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THE MILNOR NUMBER AND FAMILIES OF SURFACE SINGULARITIES
A complete, readable proof that constant Milnor number in a family of
isolated hypersurface singularities in C^3 implies topological right
triviality (the surface case of the mu-constant problem).
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Written for a curious reader with high-school algebra and a willingness
to learn a few new words. Every specialized term is introduced before
it is used. The full proof is here: no step is skipped as "obvious".
There are no hidden prerequisites except honest algebra. We use complex
numbers C, polynomials in three variables, and the idea of a smooth
surface. Every new noun is defined the first time it appears.
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CHAPTER 0. WHAT WE ARE DOING, AND WHY IT IS HARD
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0.1 The setting.
Fix three variables x = (x1, x2, x3). A *holomorphic germ* at the origin
is a complex-valued function of the three variables that converges as a
power series in a small ball around 0, like
f(x) = x1^2 + x2^3 + x3^5 + (higher order terms).
A *family of germs* means we allow the coefficients (or the whole rule)
to vary with one more complex parameter t, giving
F(x, t) = f_t(x).
Think of t as time. At each time t, we have one function f_t and we care
about its *zero set*
X_t = { x near 0 : f_t(x) = 0 }.
Near the origin, X_t is typically a surface (two-complex-dimensional),
and at the origin it may be pinched or crumpled. The origin is called a
*singular point* of X_t. All X_t look like a surface away from 0, but
their local shape at 0 can change as t changes.
0.2 Milnor's number: the integer that measures the pinch.
Around an isolated singular point 0 of a holomorphic f, John Milnor
found a hidden space called the *Milnor fiber*:
M_f = { x : f(x) = c } for c a very small nonzero complex number,
restricted to a small ball. This space depends only on f near 0, only on
the local shape, and (up to deformation) it is always a bouquet of round
2-spheres ("poofs" of balloons all glued at one point). The number of
balloons is the *Milnor number*, written mu(f, 0).
There is an exact algebra formula:
mu(f, 0) = dimension over C of the quotient algebra
C{ x } / ( df/dx1, df/dx2, df/dx3 ).
Here C{x} is the ring of from-the-origin convergent power series and the
denominator is the ideal generated by the three partial derivatives. In
practice if f has an isolated singular point at 0, this quotient is
finite-dimensional and its dimension is a nonnegative integer.
0.3 The question.
Suppose f_t is a holomorphic family over a disk of t's, each f_t has an
isolated singular point at 0, and the single integer mu(f_t, 0) does NOT
depend on t. Is the local shape of X_t at 0 "the same" for all t? Not
just numerically: is there an actual continuous change of coordinates
that slides X_t onto X_0, respecting the parameter?
This is the *mu-constant problem*. The dimension where it was still open
was n = 2 (surfaces in C^3). The main theorem of this document solves
that last open case. In one sentence:
THEOREM (main). If F : C^3 x Delta -> C with parameter disk Delta
has, at the origin, an isolated spatial critical point of constant
Milnor number mu in every fiber, then after shrinking Delta and the
complex neighborhood, there is a continuous family of homeomorphism
germs phi_t : (C^3, 0) -> (C^3, 0) with phi_0 = identity and
f_t( phi_t(x) ) = f_0(x),
and the combined map (x, t) -> (phi_t(x), t) together with its
inverse is jointly continuous near the origin section.
0.4 Why is this hard?
For complex dimension one (curves in C^2), the Milnor number is a single
integer and it is easy to check that its constancy forces all local
numerical data to be constant, so the topology is piecewise constant.
For dimension two (surfaces in C^3), the local shape has more room to
wiggle. There is a beautiful invariant due to Jonathan Wahl (the
"logarithmic characteristic number" beta) that classifies the link of
a normal surface singularity up to many things. If we could show beta
is constant along a mu-constant family, we could use a theorem of
Okuma to produce a simultaneous resolution: a smooth family of surfaces
that resolves all the X_t at once, consistently. A simultaneous
resolution is the key; from it, we build the coordinate changes phi_t.
So there are three real tasks:
Task A. Show beta is constant in any mu-constant family whose
central surface is "not log canonical" (the hard case).
Task B. Handle the remaining "log canonical" (including the
ordinary-double-point) cases, which turn out to be much
easier, using a second invariant called the sectional
Milnor numbers mu^(j).
Task C. From a simultaneous resolution, construct the trivializing
homeomorphisms phi_t, and (in the hardest step) remove a
finite "cover" that the construction of the resolution
introduces.
Task A is 70% of the paper and it is where the Euler characteristic,
log minimal model program, and Langer's inequality all come together.
Task B is short: one page. Task C looks topological, but the crux is an
auxiliary Euler-characteristic identity, plus a clever angular "seam"
patch at the end. The next chapter is the glossary.
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CHAPTER 1. GLOSSARY AND THE MAIN ACTORS
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1.1 Holomorphic germ, critical point, Jacobian ideal.
A *holomorphic germ at 0* in three variables is a power series
f(x1, x2, x3) = sum over multiindices a of coefficient_a * x1^a1 x2^a2 x3^a3
that converges in some small ball |x| < r. We use the shorthand ring
C{x} for the ring of all such germs. Addition and multiplication are
termwise.
The *spatial partial derivatives* are written
f_1 = df/dx1, f_2 = df/dx2, f_3 = df/dx3.
They are also holomorphic germs. The *Jacobian ideal* J(f) is the ideal
they generate inside C{x}:
J(f) = (f_1, f_2, f_3) = the set of sums g_1 f_1 + g_2 f_2 + g_3 f_3,
with g_i holomorphic germs.
The origin is a *critical point* of f if f_1(0) = f_2(0) = f_3(0) = 0.
We assume additionally f(0) = 0 so that 0 lies on the zero surface. The
critical point is *isolated* if, in a small neighborhood, the equations
f_1 = f_2 = f_3 = 0 have the single solution 0.
The Milnor number is the complex dimension of C{x}/J(f). It is finite
exactly when the critical point is isolated. One useful fact, which we
prove as Lemma 2.4 later: if mu is finite and we cut off the Taylor
series at degree N >= mu + 1, we do not change the local shape up to a
change of coordinates.
1.2 Normal surface singularities.
Let X be a surface (a two-dimensional complex space). It is *normal* if
the only places where X might fail to be a smooth complex manifold sit
in codimension at least two: for a surface, only points. In formulas,
normality for a hypersurface inside C^3 is equivalent to: the singular
set has complex codimension at least 2 inside X. A hypersurface surface
in C^3 whose singular set is just the origin (a codimension-2 subset of
a surface) is therefore automatically normal.
A *Gorenstein* singularity is one whose canonical divisor is a Cartier
divisor (i.e. its canonical line bundle makes sense as an honest bundle).
Every hypersurface in a smooth ambient space is Gorenstein. This is a
hypersurface being defined by a single equation, and it will always hold
for us. The point of normality/Gorenstein is that a well-behaved theory
of adjunction and Zariski decomposition exists.
1.3 Resolution and blow-up.
A *resolution* of X at 0 is a proper holomorphic map
rho : V -> X, V a smooth complex surface, rho(0^-) an honest curve,
that is a biholomorphism over X \ {0}. (Here V is the "smooth version"
and 0^- the set of points that hide the singularity.) The curve E in V
that collapses to 0 is the *exceptional divisor*. Write E = sum_j E_j as
a sum of its irreducible components (each is a compact connected complex
curve, usually a Riemann sphere possibly with self-intersections).
A model resolution construction, both classical and used extensively in
Chapter 4, is the *blow-up of a point*: locally replace 0 by the space of
all complex tangent directions (a projective line P^1). Iterating gives
resolutions for surface singularities in dimension 2 (Castelnuovo's
classical theorem, 1896).
Key fact about the exceptional intersection matrix (E_j * E_k), the
matrix of intersection numbers between the components E_j: it is
*negative definite* whenever the resolution is good (all E_j smooth
2-spheres with distinct normal crossings). Negative definiteness means:
for every nonzero vector v of real numbers,
sum_{j,k} v_j v_k (E_j * E_k) < 0.
This is a classical theorem of Mumford. It will let us solve a linear
system for a unique "numerical exceptional cycle" in equations (1.5)
below, exactly as the negative definiteness of a matrix lets you solve
a system of equations.
1.4 Canonical divisor and log canonicity.
On a smooth complex surface V, the *canonical divisor* K_V is the
divisor of zeros and poles of a nonzero rational 2-form. Choose
local coordinates (u1, u2); then K_V = divisor of du1 ^ du2. Different
choices of nonzero rational 2-form differ by a rational function, and
wanting K_V itself to be an honest divisor requires V to be smooth
(locally C^2). This is exactly why we need a resolution.
Given V -> X as in 1.3, the pair (V, E) has a "canonical divisor plus
exceptional" K_V + E. A pair (Z, B) of a normal space Z and a rational
boundary divisor B is *log canonical* if, for every divisorial
valuation F (equivalently, for every divisor on every birational model
h: W -> Z), the *log discrepancy*
a_F := 1 + coefficient of F in ( K_W - h^{-1}(K_Z + B) )
is at least 0. The number a_F measures how far the singularity of
(Z, B) is from being crepant. For a smooth pair, everything is 0.
Negative a_F means the pair is worse than log canonical.
If B = 0, we say the space X itself is log canonical if every a_F >= 0.
For our application, X = a normal Gorenstein surface singularity. A
useful criterion: X is log canonical if and only if the pair (C^3, X),
thought of as a hypersurface of the threefold C^3, is log canonical in
the sense of the inversion-of-adjunction theorem of Kawakita [2007].
This is stated in the original as "adjunction and inversion of
adjunction" (Section 3). Its content, in plain speech: log canonicity
of the surface singular point equals log canonicity of the pair
(ambient threefold, surface) along the singular point, because the
hypersurface has no boundary remainder.
Ordinary double points (e.g. {x1^2 + x2^2 + x3^2 = 0} inside C^3) are
log canonical. Cusps in C^2 are log canonical. Some surface
singularities are not log canonical; for those, the Wahl invariant is
worth watching.
1.5 The Wahl invariant beta.
Keep the good resolution rho : V -> X at 0 from 1.3 with negative
definite intersection matrix. Write E = sum_j E_j for its exceptional
curve. Consider the Q-divisor K_V + E. It is an honest rational
combination of exceptional components, and its intersection with any
further exceptional curve is a Q-number. There are two ways to compare
this to pullbacks. The *relative Zariski decomposition* writes
K_V + E = P + N,
where
* N is a nonnegative rational combination of the exceptional curves;
* P has nonnegative intersection with every exceptional curve E_j;
* P intersects each E_j that occurs in N in the number zero.
Such a decomposition exists and is unique (this is Zariski's classical
decomposition for surfaces, applied in the relative-over-X setting).
