed
+ and transport is short. Done.
+
+ Finish the chain.
+ (a) Transport the model from W into a nearby V_hat via Lemma
+ 11.26: p_1 <= h_1 < c is kept, avoidance of l union Q is
+ kept.
+ (b) Apply the lower Kähler cut of Lemma 11.9 (which requires
+ c < s and the invariant form of 11.4): the balls move into
+ the projective bundle Y of (11.8), off the infinity section
+ and off the two forbidden vertical submanifolds over D cap E
+ and D cap H_inf. The cut form is invariant and is in the
+ class of Lemma 11.4.
+ (c) Apply the normal-cone transfer Proposition 8.3 with
+ (T, D) = (Bl_o PP^3, quadric D), forbidden set F = E union
+ H_inf (whose D-traces are the line l and conic Q handled in
+ step (a)). Component classes (11.7) and (11.8) are ample
+ for 0 < c < s by Lemma 11.4. So the packing transfers into
+ T minus (E union H_inf), with a Kähler form in class H - aE,
+ with neighborhood embeddings.
+ (d) By the shell lemma 8.5, T minus (E union H_inf) with that
+ form is symplectomorphic to { a < pi sum_{j=1}^3 |z_j|^2 <
+ 1 }.
+ Add the central closed ball of capacity A < a. It is disjoint
+ from the shell packing. Every capacity had strict slack
+ (A < a, rho_i < r_i' < r_i < s) and every transfer was defined on
+ a neighborhood of a compact set, so the closed balls land inside
+ the open unit ball with neighborhood embeddings. This proves
+ Case 3.
+
+
+12. ASSEMBLY: THE THEOREM
----------------------------------------------------------------
+We now combine everything.
-Ball of capacity R:
- B^{2n}(R) = { z in C^n : pi sum |z_j|^2 <= R }.
- Radius sqrt(R/pi). Volume R^n / n!.
-
-Capacity, symplectic:
- The natural "area" unit. A capacity-one ball has a complex
- line through the origin of symplectic area exactly one.
-
-Symplectic form omega_0:
- The real 2-form sum_j dx_j ^ dy_j on C^n = R^{2n}. It
- measures signed area on each complex coordinate plane and
- sums those areas.
-
-Symplectic embedding:
- A smooth injection f : U -> U' with f^* omega_0 = omega_0.
-
-Volume-preserving embedding:
- A smooth injection with f^* vol = vol. Much weaker than
- symplectic in dimension >= 2n >= 4. The gap between the
- two is what symplectic geometry is about.
-
-Gromov's two-ball obstruction:
- In any n >= 2, symplectic balls of capacities R_1, R_2
- cannot both fit disjointly into capacity R unless
- R_1 + R_2 < R.
-
-J-holomorphic curve:
- A map from a Riemann surface (usually S^2) into a
- symplectic manifold that is holomorphic with respect to a
- compatible J. These are the "rigid rulers" that detect
- non-volume obstructions.
-
-Conjecture A (Siegel-Yao 2025):
- For n >= 3, volume + two-ball is sufficient. Now a
- theorem.
-
-Toric domain:
- V = { z in C^m : p(z) in Delta } for a "downward" polytope
- Delta in the nonnegative orthant. Standard form:
- omega_V = sum dp_j ^ d theta_j.
-
-Outer face:
- A face of Delta of the form b_nu . p = c_nu with c_nu > 0.
- The coordinate faces p_j = 0 are not outer.
-
-Hamiltonian comparison (Section 5):
- The lemma that upgrades a positive-mean inequality into a
- strict pointwise inequality via increasing rearrangements
- and a contractive fixed-point argument. Output: K (toric)
- and q (Hamiltonian) with (K-P) o q < K.
-
-Increasing rearrangement:
- Given a convex function f on [0,1], its increasing
- rearrangement g is the (unique up to values) function with
- g nondecreasing and the level sets of g having the same
- length as those of f. Formula:
- g(h) = min_{0 <= a <= 1-h} max{ f(a), f(a+h) }.
-
-Surface packing lemma (Section 6):
- Packs disjoint closed balls in a surface-plus-toric model,
- given a concave P with positive mean and a boundary
- positivity property, and given a positive-genus punctured
- base with a handle-form beta of period 1.
-
-Handle form beta:
- A closed one-form on the surface with int(beta) = 1 around
- an essential circle in the handle. This is the mechanism
- that creates disjoint "layers" along one coordinate.
-
-Deformation to the normal cone:
- A one-parameter family over A^1 whose fiber at t != 0 is
- a given T and whose fiber at t = 0 splits into the blowup
- Bl_Z T and the projectivized normal bundle
- PP_Z(1 + N_{Z/T}).