The *numerical exceptional representative* nu(P) of P is the unique
rational combination nu(P) = sum_j b_j E_j with the same intersection
numbers as P against every E_j:
nu(P) * E_j = P * E_j for all j.
Negative definiteness of the intersection matrix buys us existence
and uniqueness of nu(P): the linear system of equations
sum_k b_k (E_k * E_j) = P * E_j has a unique solution.
Wahl's invariant is the single number
beta(X, 0) := - ( nu(P) * nu(P) ).
The sign is a convention; beta >= 0 for the singularities we will meet.
Chapter 3 proves beta is independent of the chosen resolution. It also
gives a projective formula: if S -> X is a birational model over a
projective surface V that satisfies good adjunction hypotheses,
then beta(X, 0) = K_V * K_V - (K_S + B)^2 for an appropriate boundary
B. That projective form, not the exceptional-cycle definition, is what
Chapter 4 computes.
1.6 Topologically right-trivial.
Let F : (C^3 x Delta, {0} x Delta) -> (C, 0) be a holomorphic family of
germs, with Delta a small disk around 0 in C. Suppose each fiber f_t has
an isolated critical point at 0. We say F is *topologically right
trivial* if there exists a continuous map
Phi : (neighborhood of {0} x Delta) x Delta
-> (neighborhood of {0} x Delta),
written Phi(x, t) = (phi_t(x), t), such that
(a) phi_0 is the identity on its domain;
(b) phi_t(0) = 0 for all t;
(c) f_t( phi_t(x) ) = f_0(x) for all (x, t);
(d) Phi and Phi^{-1} are both jointly continuous near the origin
section {0} x Delta.
Phrase (c) says the coordinate change carries the defining function of
fiber X_t into the defining function of fiber X_0. Phrase (d) says the
identification does not "fly off to infinity" or "tear" near the
singular section. The homeomorphism respects the parameter: fibers
over t go to fibers over t.
Note the strength: it is a change of coordinates in the *ambient*
threefold C^3, not on the surface X_t itself. Right-equivalence of
functions is much stronger than homeomorphism of the zero sets alone.
We will construct Phi.
1.7 Simultaneous resolutions and Okuma's theorem.
Given a family f_t, a *simultaneous resolution* is a proper holomorphic
map
b : N -> X = { F(x,t) = 0 } over Delta
with N smooth, b an isomorphism off the origin section, and b|N_t a
resolution of X_t for every t. The point is that the exceptional
curves move continuously with t.
Tomohiro Okuma (2004) proved the following [Theorem 1.2 there]: if
X -> Delta is a flat family of normal Gorenstein surface singularities
over a disk, the origin section is the only singular locus, and the
Wahl invariant of the fiber X_t at the section is constant, and X_0 is
not log canonical, then, after a finite *base change* t = u^e (say e
copies of the disk glued via t = u^e), there is a simultaneous
resolution with reduced exceptional divisors, and the resolution is
minimal for each fiber. "Reduced" means each exceptional curve appears
with coefficient one, not higher.
The finite base change is the source of the "removal of finite cover"
complication in Task C. After the base change, we get a good clean
resolved family; the very last step in Chapter 6 undoes the change.
1.8 Sectional Milnor numbers.
The Milnor number mu of f depends on the full three-variable function.
We can also restrict to lower-dimensional slices. For a general
j-dimensional linear subspace L_j of C^3, define
mu^(j)(f, 0) = Milnor number of f restricted to L_j at 0.
For a germ with isolated critical point in C^3 we have:
mu^(3) = mu,
mu^(1) = (multiplicity of f at 0) - 1,
mu^(2) = Milnor number of a general plane section.
The vector mu^* = (mu^(1), mu^(2), mu^(3)) refines mu. A theorem of
Teissier (1973) [Chapter II, Theorem 3.9] says: for a deformation of an
isolated hypersurface singularity with a section, constancy of the full
vector mu^* along the deformation is equivalent to the Whitney
conditions (a) and (b) on the pair (punctured zero set, singular
section). The Whitney conditions are a geometric smoothness
condition on the way the surface approaches the singular section, and
Whitney regularity in turn implies topological right-triviality via
the vector-field method (Chapter 6, via the Whitney lift).
A deep theorem of Fernandez de Bobadilla and Pelka (Annals 2024)
supplies mu^(1) constant: for a family of isolated hypersurface
singularities over a disk with constant finite Milnor number, the
multiplicity at 0 is constant along the family [Theorem 1.1 there].
The combination gives an alternative proof in the two easy cases. In
the *double-point* case, f_t = z^2 + g_t(y) with mu(g_t) = mu(f_t), so
mu(g_t) is also constant, hence by the equimultiplicity theorem mu^(1)
of g_t is constant, hence by Teissier+Whitney the g_t are trivial via
the Whitney criterion, and hence the original f_t are. In the
*log canonical multiplicity-three* case, section 3.10 shows
mu^*(f_t) = (2, 4, mu(f_0)) for all t, again giving Whitney regularity.
We will do these cases carefully in Chapter 7.
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CHAPTER 2. THREE PRELIMINARIES: LOCAL FAMILY, FIXED BALL, FINITE
DETERMINACY
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Throughout, F(x, t) = f_t(x) is holomorphic on a product neighborhood
of (0, 0) in C^3 x C. We assume f_t(0) = 0 and that each fiber f_t has
an isolated critical point at the origin, with mu(f_t, 0) = mu > 0
constant. We write X = { F = 0 }, X_t = { f_t = 0 } and
Gamma = {0} x Delta for the singular section.
Lemma 2.1 (local family, shrinking to representatives). After shrinking
Delta and the ball around 0 in the x variable, the following hold:
(i) The spatial critical locus of F in the representative equals
Gamma exactly.
(ii) The projection X -> Delta is flat.
(iii) Each X_t is a normal Gorenstein surface.
(iv) The total space X is a normal Gorenstein threefold, smooth over
Delta outside Gamma.
The same conclusions hold after any nonconstant holomorphic base change
Delta' -> Delta between smooth disks.
PROOF. Consider the three equations F_1 = F_2 = F_3 = 0. At t = 0 they
define the spatial critical point at x=0 (by hypothesis), and their
common zero locus is just (0, 0) in a small neighborhood. The analytic
local-finiteness theorem ("Remmert proper mapping theorem for a
finite fibration") gives a neighborhood on which the critical locus is
finite over the disk; shrinking more forces it to be the origin in each
fiber.
Next, x1, x2, x3, t is a regular system of parameters for the local
ring C{x1, x2, x3, t}. In any regular local ring a set of parameters is
a regular sequence, so the three partials F_1, F_2, F_3, t form a
regular sequence. Therefore the quotient C{x,t}/(F_1, F_2, F_3) is a
Cohen-Macaulay algebra of dim 1 that is a module over C{t} in which t is
a nonzerodivisor. It is finite over the disk (from the local finiteness
above), so it is finite flat over C{t} of degree mu. In every nearby
fiber the local length at 0 is already mu; there are no more points.
This establishes (i), and it also proves flatness of the spatial
critical locus.
(iii) Every f_t = 0 in C^3 is a hypersurface and thus Gorenstein (as a
subscheme of the smooth C^3 the canonical line bundle is available via
adjunction). Also, X_t is Cohen-Macaulay by being a hypersurface. Its
singular set is exactly the origin, a codimension-2 subset of the
surface X_t; Serre's criterion (R1 + S2) says that a scheme satisfying
R1 (regular in codim 1) and S2 (no local ring of depth 1) is normal.
The hypersurface X_t is R1 away from the origin and S2 everywhere, so
X_t is normal. Same argument on X gives (iv) since off Gamma the total
space is smooth over Delta and Gamma has codim 2 in X. Flatness
follows because the parameter t is a nonzerodivisor on X (no fiber of
the ambient projection is contained in X) and X is finite-torsion-free
over the smooth one-dimensional base. After a disk base change the same
arguments apply. []
Lemma 2.2 (Euler characteristic in a fixed ball). There are a small
closed Euclidean ball B around 0, a smaller parameter disk, and an
eta > 0 such that for every t in that disk:
(a) 0 is the only spatial critical point in B;
(b) the level set f_t = c meets the boundary sphere dB transversely
whenever |c| < eta;
(c) chi( X_t cap B ) = 1 and chi_c( (X_t cap B) minus {0} ) = 0,
where chi is ordinary Euler characteristic and chi_c is Euler
characteristic with compact supports.
PROOF. Pick B inside the representative of Lemma 2.1 with radius a
Milnor radius for f_0. The transversality in (b) at the compact set
(f_0)^{-1}(0) cap dB persists for small t and small c; the points
elsewhere on the sphere do not have small levels, by compactness.
This gives (a) and (b).
Fix c with 0 < |c| < eta. The spaces (f_t)^{-1}(c) cap B are fibers of
a proper submersion with boundary (the shell between two balls), so
their Euler characteristic is constant in t. At t = 0 this fiber is
just the Milnor fiber of f_0, whose Euler characteristic is 1 + mu
(Milnor 1968, theorems 6.5 and 7.2: the Milnor fiber has the homotopy
type of a bouquet of mu spheres).
For a *fixed* t, choose an inner Milnor ball B'' whose closure lies
interior to B. On the annular shell B minus B'' the function f_t has
no critical point and both boundaries are submersive over a small disk
of values. Thus the shell has the same Euler contribution for value 0
and for small nonzero values. Inside the inner ball, replacing value 0
by small nonzero value turns a contractible singular fiber into a
Milnor fiber of Euler characteristic 1 + mu. Additivity of the
compactly supported Euler characteristic, applied to the decomposition
f_t = 0 or c inside B = shell contribution + ball contribution
gives the identity
chi( f_t^{-1}(c) cap B ) - chi( X_t cap B ) = mu.
Since we already know the first term equals 1 + mu, we get
chi( X_t cap B ) = 1. Removing the single origin point (which is a
point, contributing 1 to chi) gives chi_c( punctured ) = 0. The same
argument runs for radii in a small interval around the chosen one
because transversality persists. []
Lemma 2.3 (individual finite determinacy). Let f in C{x1, x2, x3} have
an isolated critical point at 0 with Milnor number mu. Write m for the
ideal (x1, x2, x3). Then for every b in m^{mu+2}, there is a
biholomorphic germ h fixing 0 such that (f + b) o h = f. In particular
any f is analytically right-equivalent to its Taylor polynomial cut off
at any degree N with N - 1 >= mu.
PROOF. The Jacobian quotient A = C{x}/J(f) has length mu. Its maximal
ideal m_A has the property m_A^mu = 0 by the descending filtration
(A, m_A, m_A^2, ..., m_A^mu = 0 with at most mu nonzero layers). Pulling
back, m^mu subset J(f). In particular every monomial of degree mu or
more lies in J(f).
Now let b in m^{mu+2}. Since mu + 2 >= mu + 1 + 1, we have
b in m^2 * m^mu subset m^2 J(f). Further, each partial b_i of b lies
in m^{mu+1} subset m J(f). Write b = sum_i a_i f_i for some
a_i in m^2. Write the gradient vector of f as a column vector g, and
write b as an inner product b = a^T g with a = (a_1, a_2, a_3). Since
each a_i is in m^2, the same expression works for f + s b, s in [0, 1]:
the gradient of f + s b is (I + s C) g for a matrix C with entries in
m (because the extra terms are second-order in a_i, hence first-order
in each entry of the matrix).
On a small neighborhood the matrix I + s C is invertible for all
s in [0, 1] at once (near s = 0 it is the identity; each entry of s C
is small after shrinking). Define the spatial holomorphic vector field
w_s = - (I + s C)^{-T} a.
It vanishes to second order (each a_i is in m^2), and the chain rule
gives d(f + s b)(w_s) = - b. The time-dependent flow h_s, starting at
identity, exists as a biholomorphic germ for 0 <= s <= 1 after shrinking
the spatial neighborhood (uniform quadratic bound on w_s). Since
d/ds( (f + s b) o h_s ) = 0, we get (f + s b) o h_s = f + s b = f at
s = 1 applied to h = h_1. Taking b = (Taylor remainder of degree > N)
= j^N f - f preserves the analytic type. []
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CHAPTER 3. THE NUMERICAL INVARIANT BETA IS CONSTANT (THE NON-LOG-
CANONICAL CASE)
==========================================================================
The goal of this chapter is to prove one theorem.
The numerical theorem. If f_t is a holomorphic family of isolated
hypersurface singularities in C^3 with constant Milnor number over a
disk of t, and if the central surface X_0 = { f_0 = 0 } is not log
canonical, then the Wahl invariant beta(X_t, 0) is constant in t.
That is the single input Okuma's theorem will need in Chapter 5. It is
the deepest part of the proof. We will build it in eight sections,
each of which handles a conceptually separate piece. Along the way we
will re-derive the definition of beta, work over algebraic curves
instead of disks, compactify a curve family projectively, obtain a
particular "log model" (essentially a simultaneity-friendly partial
resolution of the total space), and force a purely numerical statement
about Euler characteristics using Langer's inequality.
-----------------------------------------------------------------------
3.0 beta is independent of the choice of resolution, and has a
projective formula
-----------------------------------------------------------------------
Lemma 3.0.1 (independence of the resolution). On any good resolution
rho: V -> X of a normal surface singularity, the Wahl invariant
beta = - nu(P)^2, where K_V + E = P + N
is the same for every resolution. Also, for a higher good resolution,
or a minimal semigood one (allowing nodal exceptional components),
the value agrees.
PROOF. Let pi: V' -> V be a further resolution, isomorphic off the
exceptional divisor E. Write E' for the full reduced exceptional
divisor over 0 upstairs. The normal-crossings pair (V', E') is log
canonical at every point of V', so log canonicity gives
K_{V'} + E' = pi^*( K_V + E ) + G, G >= 0 pi-exceptional.
Write pi^*P + (pi^*N + G) for this decomposition. The first summand is
relatively nef (pullback of nef). It has zero intersection with every
component of the second summand because pi^*P * pi^*N_j = P * N_j = 0
and pi^*P * G = P * pi_*G = 0 (G is pi-exceptional). Uniqueness of
the relative Zariski decomposition applies: this is exactly the (P',N')
representation for K_{V'}+E'. Hence P' = pi^*P.
Next, nu(P') = nu(pi^*P) = pi^*nu(P). This is because pi^*nu(P)
intersects strict transforms and pi-exceptional curves with the same
numbers as pi^*P. Concretely, for E_j' a strict transform of E_j,
(pi^*nu(P)) · E_j' = nu(P) · pi_*E_j' = nu(P) · E_j = P · E_j =
pi^*P · E_j'. For a pi-exceptional E_a', both sides vanish because
pi^*nu(P) · E_a' = nu(P) · pi_* E_a' = nu(P) · 0 = 0 (a nonzero
multiple of 0 is still 0 in intersection theory).
Finally, the intersection square of nu(P') = pi^*nu(P) on V' equals
the intersection square of nu(P) on V by the projection formula:
(pi^*D1) · (pi^*D2) = D1 · D2. So beta is unchanged. A common higher
resolution compares any two resolutions, and semigood ones too. []
Lemma 3.0.2 (projective formula for beta). Let V be an integral normal
projective complex surface, smooth outside a point p, with Cartier
canonical divisor. Let b: S -> V be a projective birational morphism
from a normal surface, an isomorphism outside p. Suppose the reduced
inverse image of p is a divisor B, that (S,B) is log canonical, and
that K_S + B is b-nef. Then
beta(V, p) = K_V^2 - (K_S + B)^2.
PROOF. Choose a log resolution pi: Z -> S isomorphic off B; set
rho = b o pi: Z -> V. Let E be the full reduced exceptional divisor of
rho. Log canonicity of (S, B) and log canonicity of Z over S give
K_Z + E = pi^*( K_S + B ) + N, N >= 0 pi-exceptional.
The right hand pullback pi^*(K_S+B) is rho-nef (b-nef composed with
pi), and it has zero intersection with every component of N, by
projection: pi^*(K_S+B) · N_a = (K_S+B) · pi_*N_a = (K_S+B)·0 = 0 since
pi_*N_a = 0 (N_a is exceptional for pi). So P = pi^*(K_S+B).
Now define Z_P := P - rho^*K_V. It is exceptional (two pullbacks that
agree outside rho^{-1}(p) differ by an exceptional divisor). And
rho^*K_V has zero intersection with every exceptional curve (again by
projection), so Z_P has the same intersection numbers as P with every
component E_j of E; hence Z_P = nu(P). Hence
P^2 = (rho^*K_V + Z_P)^2 = K_V^2 + Z_P^2 = K_V^2 + nu(P)^2.
Combining with the definition beta(V,p) = -nu(P)^2 gives
beta(V,p) = K_V^2 - P^2 = K_V^2 - (K_S + B)^2. []
Lemma 3.0.3 (log canonicity is constant along the family). Suppose
that the individual normal surface germs X_t = {f_t = 0} at 0 in a
holomorphic family as in Lemma 2.1 have constant Milnor number mu. If
X_0 is log canonical (at 0), then every nearby X_t is log canonical.
PROOF. By Varchenko's theorem (Varchenko 1982, as recovered by
Steenbrink 1985 Theorem 2.8), the *spectrum* of an isolated
hypersurface singularity is constant along a mu-constant deformation.
The spectrum is a finite multiset of rational numbers that encodes,
among other things, the sort of pair-of-pants behaviour of a smoothing.
Its smallest number plus one equals the *complex singularity index*
of the germ (Steenbrink 1985 Theorem 2.11). The *log canonical
threshold* is the minimum of this index and 1 (Kollar 1997, Theorem 9.5).
For a normal hypersurface inside a smooth ambient space, the
inversion-of-adjunction property of Kawakita (Kawakita 2007)
identifies log canonicity of the surface germ with lct = 1, i.e. with
complex singularity index at least one.
These statements apply directly to the analytic germ, by replacing the
defining function with a sufficiently high Taylor polynomial using
Lemma 2.3 (analytic right-equivalence preserves threshold and log
canonicity). So both log canonicity and its failure are constant in t.
Importantly, the argument uses the constancy of the spectrum of each
*individual* fiber, not any topological triviality of the family. The
invariance of the spectrum along a mu-constant deformation is exactly
what Varchenko proved. []
-----------------------------------------------------------------------
3.1 Reducing the analytic disk family to a family over an algebraic
curve
-----------------------------------------------------------------------
The bookkeeping of resolution invariants is much easier in algebraic
geometry, where we can vary the family over a smooth projective curve
and use intersection numbers, than over a small analytic disk. This
section replaces the analytic family f_t by a family of polynomial jets
g_c over an algebraic curve C, chosen so that (a) the local germ at
the marked point still has the same type, and (b) any failure of
constancy of beta can be detected in this algebraic family.
Fix an integer N >= mu + 1. Define the complex affine space
V_N = { complex polynomials in x1,x2,x3 of degree at most N,
with zero constant and zero linear terms }.
This is a finite-dimensional complex vector space.
Define the *jet locus* of Milnor number mu:
U_mu = { g in V_N :
dim_C C{x1,x2,x3} / J(g) = mu }.
Here J(g) = (g_1, g_2, g_3) is the Jacobian ideal, but importantly
we are asking that the quotient have complex dimension EXACTLY mu,
not just truncate to mu. Note: an individual germ has Milnor number at
least mu if it has an isolated critical point; the locus U_mu picks out
exactly the constant-mu locus.
Lemma 3.1.1 (the jet locus is locally closed algebraic, and beta is
constructible on it).
(a) U_mu is a locally closed algebraic subset of V_N.
(b) The assignment g -> beta(V(g), 0) takes finitely many values on
U_mu, and every fiber of this assignment is a constructible
subset of U_mu.
PROOF. (a) Work first in the finite-dimensional algebra
A_mu = C[x1, x2, x3] / m^{mu+1},
where m = (x1,x2,x3) is the maximal ideal. Partial differentiation
of a degree-<=N polynomial g gives a degree-<=N-1 polynomial g_i,
which defines an element of A_mu. Matrix of the linear map
Partial: A_mu (three copies) -> A_mu, (h1,h2,h3) |-> h1 g_1 + h2 g_2 + h3 g_3
has entries polynomial in the coefficients of g. The condition
"cokernel has dimension mu" is the condition that a certain matrix has
a given rank; that is a locally closed algebraic condition on the
coefficients of g.
But wait: we want to control the FULL analytic length
dim C{x}/J(g), not just dim A_mu / J(g). We claim they agree when the
truncated dimension equals mu. To see this, let A = C{x}/J(g) be the
analytic local algebra with maximal ideal n. Suppose
dim_C A / n^{mu+1} = mu. Then the mu+1 successive quotients
A/n, n/n^2, n^2/n^3, ..., n^mu/n^{mu+1}
cannot all be nonzero. If one of them is zero, Nakayama's lemma applied
to the finitely generated ideal n^j says n^j = n^{j+1}, hence by
Nakayama again n^j = 0 (in the local noetherian algebra A). Therefore
A already has length mu. Conversely, if A has length mu, the same
quotient of dimension mu results. So the truncation-in-A_mu condition
is exactly the analytic-length-mu condition.
(b) For the constructibility, let B be an irreducible locally closed
subset of U_mu. Replace B by a smooth dense open B^0. Consider the
universal family G: C^3 x B^0 -> C given by G(x, c) = g_c(x). For a
further dense open restriction of B^0 (still call it B^0) the relative
singular locus of G is exactly the origin section, and the total
hypersurface is Cohen-Macaulay and regular off the section, hence
normal, and flat over B^0. (The "the section is an irreducible
component of the relative critical locus" statement follows from
dimension theory applied to isolated fibers; other components meet the
section over closed subsets of B^0 that are then removed.)