- Used to move packings between these pieces.
-
-Normal-cone transfer (Section 7.2):
- The lemma that transports a compact packing from
- PP_Z(1 + N_{Z/T}) into T itself, via the deformation to
- the normal cone, while avoiding F subset T.
-
-Moser isotopy (Section 7.1):
- Given a smooth path of symplectic forms omega_t = omega_0 +
- d alpha_t, integrate the vector fields X_t defined by
- iota_{X_t} omega_t = -alpha_t into an isotopy psi_t with
- psi_t^* omega_t = omega_0. The engine behind "symplectically
- equalize two forms in the same cohomology class."
-
-Relative Kähler Moser:
- The version of Moser that preserves a list of complex
- submanifolds as sets. Uses a normal first-jet correction
- along each S_j.
-
-Kähler cut (symplectic cut):
- Take V x C, act by the standard diagonal circle, reduce at
- a circle-invariant level p_1 + pi |v|^2 = c. The result
- is a symplectic quotient. Used in Case 3 to build the
- critical projective bundle Y over the two-point blowup of
- PP^2.
-
-Complete intersection of quadrics:
- The set { Q_1 = ... = Q_{n-1} = 0 } in PP^n for n-1
- general homogeneous quadrics. Smooth of degree 2^{n-1},
- genus 1 + (n-3) 2^{n-2}, normal bundle
- O_C(2)^{oplus (n-1)}. This is the "handle" for Cases 1
- and 2.
-
-Shell:
- { z in C^n : a < pi sum |z_j|^2 < 1 }. Symplectically
- equal to Bl_o PP^n minus (E union H_infty) for suitable
- standard classes. The target of the large-ball reduction.
-
-
-12. A SUGGESTED READING ORDER FOR THE PAPER
+ CLAIM 12.1 (sufficiency). Let n >= 3, and let R_1,...,R_k > 0
+ with sum_i R_i^n < 1 and R_i + R_j < 1 for all i < j. Then the
+ closed balls of capacities R_i pack into Int B^{2n}(1) with
+ neighborhood embeddings.
+ Proof. By 8.5.1 normalize the target to capacity 1. If k = 1,
+ then R_1 < 1 and inclusion packs it (Section 4.1). Assume
+ k >= 2. By 4.2 at most one capacity is >= 1/2.
+ - If every capacity is < 1/2, Section 9 (Case 1) applies: all
+ its hypotheses (rho_i < 1/2, sum rho_i^n < 1) are exactly
+ the assumed ones.
+ - Otherwise exactly one capacity, A, is >= 1/2.
+ If n >= 4, Section 10 (Case 2) applies: its hypotheses
+ A^n + sum rho_i^n < 1 and A + rho_i < 1 (all i) are
+ exactly the assumed ones.
+ If n = 3, Section 11 (Case 3) applies with the same
+ hypotheses.
+ In each case the preceding section produces the packing with
+ neighborhood embeddings. So sufficiency holds.
+
+ THEOREM 12.2. Theorem 1 holds: for every n >= 3, k >= 1, and
+ positive R, R_1,...,R_k, the packing exists iff both the volume
+ inequality and all pairwise inequalities hold.
+ Proof. Necessity is Section 3 (volume plus Gromov's two-ball
+ rigidity, then all pairs). Sufficiency is Claim 12.1. Together
+ they prove the "if and only if" in every n >= 3. QED.
+
+
+13. DIMENSION FOUR, AND DIMENSION ONE
----------------------------------------------------------------
-
-The technical paper (symplectic_ball_packings_original.txt) is
-organized in this order, with the main files as follows:
-
- intro.tex statement of Theorem + history + sketch
- comparison.tex Hamiltonian comparison; rearrangement; disks
- surface.tex surface packing; handle form; layers; Moser
- geometry.tex Kähler Moser; normal-cone transfer; explicit
- bundle form; blowup-complement-is-shell
- applications.tex Case 1 (all small); Case 2 (n >= 4, large)
- sixdim.tex Case 3 (n = 3, large); cubic; Kähler cut
- conclusion.tex sufficiency assembly
- bibliography.tex references
-
-A good first-pass reading order:
-
- 1. intro.tex, sections 1-2 (statement + easy necessity).
- 2. comparison.tex, statement of Lemma cmp:comparison and
- its fixed-point proof.
- 3. surface.tex, statement of Lemma surf:packing and Step 5
- (how the walls of a layer hold a closed ball).