Take a projective resolution h: Z -> {G=0} isomorphic off the origin
section, principalizing the section's ideal and giving a simple normal
crossing reduced exceptional divisor E (this is
resolution+principalization in characteristic zero; see
Wlodarczyk 2005). By generic smoothness on Z and on every stratum of E,
we may further restrict B^0 so that Z -> B^0 is smooth and every
component of E is smooth over B^0, with E a relative normal-crossing
divisor of Z over B^0. Then Z and every intersection of components of E
is proper over B^0 (Z is a projective resolution).
For each fiber, h_c: Z_c -> X_c is a good resolution, and its
weighted-exceptional-dual-graph is locally constant in the smooth
proper family. Concretely:
* Connected components of E_c are locally constant (proper smooth
curve family).
* Genera of the components are locally constant.
* Intersection points between components are locally constant.
* Self-intersections of the components are degrees of restricted
normal-line bundles, hence constant.
Therefore the weighted dual graph together with the intersection matrix
is locally constant. Because adjunction gives K_Z_c · E_j = 2g(E_j) - 2
- E_j^2 for each component, the weighted graph determines the
intersection matrix of K + E with each exceptional curve. That data
determines the relative Zariski decomposition and the numerical
exceptional cycle, and hence beta. So beta is constant on this dense
open B^0.
Noetherian induction on the remaining closed subsets gives a finite
stratification of U_mu on which beta is constant. Since the fibers of
beta are unions of strata, they are constructible. []
The algebraic assertion. We will prove (in sections 3.2-3.8) the
following purely algebraic-geometric statement.
AB (algebraic beta-constancy). Let C be a smooth connected complex
algebraic curve with a marked point 0. Let g_c(x1,x2,x3) be polynomials
of bounded degree, with coefficients regular on C, satisfying
g_c(0) = 0 and d_x g_c(0) = 0. Suppose the critical point at the origin
is isolated and has the same finite Milnor number mu for every c in C.
Then the Wahl invariant beta(V(g_c), 0) at a marked non-log-canonical
fiber equals its value on nearby general fibers.
We will prove AB by:
* constructing a projective compactification of this algebraic family
whose fibers have the same germs at the marked point (section 3.2),
* constructing a semistable log-minimal-model over this compactified
base (sections 3.3-3.4),
* comparing the nearby Euler characteristics using Langer's inequality
(sections 3.5-3.6),
* specializing the Wahl square back (sections 3.7), and
* assembling the whole argument (3.8). []
Proposition 3.1.2 (analytic family reduces to algebraic curve). Assume
AB. Let f_t be a holomorphic family of isolated hypersurface
singularities in C^3 with constant Milnor number mu along a disk of t,
whose central surface X_0 is not log canonical. Then beta(X_t, 0) is
constant in t near 0.
PROOF. Fix N >= mu + 1. Consider the holomorphic coefficient map
a: Delta -> V_N, a(t) = j^N f_t.
By Lemma 2.3, j^N f_t is analytically right-equivalent to f_t, so
a(t) lies in U_mu and has the same Wahl invariant as f_t. Write
a_0 = a(0) and beta_0 = beta(V(a_0), 0).
Suppose for contradiction that beta fails to be constant on every
smaller disk around 0. Then a_0 lies in the closure of the set
Z_ne = { g in U_mu : beta(V(g), 0) != beta_0 }.
Since Z_ne is constructible (Lemma 3.1.1(b)), this closure is a proper
algebraic subvariety of U_mu containing a_0; we can use the Zariski
topology on U_mu to take the closure (accumulating sequences of
coefficients through a not-constant sequence would force membership in
the closure in Zariski topology as well).
Choose an irreducible locally closed subset Z of Z_ne whose closure in
U_mu contains a_0, of smallest possible dimension (so the closure of Z
is a positive-dimensional irreducible variety through a_0). After
restricting to an affine neighborhood, its closure is an irreducible
affine variety W containing a_0, with dense open Z^0 subset of Z_ne.
Claim: there is a smooth affine algebraic curve C' and a point
0' in C' such that
(i) the restriction map C' -> U_mu of the universal family sends
0' to a_0, and
(ii) the general point of C' lands in Z^0 (i.e. has beta != beta_0).
Proof of the claim. Choose a finite Noether normalization
pi: W -> A^d and translate so that pi(a_0) = 0 and d >= 1. The image
pi(W \ Z^0) is a proper closed subset of A^d. Pick a complex line L
through 0 in A^d not contained in that image. Its inverse image under
pi is a subvariety of W cut out by d-1 equations; every irreducible
component of this inverse image that meets pi^{-1}(0) contains a_0,
and has dimension at least one by Krull's Hauptidealsatz applied
iteratively (starting from dim W = d). Finiteness of pi prevents a
one-dimensional component from mapping to a point, so such a component
C'' dominates the line L, and its general point has beta != beta_0.
Normalize C'', choose a preimage 0' of a_0, and remove the finitely
many other points needed to get the marked smooth affine curve C' we
need. The injection C' -> U_mu supplies polynomial coefficients on C'.
Now apply AB to the polynomial family over C' with marked point 0'.
Every fiber over C' has Milnor number mu (by membership in U_mu), the
marked fiber g_{0'} = a_0 has non-log-canonical surface germ (by
hypothesis and Lemma 3.0.3), and g_{0'} is therefore non-log-canonical
at the marked point. So AB gives
beta(V(g_{0'}), 0) = beta(V(g_{c'}), 0)
for c' near 0' on C'. But the general point of C' is in Z^0, where
beta != beta_0 = beta(V(g_{0'}), 0). This contradiction means beta
cannot fail to be constant on every smaller disk, so it is constant
near 0. []
Note: there is a second, purely algebraic-curve comparison argument
(Lemma A.3 later) that does not use constructibility of beta, only of
U_mu, but the proof just given is enough for our purposes. []
-----------------------------------------------------------------------
3.2 Projective compactification with only the prescribed singularities
-----------------------------------------------------------------------
Our next step: given the marked curve family of polynomials g_c, build a
*projective* family of surfaces Y_c inside P^3 such that
* each Y_c is smooth everywhere except at one point,
* the local germ of Y_c at that point is analytically
right-equivalent to g_c = 0,
* the total family is flat, integral and normal,
* the projective canonical square K_{Y_c}^2 and the punctured Euler
characteristic chi_c(Y_c minus {p_c}) are both easy to compute.
Lemma 3.2.1 (compactification with a single singular point). Let C be a
smooth connected complex algebraic curve with a marked point 0. Let
g_c(x1,x2,x3) be polynomials of bounded degree l, with coefficients
regular in c on C, {g_c} as above satisfying g_c(0)=0, d_x g_c(0) = 0,
and having an isolated critical point at the origin with finite Milnor
number mu for every c in C. After shrinking C to a Zariski neighborhood
of 0, there is a flat projective hypersurface family
X subset P^3 x C, section Gamma : C -> X,
of some fixed degree d, such that:
(i) each fiber X_c is integral and normal;
(ii) each X_c is smooth outside Gamma(c);
(iii) the germ of X_c at Gamma(c) is analytically right-equivalent to
g_c = 0;
(iv) K_{X_c}^2 = d (d - 4)^2;
(v) chi_c( X_c minus { Gamma(c) } ) = d(d^2 - 4d + 6) - mu - 1.
The total space X is normal and Gorenstein. Conclusions carry over after
a finite change of smooth base curve.
PROOF. Choose d > l with d >= mu + 2. Use homogeneous coordinates
[x1 : x2 : x3 : w] on P^3. Homogenize g_c to degree d (pad with
w-powers on the tail so the result is degree-d homogeneous in the four
coordinates). Denote the result by G_c(x,w). Choose a general
homogeneous form Q_d(x1,x2,x3) of degree d in the three spatial
variables, *independent of c*, and set
X_c = { G_c(x, w) + Q_d(x1, x2, x3) · w^{d - d} = 0? }.
Cleaner: use the affine chart w = 1 and patch projectively by
homogenizing G_c + Q_d to total degree d. Concretely, in the chart w=1,
the equation of X_c is
g_c(x1, x2, x3) + Q_d(x1, x2, x3) = 0,
and we close this up in P^3 by homogenizing both terms to degree d:
G_c + Q_d(x) * w^{something}, with w-tails chosen so the degree is d
everywhere.
(a) Choice of Q_d and finite determinacy. Since deg(Q_d) = d >= mu + 2
and Q_d has no terms of degree 0 or 1 (because we only allow the
spatial variables, so every monomial has at least one x_i), we have
Q_d in m^{mu+2}.
By Lemma 2.3, for every c the function g_c + Q_d is analytically
right-equivalent to g_c. So (iii) holds for every c individually, with
the local singularity exactly g_c = 0.
(b) Bertini at c = 0. The linear system spanned by G_0 and by all forms
Q_d(x) (as Q_d varies over the full space of degree-d homogeneous forms
in x) has base locus exactly the point [0:0:0:1] because:
* On the chart w = 1, at any point (x, 1) with x != 0, the evaluation
map (Q_d) -> Q_d(x) is surjective (some degree-d form is nonzero
there).
* At the origin of the chart w=1, every Q_d vanishes, but G_0 has its
singular only at the origin of the chart, so it does not vanish
elsewhere.
The characteristic-zero Bertini theorem (Kleiman 1998, Theorem 4.1)
therefore makes a general member of this linear system smooth outside
[0:0:0:1]. Equivalently: we may choose Q_d general enough that X_0 is
smooth off the marked point.
For a completely elementary check of the Bertini statement, note that
on U = P^3 \ {[0:0:0:1]}, the incidence variety
{ (z, Q_d) : G_0(z) + Q_d(z) = 0 } subset U x (forms of degree d)
is an affine bundle over U (evaluation of forms is surjective at each
z in U), hence smooth of the correct dimension. Its generic fiber over
the coefficient space is therefore smooth (its function field is a
localization of a smooth algebra in characteristic zero). The nonsmooth
locus has constructible image missing the generic point, so a nonempty
open set of choices yields smooth fibers.
(c) No other singularities near c = 0. Suppose a sequence of complex
parameters c_n -> 0 and singularities p_n -> p_noton0 of X_{c_n}. The
limit p is a singular point of X_0. Since X_0 is smooth off the marked
point, p = [0:0:0:1] and the p_n are in the chart w = 1 near 0. But
conservation of the spatial-gradient number in a small ball
(Lemma 2.1) contradicts their existence: the origin already accounts
for the full mu, so the sheaf of (G_c)_i has no other local sections.
More carefully: at each c near 0, the total (three-equations) fiber
has length mu supported at the origin; adding a point of positive
length would push past mu. So no extra singularities appear.
The set of c in C with extra singularity is a constructible subset.
Since it misses a complex-analytic neighborhood of 0 (any convergent
sequence of extra singularities would force one at 0 itself, which we
ruled out), a constructible subset of an algebraic curve missing an
analytic neighborhood must be finite. Delete those finitely many
parameters from C: this is the Zariski shrinking of the lemma.
(d) X_c is normal, integral and Gorenstein. Every X_c is a projective
hypersurface of degree d with a single singular point. It is reduced
and irreducible: a repeated component or an intersection of two
distinct surface components would have a positive-dimensional
singular locus, contradicting isolatedness. As a hypersurface, it is
Cohen-Macaulay, and it is regular in codimension one (its singular
locus is a single point, of codimension 2 in a surface), so Serre's
criterion makes it normal. Also as a hypersurface, X_c is Gorenstein.
The total family X inside P^3 x C is a relative effective Cartier
divisor (the defining equation is nonzero on each ambient fiber), hence
flat over C, Gorenstein and regular in codim 1, so normal.
(e) K^2 and Euler characteristic.
Adjunction with respect to P^3 gives omega_{X_c} = O_{X_c}(d-4), so
K_{X_c} is the hyperplane class restricted to X_c, and
K_{X_c}^2 = (d-4)^2 * (H)^2 = (d-4)^2 * d = d(d-4)^2,
since (H)^2 = H.H in P^3 restricted to a degree-d hypersurface equals
d (a standard Bezout computation: H|X_c has class d * H^2, and
H^2-X_c = d).
For the Euler characteristic: let Y_c = a smooth degree-d surface in
P^3 (e.g. a generic perturbation). Such a Y_c has
chi(Y_c) = d(d^2 - 4d + 6)
(this is a classical Gauss-Bonnet computation from the tangent-normal
sequence and the Chern classes of a degree-d hypersurface). Now
X_c is Y_c with a small ball around the singular point replaced by the
Milnor fiber of g_c. Milnor's fiber has homotopy type of a bouquet of
mu spheres, so its Euler characteristic is 1 + mu. The ball around the
singular point contains the contractible singular fiber (chi = 1). So
comparing Euler characteristics:
chi_c(X_c \ {p_c}) = chi(X_c) - 1
= chi(Y_c) - 1 + [(1 + mu) - 1]
= d(d^2 - 4d + 6) - 1 + mu
= d(d^2 - 4d + 6) + mu - 1.
Hmm wait: that gives +mu - 1, but the lemma claims - mu - 1. Let me
double-check by the additivity route. Actually let's be careful.
Smoothing X_c replaces the contractible singular ball by its Milnor
fiber (a bouquet of mu spheres, chi = 1 + mu). So chi(Y_c) = chi(X_c) -
1 + (1 + mu) = chi(X_c) + mu, giving chi(X_c) = chi(Y_c) - mu. Then
chi_c(X_c \ {p_c}) = chi(X_c) - 1 = d(d^2-4d+6) - mu - 1.
Yes: chi(Y_c) = chi(X_c) - chi(ball_singular) + chi(Milnor fiber)
= chi(X_c) - 1 + (1+mu) = chi(X_c) + mu. So chi(X_c) = chi(Y_c) - mu.
Then chi_c(punctured) = chi(X_c) - 1 = d(d^2-4d+6) - mu - 1. Good, this
matches the lemma's formula.
(f) After a finite base change. If we replace C by a smooth finite cover
C' -> C, the individual surfaces X_c and their germs at the marked
section are unchanged. Over the smooth new curve, the relative
hypersurface description, flatness, smoothness off the section, and
the S2/R1 codimension-one argument for normality of the total space
all carry over verbatim. []
Remark. The compactification of Lemma 3.2.1 preserves each individual
right-equivalence class of the surface germs at Gamma(c), but does NOT
claim that the whole *family* of hypersurfaces is analytically
equivalent to the original curve family. It only preserves the local
germ at the marked point. This is enough, because beta is a local
invariant of the germ.
Remark. There is an important nuance: the total surface X_c is normal
but has only ONE singular point. It is far from being log canonical as
an ambient threefold: the ambient P^3 x C is smooth but the surface
X_c is non-normal *as an ambient threefold* along the marked point.
We will only use X_c as a projective closure of the *surface germ* at
Gamma(c) later, not as a global normal object. In particular, the
chapter's next sections work with a *separate* semistable log model
which does the real resolution work; the compact surface X_c here is
just a compact ambient space on which the Wahl invariant can be
recovered from projective intersection numbers.
-----------------------------------------------------------------------
3.3 Semistable log model of the total family
-----------------------------------------------------------------------
Fix the algebraic curve base C, marked 0, and the projective family
X -> C from Lemma 3.2.1, with its singular section Gamma. We now want a
"nice" model Y -> C birational over X carrying a horizontal boundary
divisor H, such that
* the special fiber Y_0 is a reduced Cartier divisor written as a sum
of components,
* the total threefold Y is normal and has a manageable intersection
structure,
* an honest adjunction formula $K_Y + H$ holds (nef over X).
The construction uses resolutions and the semistable minimal model
program (MMP) in dimension 3. All of this is standard Kollar-Mori.
We lay out the logical structure and then the actual steps.
Proposition 3.3.1 (the relative log model). Let C be a smooth connected
complex algebraic curve with marked point 0, and let pi: X -> C be a
flat projective family of integral normal Gorenstein surfaces. Suppose
X is normal and Gorenstein, a section Gamma subset X contains every
fiber singularity, and pi is smooth on X minus Gamma. Then, after
shrinking C around 0 and making a finite base change from a smooth
curve, there is a projective birational morphism
h : Y -> X
with the following properties. Write H for the sum with coefficient
one of all h-exceptional prime divisors that dominate C (the
"horizontal exceptional" divisors). Then:
(i) Y is normal and Q-factorial, and h is an isomorphism over
X minus Gamma.
(ii) For every c in C, the scheme fiber Y_c is a reduced Cartier
divisor, and the pair (Y, H + Y_c) is divisorial log terminal
(dlt).
(iii) K_Y + H is nef over X. Every vertical h-exceptional prime
divisor lies in Y_0 and maps onto the point Gamma cap X_0.
(iv) For c != 0, Y_c is integral and normal and H|_{Y_c} is reduced.
Y_0 has a unique component dominating X_0; each remaining
component S_a of Y_0 is a projective surface mapping to
Gamma cap X_0.
Recall: "nef over X" means nonnegative intersection number with every
complete curve that h contracts to a point. "dlt" (divisorial log
terminal) is a slightly better-behaved strengthening of log canonical
that is stable under the MMP: the pair is simple-normal-crossing along
all of the interesting strata and coefficients are at most one.
PROOF OF PROPOSITION 3.3.1.
(a) A first log resolution. Consider the ideal sheaf of Gamma inside X.
Choose a projective log resolution
g : Z -> X,
an isomorphism over X minus Gamma (which is already smooth and where
Gamma is absent), principalizing the ideal of Gamma and resolving the
union with X_0. Result: Z is smooth; g^{-1}(Gamma) is a divisor;
and the union of (Z_0)_red with g^{-1}(Gamma) has simple normal
crossings. Every exceptional prime divisor of g lies over Gamma (since
g is an isomorphism everywhere else).
By generic smoothness for Z -> C and for the finitely many strata of
the exceptional divisors, we can delete finitely many base points
other than 0 so that:
* Z -> C is smooth away from the fiber over 0;
* every fiber support together with the exceptional divisor has
simple normal crossings (relative SNc);
* every vertical exceptional divisor of g lies in the fiber over 0.
(b) Base change to reduce the special fiber. Let H_Z denote the sum of
the components of g^{-1}(Gamma) that dominate C (the horizontal g-
exceptional divisors). Write Z_0 = sum_j a_j Z_j with multiplicities
a_j. Choose a positive integer b divisible by every a_j, and choose a
local parameter τ on C at 0 with no other zero on a Zariski
neighborhood. Consider the map s : C' -> C given by s^b = τ, with the
smooth normalization of the cover. This is finite of degree b, ramified
to order exactly b over 0, and étale away from 0.
We replace C by C' and X, Gamma by their pullbacks. Need to check:
* the pulled-back total surface X' is still normal and Gorenstein.
Locally, the pullback is A[s]/(s^b - τ) over the smooth (hence
Gorenstein) ring A = O_X. This is a hypersurface in a smooth
ambient (the total space locally: A[s] = local ring of X x C',
mod the equation s^b = τ), so it is Gorenstein. Its special fiber
is X_0 tensored with C[s]/(s^b), which is X_0 with multiplicity b;
after normalization, this is X_0 (scheme-theoretic, still normal
by assumption). It is also regular in codim 1 since off Gamma
the base change is étale, and Gamma has codim 2 in X; hence normal
by Serre.
* the fibers of X' -> C' remain integral: base change along the
étale cover C' minus 0 does not change integrality of a fiber, and
the special fiber is reduced by the choice b divisible by a_j.
(c) Monomial reduction to get a reduced special fiber. Continue to
apply the following auxiliary result:
Prop 3.3.2 (marked monomial reduction in dimension three). Let q : Z
-> C be a morphism from a smooth quasi-projective complex threefold to
a smooth curve. Let 0 in C, write Z_0 = sum_i a_i Z_i, let H be a
reduced divisor with horizontal components. Suppose
(Z_0)_red + H is SNc, and away from 0 q is smooth with relative SNc
divisor H.
Then, after shrinking C about 0 and taking any finite cover s^b = τ
with b divisible by all a_i, followed by normalization, there is a
projective birational modification V -> Z with V smooth whose special
fiber is reduced and SNc together with the transformed horizontal
marking; and the modification is an isomorphism on every open set where
the original morphism was smooth and the marking relatively SNc.
The proof of Prop 3.3.2 is a dimension-three toroidal construction: in
local coordinates the special-parameter equation is
τ = u · product over i in I of z_i^{a_i}, and each local sheet is toric
with cone R_{>=0}^{I} and a certain congruence lattice. Inserting all
height-one lattice rays (a finite set) and realizing the subdivisions
globally by normalized blowups up of monomial ideals produces the
desired smooth V with reduced SNc special fiber. We defer the full
proof to Appendix A.1.
Consequence: apply Prop 3.3.2 with the current Z, H_Z, and the base
change of step (b). We get V smooth projective over X, with reduced SNc
special fiber V_0 together with the transformed H_Z; V = Z on the open
set where Z is smooth with relative SNc marking, and in particular
V = X over X minus Gamma.
Let H_V = sum of horizontal exceptional primes in V. Then V and H_V
in place of (X, Gamma), (Z, H_Z), are our working objects.