- 4. geometry.tex, statement of Lemma geo:transfer and Lemma
- geo:shell ("normal-cone transfer" and
- "blowup-complement-is-shell").
- 5. applications.tex, Proposition app:small (Case 1).
- 6. applications.tex, Proposition app:large (Case 2).
- 7. sixdim.tex, Proposition six:large (Case 3).
- 8. conclusion.tex.
-
-Sections 3-4 of the paper play a supporting role (relative
-Kähler Moser, projective transfer). They can be skimmed on a
-first reading.
-
-For a truly thorough reading, pay attention to:
- - the two "regularity" tricks in the surface packing lemma
- (Step 2: relative area primitive; Step 3: layer path);
- - why the concavity of P is preserved under the fixed-point
- iteration;
- - why the cubic model in Case 3 needs deg O_C(3) = 9 and
- not 3.
-
-Each of these is where a sloppy proof would break.
-
-
-13. THE ONE-LINE MORAL
+The theorem is stated for n >= 3 (real dimension 6 and up). The
+two excluded cases are genuinely different, not laziness.
+
+n = 1 (real dimension 2).
+ Here omega_0 = dx ^ dy is just the ordinary area form, and
+ "symplectic" = "area preserving". Packing is pure area:
+ the k disks fit iff R_1 + ... + R_k < R. No pairwise condition
+ is needed (2-dim packing of round disks is flexible). The
+ second inequality is vacuous. Nothing deep happens.
+
+n = 2 (real dimension 4).
+ This is the rich case. Gromov (1985) proved the two-ball
+ obstruction R_1 + R_2 < R here too. But McDuff and Polterovich
+ (1994) showed volume + pairwise is NOT enough: there are
+ infinitely many further obstructions, coming from exceptional
+ classes of blow-ups of CP^2 and the J-holomorphic curves that
+ represent them. These act as extra rigid rulers that forbid
+ configurations the two-ball rule would allow. Partial
+ understanding: Biran (1997, 1999) proved packing STABILITY
+ (asymptotically, on a closed rational 4-manifold, many equal
+ balls fill almost the whole volume); Buse-Hind (2011, 2013)
+ extended equal-ball stability to higher dimensions and to all
+ closed rational symplectic manifolds; Buse-Hind-Opshtein (2016)
+ proved strong stability for unequal balls, provided every
+ capacity is sufficiently small. But the EXACT criterion (which
+ configurations fit) is still open in dimension four.
+ Our proof uses, in Cases 2 and 3, a base surface of positive
+ genus as the engine (the "handle"). In dimension 4 the natural
+ analogue has no room to build such a handle with the required
+ volume, and the extra J-holomorphic obstructions are real. So
+ dimension 4 needs different ideas.
+
+The moral: once there are at least 3 complex dimensions, the extra
+"floppiness" makes every obstruction disappear except volume and
+the two-ball rule.
+
+
+14. WHAT THE PROOF GIVES, AND WHAT IT DOES NOT
----------------------------------------------------------------
+14.1 It is an EXISTENCE proof.
+ It produces a packing when the two inequalities hold, and proves
+ no packing exists otherwise. It does NOT give:
+ - an algorithm to construct or draw the packing;
+ - the shapes of the packed ball images (they are wildly bent);
+ - effective numerical bounds on how small the neighborhoods
+ around the closed source balls can be.
+ That last point is why every "embedding" in this file means
+ "symplectic embedding defined on an OPEN NEIGHBORHOOD of the
+ closed source ball", and why the target is always the OPEN
+ interior: no image may touch the boundary, and every image has
+ a little room around it. Under finite composition of our
+ constructions, the room is preserved (shrink neighborhoods at
+ each step).
+
+14.2 The three explicit models, in one table.
+ In all rows: target = unit ball (or shell), H(p) is the limiting
+ base area, "cap" is (r_i - sum p_j)_+, and r_i > R_i are padded
+ capacities. Each row's H has slopes whose drop keeps P concave.
+
+ CASE 1 (every R_i < 1/2), Section 9.
+ base C : complete intersection of n-1 quadrics in PP^n,
+ deg 2^{n-1}, genus 1+(n-3)2^{n-2}, normal
+ O_C(2)^{oplus(n-1)}.
+ H(p) : 2^{n-1} (1 - 2 sum_{j=1}^{n-1} p_j).
+ truncate : sum p_j <= h < 1/2.
+ volume : J_m(h) = (2^{n-1} h^{n-1}/(n-1)!)(1 - 2(n-1)h/n)
+ -> 1/n! as h -> 1/2.
+ transfer : normal-cone (PP^n, C) -> PP^n minus H_inf -> ball.