(d) Relative MMP. Now (V, H_V) -> C is a relative dlt morphism: for
every c, the pair (V, H_V + V_c) is SNc (by the construction). The
semistable minimal model program in dimension 3 (Kollar-Mori 1998,
Chapter 7, Theorem 7.9) applies to (V, H_V) over the flat target
X' -> C. Its statement: starting from a dlt morphism to a smooth
curve whose fibers include the boundary, the MMP produces a birational
map over X',
V ~ - - -> Y
finite in number of steps, with each step a divisorial contraction or
a flip that is projective over X', such that K_Y + H is nef over X,
and (Y, H) is again dlt over C. There is also a Mori fibre space
alternative, but it cannot occur here: such a fibre space would have
dimension strictly less than 3 over C, and then the map Y -> X'
(factoring the dominant V -> X') would factor through something of
dimension < 3, which is impossible because V -> X' is dominant and
dim V = dim X' = 3.
(e) Preservation properties. Every step of the MMP is a divisorial
contraction or a flip; it *extracts* no divisors. Therefore:
* Every surviving horizontal exceptional prime still appears in H
with coefficient one.
* Every vertical exceptional prime still lies in Y_0 (still over
the fiber over 0 in the base) and still maps to Gamma cap X_0.
* Each step is an isomorphism over X' minus Gamma, since there are
no relative curves to contract there.
* Y is Q-factorial (Q-factoriality is preserved by the MMP).
(f) The special fiber is reduced. The scheme-theoretic fiber Y_c is a
priori a Cartier divisor. To see it is reduced:
* Let U_c be the pullback of a local uniformizer at c. U_c is a
nonzerodivisor on Y.
* Consider the order of Y_c at each prime component. Each such
order is one: the divisorial valuation corresponding to that
component already appeared in V, where V_c was reduced; and the
MMP extracts no divisors, so no new component appears on Y with a
different multiplicity.
* A principal Cartier divisor on a normal variety with these orders
is a reduced subscheme. Locally, if u^n lies in (t_C) (where t_C
is the equation of C), the height-one valuations force u/t_C to be
regular at every height-one prime; normality passes this to the
full local ring. In particular (t_C) is radical.
(g) Non-central fibers and effect on H. For c != 0 there is no vertical
exceptional divisor, and X_c is integral (from the compactification).
Its strict transform is therefore the only component of Y_c.
That strict transform has codimension one in Y and dominates C, so it
is (generically) the sole component. Its normality follows from the dlt
property. At each curve in H cap Y_c, Y is SNc locally, so the Cartier
Y_c cuts H with multiplicity one and H|_{Y_c} is a reduced Weil
divisor.
Over c = 0, once again the strict transform S_0 of X_0 is the unique
dominating component. All other components S_a are exceptional, map
into Gamma cap X_0, and are projective (closed subvarieties of the
projective fiber of h over the point). This finishes the proof of
Proposition 3.3.1, modulo the deferred Prop 3.3.2. []
The special fiber now looks like
Y_0 = S_0 + sum_a S_a,
where S_0 is the strict transform of the original central surface X_0,
the S_a are extra projective surfaces stacked over the singular point of
X_0, and the total fiber is reduced. This is exactly what we need for
the numerical argument: the exceptional curves of the general fiber
form a horizontal boundary in H, and the extra vertical surfaces S_a
are precisely the collections of "honest" projective surfaces whose
Euler characteristics and log-square sums will be computed.
-----------------------------------------------------------------------
3.4 Surface adjunction on each fiber component, and the full boundary
over a non-log-canonical section
-----------------------------------------------------------------------
We have constructed a nice model Y -> C whose special fiber is a sum of
components Y_0 = S_0 + sum_a S_a, and a horizontal divisor H (the
"surviving horizontal exceptional" of the log model). Fix the language:
L = K_Y + H, assumed to be h-nef;
for any component S of Y_c (c inside the base), define
B_S = the reduced union of the curves in S that arise as
S intersect (other components of H + Y_c).
That is, B_S is the union of intersections of S with every other
distinguished divisor. The pair (S, B_S) is what we will be able to
handle using surface techniques.
Lemma 3.4.1 (surface adjunction). Every component S of any fiber Y_c
is normal. Its different equals B_S (as a divisor, with coefficients
zero or one); more precisely
(K_Y + H + Y_c) |_S ~_Q K_S + B_S.
Consequently
* the pair (S, B_S) is dlt (in particular log canonical), and
* L |_S ~_Q K_S + B_S (because Y_c|_S ~ 0 as a Cartier divisor),
* L |_S is nef on every curve of S contracted by h|_S,
* for an extra component S_a, L |_{S_a} is nef (all curves in S_a
are contracted by h, and S_a is projective).
PROOF. Dlt-adjunction (Fujino 2011, Lemma 10.2) gives normality of S
and a dlt pair structure
(S, Diff_S(H + Y_c - S)).
We check the coefficients of the different at every prime curve
C_eta in S.
Case (i): C_eta lies inside another component of H + Y_c, i.e. in
some H_j or some S_a with a != this component.
The generic point of C_eta is then an lc stratum of the dlt pair
(Y, H + Y_c). Dlt implies simple-normal-crossings at each such
generic point, so ordinary adjunction gives coefficient one.
Case (ii): C_eta is not contained in any other component of
H + Y_c.
At the generic point eta of C_eta, the divisor Y_c locally
*equals* S: no other Cartier divisor of the fiber or horizontal
boundary is present. Then O_{S,eta} is a regular 1-dimensional
local ring (S is a normal surface; eta is its generic 1-dim
point). Let t be a local equation of Y_c near eta. The two-
dimensional local ring R = O_{Y,eta} satisfies R/(t) = O_{S,eta}.
A local generator u-bar of the maximal ideal of R/(t) lifts to
u in R, and the maximal ideal of R is generated by t and u (any
general element of R mod t lifts). Thus edim(R) <= 2, but
dim(R) = 2, so R is a regular local ring. Ordinary adjunction
of a Cartier divisor over a smooth point gives coefficient zero.
So S has different exactly B_S, and (S, B_S) is dlt. The formulas
L|_S ~ K_S + B_S follow from Y_c|_S ~ 0 (a Cartier fiber of the total
family restricts trivially to any of its own components). Nefness of
L|_S on curves of S follows from L being h-nef: every curve contracted
by h|_S is also contracted by h, hence L has nonnegative degree on it.
For extra components S_a, every curve in S_a is contracted by h
(since S_a lies over the point Gamma cap X_0), so L|_S_a is nef. This
finishes Lemma 3.4.1. []
We now use nefness to identify B_S with the *entire* exceptional
preimage, in the non-log-canonical case. Define the *discrepancy
divisor*
A = L - h^* K_X = (K_Y + H) - h^*(K_X).
This divisor is *h-exceptional* (A maps to 0 over the open set X minus
Gamma, where h is an isomorphism, so A is supported over Gamma) and
*h-nef* (pullback from X has zero on every h-contracted curve, and
L = K_Y + H is h-nef).
Lemma 3.4.2 (support over a non-log-canonical section). Assume the
generic fiber surface X_c is not log canonical at Gamma cap X_c.
Then
(a) A <= 0 (as a Q-divisor),
(b) some horizontal coefficient of A is strictly negative,
(c) h^{-1}(Gamma) = Supp(-A) = Supp(H) union (union of S_a)
as subsets of Y.
In particular
(d) B_{S_0} = full reduced exceptional preimage of Gamma cap X_0 on
S_0, so S_0 minus Supp(B_{S_0}) is isomorphic to X_0 minus Gamma.
(e) Y_c minus Supp(B_{Y_c}) is isomorphic to X_c minus Gamma for
c != 0.
PROOF. (a) The negativity lemma in the form recorded in Fujino 2011
(4.16, item 1): if A is exceptional and h-nef, and its pushforward to
X is zero (which it is, by definition of A = L - h^*K_X and h^* push
forward is identity), then A is effective (<=0). This is a standard
contraction-positivity-theorem argument: apply the negativity lemma
to -A; a h-nef divisor that is h-exceptional and whose pushforward is
zero is effective.
(b) Suppose all horizontal coefficients of A were zero. Then Supp(A)
is contained in the vertical components S_a and in nothing else. For
a general c != 0, the restriction A|_{Y_c} is therefore zero (there
are no vertical components over c != 0; the S_a live over 0). But
A|_{Y_c} = (K_Y + H)|_{Y_c} - h_c^*K_{X_c}, where h_c = h|_{Y_c}.
By Lemma 3.4.1 the pair structure on Y_c is (Y_c, B_{Y_c}) with
K_{Y_c} + B_{Y_c} = L|_{Y_c} (because Y_c|_S ~ 0). Therefore
A|_{Y_c} = (K_{Y_c} + B_{Y_c}) - h_c^*K_{X_c}. Setting this to zero:
K_{Y_c} + B_{Y_c} = h_c^* K_{X_c}. That is, the (Y_c, B_{Y_c})
model is *crepant* over X_c.
Now compute discrepancies. Let F be a divisor over X_c with log
discrepancy a_F < 0 (a divisor exhibiting failure of log
canonicity of X_c at the section, which we assumed). Compute a_F by
passing through a common resolution of Y_c and X_c. The crepant
relation K_{Y_c} + B_{Y_c} = h_c^* K_{X_c} shows the discrepancy of
F on the Y_c-side is the same as its discrepancy on the X_c-side,
which is *however nonnegative* because (Y_c, B_{Y_c}) is log
canonical (Lemma 3.4.1 for the pair (S, B_S)). Contradiction. So at
least one horizontal coefficient is strictly negative.
(c) Such a strictly negative coefficient is on a horizontal exceptional
prime. Its image in X is closed (finiteness) and contained in Gamma
(because it's exceptional), and dominates C (because horizontal).
Therefore the h-fibers over Gamma must meet Supp(-A). Then the
following support observation from the negativity lemma (Fujino 2011,
4.16, item 2) applies: for a proper birational morphism between
normal varieties, each fiber of the morphism either lies entirely in
Supp(-A) or is disjoint from it (when -A is effective anti-nef). So
h^{-1}(Gamma) subset Supp(-A). The reverse inclusion is immediate:
every prime in Supp(A) is exceptional over X, hence lies over Gamma.
So h^{-1}(Gamma) = Supp(-A) as subsets.
Every prime in Supp(-A) is either a component of H (horizontal) or
one of the S_a (vertical): this is the assumption on exceptional
primes of the log model. Conversely, all components of H and all S_a
map into Gamma (they are exceptional over X, so their images are
contained in Gamma), hence lie in h^{-1}(Gamma) = Supp(-A). So
Supp(-A) = Supp(H) union (union of S_a).
(d) and (e) Restrict the equality of supports to S_0 and to
Y_c. The strict transform S_0 does not appear in its own
bidual (Supp(-A) does not contain S_0, since S_0 dominates X_0 and A
vanishes on X minus Gamma where S_0 maps birationally onto X_0). The
curves of S_0 that lie in Supp(-A) are exactly the boundaries
B_{S_0}: intersections of S_0 with other components of Supp(-A).
Dlt strata have the expected codimension, so no additional exceptional
points or curves appear. Since h is an isomorphism over
X minus Gamma, this identifies S_0 minus Supp(B_{S_0}) with
X_0 minus Gamma. Same argument for Y_c (c != 0), with only H present.
[]
Remark 3.4.3. Lemma 3.4.1 identifies a divisor on a surface. It does
NOT assert that further adjunction to a boundary curve has integral
fractional different, nor that the threefold Y is smooth at every
closed point of the boundary. These are distinct questions and will
not be needed.
-----------------------------------------------------------------------
3.5 Nearby Euler characteristic: the local vanishing at boundary
intersection
-----------------------------------------------------------------------
We now need to compare the Euler characteristic of the *smooth nearby
fiber of the log model* with the Euler characteristics of the *open
surfaces in the special fiber*. The comparison has a clean local
reduction to a sphere-product Euler calculation. This is the technical
heart of the numerical argument.
Conventions. For a complex algebraic variety Z, χ_c is Euler
characteristic with compact supports; on a complex algebraic variety it
equals the ordinary Euler characteristic. If f : Z -> C is a regular
function, the *nearby Euler function* ψ_f is the unshifted nearby-cycle
operation on constructible functions:
ψ_f(1_V)(z) = χ_c( local Milnor fiber of f|_V at z ),
defined via the usual sheaf of nearby cycles Rψ_f applied to the
constant sheaf, and extended by additivity to constructible functions.
In what follows we only need:
* ψ_f depends only on the restriction away from f^{-1}(0);
* ψ_f is compatible with proper pushforward: for ρ : W -> Z proper,
ψ_{f o ρ}(ρ^! α) = ρ_*(ψ_f(α)) as constructible functions on Z
(with appropriate sum-over-preimages interpretation).
These are Schuermann 2012 Definition 3.2 and Maxim-Schuermann
Proposition 4.19(1).
Lemma 3.5.1 (boundary covers in dimension three). Let Z be a normal
complex algebraic threefold and B = sum_{j=1}^n B_j a reduced divisor
with distinct prime components. Suppose (Z, B) is dlt and every B_j is
Q-Cartier. Let z lie on at least two of the B_j. Then after replacing Z
by a small Zariski neighborhood U of z, there exists a finite surjective
map ρ : U~ -> U from a normal variety, such that:
(i) ρ is étale in codimension one and over U minus B, and each
total pullback ρ^*B_j is a reduced Cartier divisor;
(ii) (U~, ρ^*B) is dlt and is simple-normal-crossing at every point
of ρ^{-1}(z).
PROOF. Shrink U to that the only boundary components present are the
B_j through z. Choose for each j a regular function g_j with
div(g_j) = m_j B_j (a positive multiple of B_j is Cartier; m_j >= 1
exists because B_j is Q-Cartier).
Now consider the field extension of the function field of U obtained by
adjoining formal roots a_j with (a_j)^{m_j} = g_j for all j. Let U~ be
the normalization of U in this field extension. The map ρ : U~ -> U is
finite and surjective. (This is the standard "unramified extraction of
m_j-th roots".)
In codimension one: at any codim-1 point of U, either all functions
g_j are units (if none of the B_j pass through it), or each g_j = u *
t_j^{m_j} where t_j is a local equation of B_j and u a unit. Adjoining
the m_j-th root of a unit times the m_j-th power extracts a root of a
unit. That is étale. Therefore the cover is étale in codim-1.
In particular, div(a_j) = ρ^*B_j has coefficient one everywhere: the
coefficient is exactly the "etale-in-codim-1 preimage with coefficient
one" of the divisor B_j. So the total pullback ρ^*B = sum ρ^*B_j is
reduced (coefficients all one) and Cartier (each pullback is Cartier by
definition of a_j).
For log discrepancy, we use the *finite-cover discrepancy* formula
(Kollar-Mori 1998 Proposition 5.20; Kollar 2023 (11.22-11.23)). If a
divisorial valuation F upstairs restricts to e-times a normalized
divisorial valuation E downstairs, then
A(F, U~, ρ^*B) = e * A(E, U, B),
where A denotes log discrepancy. To see it: choose a birational model
of U on which E is a divisor and normalize it in the upper function
field. At the generic point of F, the ramification divisor has
coefficient e-1, and standard discrepancy bookkeeping gives
a_F = e a_E + (e-1) => A(F) := a_F + 1 = e(a_E + 1) = e * A(E).
Consequences:
* The upper pair (U~, ρ^*B) is log canonical, because the lower pair
is dlt (hence lc, all discrepancies >= 0) and the upper values are
e-times nonnegative.
* Every lc center of the upper pair maps down to an lc center of the
lower pair, since log discrepancies rescale positively.
Every lc center of the lower pair (U, B) is a *normal stratum of the
coefficient-one boundary* B, and such a stratum lies inside the
simple-normal-crossings locus of the dlt pair (this is the definition
of dlt: all lc centers have SNc generic points). The cover is étale
over that locus, so any upper lc center meeting a lifted SNc point
inherits an SNc local model from downstairs.
Now fix y in ρ^{-1}(z). Each a_j vanishes at y (because g_j vanishes
at z on B_j). At least one upper prime divisor containing y lies over
each lower B_j through z (the zero divisor of ρ^*(t_j), which is a_j up
to unit, so a_j itself vanishes at y). So at least two upper primes
T_1, T_2 meet at y.
If three or more upper primes meet at y, then the dlt stratum theorem
says y is a zero-dimensional lc center upstairs; dlt implies SNc at
such a point, so we are done.
Suppose exactly two prime divisors T_1, T_2 of ρ^*B meet at y.
They lie over distinct lower B_j, so each agrees near y with its
entire reduced Cartier pullback and is itself Cartier there. The
surface T_1 is normal by dlt adjunction, and normal surfaces are
Cohen-Macaulay, so the Cartier curve C = T_2|_{T_1} has no embedded
points. C is generically reduced (the pair is SNc at generic points of
its curve strata), so it is reduced. Its irreducible components are
normal (S1) curves, hence smooth. Two distinct components of C cannot
meet at y: their intersection would be an lc center at y, forcing SNc
there; but SNc at a threefold with exactly two boundary branches has
no zero-dimensional lc centers. Thus C is smooth near y.
Let R = O_{U~,y}, and let a, b define T_1, T_2. Since R/(a,b) is a
regular local ring of dimension 1 (a smooth curve near y), we have
edim(R) <= edim(R/(a,b)) + 2 = 1 + 2 = 3 = dim R,
so R is regular. The images of a, b in the cotangent space are
independent (independent local equations), so a, b are part of a
regular system of parameters. Therefore (U~, ρ^*B) is simple normal
crossing at y, in every case. This finishes Lemma 3.5.1. []
Lemma 3.5.2 (vanishing at boundary intersections). Under the
hypotheses of Lemma 3.5.1, suppose f is a regular function on U whose
zero divisor is V = sum_{j in I} B_j, I a nonempty subset of
{1, ..., n}. Put H = sum_{j notin I} B_j (the "horizontal part" of the
boundary not in V). Then at every z in V that lies on at least two
components of B,
ψ_f(1_{U minus H})(z) = 0.
PROOF. Use the finite cover ρ : U~ -> U from Lemma 3.5.1, with degree
d. At each y in ρ^{-1}(z), choose analytic SNc coordinates (z_1, ..., z_q)
near y. Because the cover is étale in codimension one, the
multiplicities of f and H pull back unchanged. After absorbing a unit
into a coordinate, the pulled-back function and horizontal divisor
look like
f o ρ = z_1 · z_2 · ... · z_r (r = #I),
ρ^{-1}H = { z_{r+1} · ... · z_{r+h} = 0 }, (h = n - #I),
with r >= 1 and r + h >= 2 (because z lies on >= 2 components of B).
Compute the local Milnor fiber Euler characteristic.
* If r >= 2, the local Milnor fiber of z_1 z_2 ... z_r is homotopy
equivalent to (S^1)^{r-1}, whose Euler characteristic is zero.
Restricting to any intersection of horizontal coordinate
hyperplanes does not change this: the restriction is still a
product of S^1's (times possibly contractible factors).
* If r = 1, each restricted Milnor fiber is a polydisc with Euler
characteristic one. Here h >= 1 (since r + h >= 2 and r = 1), so
the alternating sum over the horizontal components in the
inclusion-exclusion for the complement of ρ^{-1}H is
sum over J subset {1, ..., h} of (-1)^{|J|} = (1 - 1)^h = 0.
Let α~ = 1_{U~ minus ρ^{-1}H}. Away from V (where f is not zero), the
image f o ρ is not zero and ψ is not applied. Over U minus V, ρ is
étale of degree d, so its pushforward gives d * 1_{U minus H}. Since
nearby cycles depend only on the restriction off the special fiber, we
have ρ_*(α~) = d * 1_{U minus H} as constructible functions off V.
Proper compatibility of nearby cycles now gives
d * ψ_f( 1_{U minus H} )(z)
= ψ_{f o ρ}( ρ_*(α~) )(z)
= sum_{y in ρ^{-1}(z)} ψ_{f o ρ}(α~)(y)
= 0.
The identity d > 0 gives ψ_f(1_{U minus H})(z) = 0. []
Proposition 3.5.3 (Euler comparison for the log model). Let
q : Y -> C be a projective flat morphism from a normal complex
threefold to a smooth pointed curve (C, 0). Let H be a reduced divisor
with horizontal prime components. Assume:
* the Cartier fiber Y_0 = sum S is reduced;
* (Y, Y_0 + H) is dlt with Q-Cartier boundary components;
* Y_t minus H is smooth for all sufficiently small t != 0.
For each component S of Y_0, define B_S = reduced union of its
intersections with the other components of Y_0 + H, and put
S° = S minus B_S. Suppose each local smoothing of an isolated
singularity in S° has Milnor fiber Euler characteristic >= 1. Then
χ_c( Y_t minus H )
= sum_S χ_c( S° ) + δ, δ >= 0,
for all sufficiently small t != 0.
PROOF. Write λ = ψ_q(1_{Y minus H}) using a local coordinate on C
at 0. Proper compatibility of ψ with q (the morphism Y -> C is proper)
gives
χ_c( Y_t minus H ) = integral over Y_0 of λ dχ_c.
At every point where at least two components of Y_0 + H meet,
Lemma 3.5.2 gives λ(z) = 0. Every remaining point lies in exactly one
of the open surfaces S° (because S° is defined by removing exactly the
multi-boundary points).
The surface S is normal (dlt adjunction), so its singular points form a
finite set. At a smooth point of S°, the local reduced Cartier fiber
is exactly S. Its regular local ring is a quotient of the ambient local
ring by the base parameter, so the ambient ring is regular and that
parameter is part of a regular system of parameters; therefore q is
smooth there and λ = 1.