+
+ CASE 2 (one A >= 1/2, n >= 4), Section 10.
+ base : C = quadrics curve inside hyperplane
+ D = PP^{n-1} inside T = Bl_o PP^n; deg 2^{n-2},
+ genus 1+(n-4)2^{n-3}.
+ H(p) : 2^{n-2} (1 - p_1 - 2 w), w = sum_{j>=2} p_j.
+ truncate : p_1 <= h_1 < s, w + h_2 p_1 <= h_2 < 1/2.
+ volume : B_q(h_2)(1-(1-h_1)^n)/n -> (1-a^n)/n! (shell).
+ transfer : cut/fiber chain -> PP^n minus (E union H_inf)
+ -> shell; add central ball of capacity A < a.
+
+ CASE 3 (one A >= 1/2, n = 3), Section 11.
+ base : plane CUBIC C through two marks q_1,q_2,
+ genus 1, deg_C O(1) = 3.
+ H(p) : 9(b - p_1 - p_2) - 2(s - p_1 - p_2)_+,
+ b = (1+s)/3, s = 1-a.
+ truncate : p_1 <= h_1 < c < s, p_1 + p_2 <= h_2 near b.
+ volume : int_{Delta_inf} H = (1-a^3)/6 (shell).
+ transfer : moment-preserving transport -> Kähler cut ->
+ normal-cone (Bl_o PP^3, quadric) -> shell;
+ add central ball A < a.
+
+ In every row the map P = H - sum_i cap_i is concave, has
+ positive integral (that integral is a volume surplus), and is
+ positive away from a central core; Part A (Section 6) upgrades
+ this to a pointwise gap, Part B (Section 7) cuts disjoint
+ layers, and Part C (Section 8) carries the packing home.
+
+14.3 What the proof uses from outside (the honest list).
+ Everything is proved here except these named standard facts,
+ each of which is a well-known theorem used as a black box:
+ - Bertini's theorem and the Koszul/adjunction computations
+ (Lemma 9.3): standard algebraic geometry.
+ - the classification of line bundles on a bordered surface
+ and on a genus-one curve (basepoint-freeness): standard.
+ - the symplectic cut / Kähler quotient (Lemma 8.5): Lerman.
+ - Moser's trick and its relative/equivariant refinement
+ (Lemma 8.1, Lemma 7.7): standard.
+ - the nested-disk rearrangement lemma (Lemma 6.10): standard
+ planar area-preserving construction (Schlenk).
+ - the invariant-cycle theorem for toric ampleness (Claim
+ 11.17): standard toric geometry (Fulton et al.).
+ - GROMOV'S TWO-BALL RIGIDITY (Section 3.2): the one
+ genuinely symplectic input, via J-holomorphic spheres.
+ With these imported, every step in between is written out.
+
+14.4 Where each piece lives (a reading map).
+ statement + conventions .......... 1, 2
+ necessity (volume, two-ball) ..... 3
+ architecture, three cases ........ 4
+ Hamiltonian flow, 1D sorting ..... 5
+ Part A: comparison lemma ......... 6
+ Part B: surface packing .......... 7
+ Part C: Moser, normal-cone, bundle
+ form, shell ............... 8
+ Case 1 (all < 1/2) ............... 9
+ Case 2 (big, n >= 4) ............. 10
+ Case 3 (big, n = 3) .............. 11
+ assembly = the theorem ........... 12
+ dimension 4, caveats, summary .... 13, 14
+
+That is the complete proof, and the honest accounting of what it
+rests on. The two-line statement it establishes:
+
+ for every n >= 3, every k >= 1, and all positive capacities,
+
+ the balls pack into Int B^{2n}(R)
+ iff sum_i R_i^n < R^n and R_i + R_j < R (i != j).
-Symplectic geometry is more than volume. It measures
-2-dimensional signed area on every coordinate plane, and this
-strict structure produces extra obstructions you might not
-expect (Gromov's two-ball rules).
-
-But those extra obstructions are tameable in dimension >= 6.
-Once n >= 3, only volume and the two-ball rule survive, and
-the paper shows they are SUFFICIENT. The construction builds
-surface-plus-toric models inside the target, reshuffles fiber
-coordinates via a Hamiltonian isotopy to produce disjoint
-layers, and transports the resulting packing back to the ball
-via normal-cone degenerations and Moser isotopies.
-
-The final answer, in every dimension n >= 3, for every k, and
-for every choice of positive capacities:
-
- pack iff (sum R_i^n < R^n) and (all R_i + R_j < R).
-
-Not one condition more. Not one condition less.
-
- --- end ---
+================================================================
+ end
+================================================================