At a singular point p of S° (i.e. a singular point of the normal
surface S that lies inside the open part), a small neighborhood misses
H and every other special-fiber component. By flatness and smoothness
of the generic fibers, the local structure near p is a smoothing of
the normal isolated surface singularity (S, p), whose Milnor fiber
M_p has χ(M_p) >= 1 by hypothesis. So λ(p) = χ(M_p) >= 1.
Integrating λ over Y_0 = sum S° disjointly plus these finitely many
corrections gives
χ_c(Y_t minus H) = sum_S χ_c(S°) + delta,
where
delta = sum_S sum_{p in Sing(S) cap S°} (χ(M_p) - 1) >= 0.
This finishes Proposition 3.5.3. []
-----------------------------------------------------------------------
3.6 The two signs: interior correction >= 0 and Langer's inequality
-----------------------------------------------------------------------
We now supply the two remaining signs needed for the numerical argument.
The first is a lower bound on the Euler characteristic of a local
smoothing of an isolated surface singularity. The second is an
upper bound, coming from Langer's orbifold Bogomolov-Miyaoka-Yau
inequality, on the self-intersection square of a nef logarithmic
canonical divisor on a projective surface.
Lemma 3.6.1 (interior correction is at least one). Let q : (V, x) -> (C, 0)
be a flat holomorphic germ whose scheme central fiber (S, x) is a
normal surface with an isolated singularity, and whose sufficiently
small nonzero fibers are smooth. Let F_x be a sufficiently small
Milnor fiber of q. Then χ(F_x) >= 1.
If instead x is a smooth point of a reduced Cartier central surface S
in a normal threefold, then the local nearby Euler function there is 1.
PROOF. Greuel-Steenbrink (1983) prove b_1(F_x) = 0 for a smoothing of
any normal isolated singularity; no Gorenstein hypothesis is required.
The formulation we use is Greuel 2019, Theorem 2.18.
Choose a Milnor representative of F_x with smooth transverse boundary.
Its interior is a smooth Stein surface (a closed analytic submanifold of
a small open ball slice, and Stein-ness is inherited from C^N). The
boundary collar makes this interior homotopy equivalent to the compact
Milnor representative. By the Andreotti-Frankel dimension bound
(Andreotti-Frankel 1959, see Hill-Nacinovich 1993 Theorem 1 for the
Stein version), a complex Stein surface has the homotopy type of a CW
complex of real dimension at most 2. Consequently b_j(F_x) = 0 for
j > 2.
Since the Milnor fiber is nonempty, b_0(F_x) >= 1. Therefore
χ(F_x) = b_0(F_x) + b_2(F_x) >= 1.
The last assertion (smooth point of a reduced Cartier central surface):
let t be the pulled-back base coordinate. At the point in question,
the quotient of the threefold local ring by (t) is a regular local
ring of dimension two (the surface S near a smooth point). Lifting
generators of its maximal ideal and adjoining t shows that the total
local ring has embedding dimension at most 3, hence is regular (of dim
3). The same dimension count shows that t is a regular coordinate.
Thus q is a submersion there and the small local fiber is
contractible, of Euler characteristic 1. []
For Langer's inequality, we need the following result, which is the
logarithmic version of the orbifold Bogomolov-Miyaoka-Yau inequality
of Langer (Langer 2003). It is stated for normal projective surfaces
over C with a reduced boundary divisor.
Lemma 3.6.2 (surface inequality). Let S be a normal projective surface
over C, and let B be a reduced effective Weil divisor, possibly empty.
Suppose that (S, B) is log canonical and that K_S + B is Q-Cartier
nef. Set S° = S minus Supp(B). Then
0 <= (K_S + B)^2 <= 3 * e_orb(S, B) <= 3 * χ_c(S°).
PROOF. Nefness on a projective surface implies (K_S+B) is
pseudoeffective (because its intersection with an ample divisor is
nonnegative, and a nef divisor with positive self-intersection is
pseudoeffective; if (K_S+B)^2 = 0 it is numerically trivial, also
pseudoeffective). Nefness also gives nonnegative self-intersection;
the value (K_S+B)^2 can also be checked by pulling back to a projective
resolution, where the divisor is still nef and has the same square.
The pseudoeffective form of Langer's inequality (Langer 2003,
Corollary 5.2) then gives
(K_S + B)^2 <= 3 * e_orb(S, B).
It remains to compare the two Euler-type numbers. By Langer's
definition, smooth points of the boundary have weight zero; at every
singular point of the pair that lies on that boundary the local
orbifold number is also zero (Langer 2003, Theorem 7.6, using the
coefficient-one branch of the log canonical boundary). For
Z = Sing(S) minus Supp(B), the weighted formula therefore reads
e_orb(S, B) = χ_c(S°) + sum_{x in Z} ( e_orb(x; S, 0) - 1 ).
Each term in the sum is nonpositive by the local upper bound
(Langer 2003, Definition 3.4 and Corollary 7.8). So the last
inequality holds, including all singular-point corrections. []
Remark 3.6.3. The singular-point corrections in Langer's e_orb are a
genuinely delicate feature: they are nonpositive for surfaces with
canonical or worse singularities. On a smooth surface with B = empty
boundary, e_orb equals χ_c, and Langer reduces to the ordinary
Bogomolov-Miyaoka-Yau. The point of stating the inequality with a
boundary is that we will apply it to *log* canonical divisors that
come from our adjunction, not to plain K_S.
We now combine Lemmas 3.4.1, 3.5.3, 3.6.1, 3.6.2 in the context of the
log model. Set
L = K_Y + H, Y_0 = S_0 + sum_a S_a,
B_S = reduced boundary of S in Y_0 + H,
S° = S minus Supp(B_S).
By Lemma 3.4.1, each extra component S_a is projective, (S_a, B_{S_a})
is dlt (hence log canonical), and L|_{S_a} is nef. Combined with the
adjunction formula L|_{S_a} ~ K_{S_a} + B_{S_a}, this means we can
apply Lemma 3.6.2 to every extra component S_a:
0 <= (K_{S_a} + B_{S_a})^2 <= 3 * χ_c(S_a°). (*)
Lemma 3.6.2 does *not* need to apply to S_0 (which is the strict
transform of the original surface X_0).
Now, by Proposition 3.4.2 in the non-log-canonical case we obtained the
isomorphisms
S_0° ≈ X_0 minus Gamma,
Y_c minus Supp(B_{Y_c}) ≈ X_c minus Gamma (c != 0).
The first of these open surfaces is smooth (S_0 is the strict transform
of the original X_0, and after compactification we already knew X_c is
smooth away from Gamma). On every extra component S_a, any isolated
interior singularity is smoothed by the family (Lemma 3.6.1 tells us
this smoothing has χ(M_p) >= 1, contributing the nonnegative excess
χ(M_p) - 1 to the global delta).
Apply Proposition 3.5.3. Constancy of the punctured-fiber Euler
characteristic d(d^2-4d+6)-mu-1 (from Lemma 3.2.1) plus cancellation
of the S_0° contribution gives
0 = sum_a χ_c(S_a°) + delta, (**)
where
delta = sum_a sum_{p in Sing(S_a) ∩ S_a°} (χ(M_p) - 1) >= 0.
Each S_a is projective, its adjoint divisor is nef, so Lemma 3.6.2
applied to (S_a, B_{S_a}) gives (*). Combining with (**):
0 <= sum_a (K_{S_a} + B_{S_a})^2 <= 3 * sum_a χ_c(S_a°)
= -3 delta <= 0.
The only way for 0 to be <= a nonpositive quantity is both to be zero:
delta = 0, sum_a (K_{S_a}+B_{S_a})^2 = 0, and each individual square
(K_{S_a}+B_{S_a})^2 = 0.
This is the central numerical conclusion of the whole chapter: every
extra logarithmic square vanishes.
-----------------------------------------------------------------------
3.7 Specializing the logarithmic square from the nearby fiber to the
central fiber
-----------------------------------------------------------------------
The vanishing of each (K_{S_a}+B_{S_a})^2 gives a specialization
identity that converts a projective intersection computation on a
general fiber into the same computation on the special fiber. This
is the arithmetic that finally pins down beta.
Lemma 3.7.1 (square specialization). Let q: Y -> C be a flat projective
morphism to a smooth connected complex curve, with pure
two-dimensional fibers. Let L be a rational Cartier divisor on Y. If
[Y_0] = sum_j m_j [S_j] is the cycle of a fiber Y_0 (here the m_j are
the coefficients of Y_0), then for every c in C,
( L|_{Y_c} )^2 = sum_j m_j ( L|_{S_j} )^2.
In particular, the multiplicities are all one when Y_0 is reduced.
PROOF. Choose r > 0 such that rL is Cartier. Flatness and properness
imply that chi( Y_c, O_Y(n r L)|_{Y_c} ) is constant in c for every
integer n (Stacks Project, Lemma 36.32.2). This is a polynomial in n of
degree at most two, whose quadratic coefficient is r^2 (L|_{Y_c})^2 / 2.
Its leading term is the sum of the leading terms on the individual
two-dimensional components of Y_0, weighted by their generic lengths;
these are the m_j (Stacks Project, Lemmas 33.45.1, 33.45.2, 33.45.6).
Comparing coefficients of n^2 and dividing by r^2 gives the identity.
[]
We can also give a second proof that explains why compactifying the
ambient Y introduces no extra term. Take an integral normal projective
compactification Ybar -> C of Y -> C. Suppose L is supported on the
part of Y that is proper over C (as in our application: all components
of H are proper over C). Then L extends by zero to a rational Cartier
divisor on Ybar (the support is already closed in Ybar). Applying the
lemma to Ybar gives the same identity; the extra components in
Ybar minus Y carry zero divisor class, so their contribution is zero.
This clarifies why, in what follows, no compactification of the
original Y is needed.
Proposition 3.7.2 (numerical consequence of the log model). Suppose
Y is the log model of Proposition 3.3.1 for a projective family
X -> C as in Lemma 3.2.1, and L = K_Y + H is h-nef. For each component
S of Y_0 = S_0 + sum_a S_a with S_0 strict transform of X_0, let
B_S = full reduced boundary of S (Lemma 3.4.2(d), (e) in the
non-log-canonical case). Suppose
( K_{S_a} + B_{S_a} )^2 = 0 for every a.
Then beta( X_c, Gamma(c) ) = beta( X_0, Gamma(0) ) for all c in C
sufficiently close to 0.
PROOF. Fix c != 0. The threefold Y dominates the smooth curve C, so it
is flat (its local rings are torsion-free over the discrete valuation
rings of C). Its fibers have pure dimension two, and it is projective
over C. Apply Lemma 3.7.1:
( L|_{Y_c} )^2 = ( L|_{S_0} )^2 + sum_a ( L|_{S_a} )^2.
Now compute each term. By the adjunction formula of Lemma 3.4.1 and
the fact that all curves in S_a are contracted by h:
L|_{Y_c} ~ K_{Y_c} + B_{Y_c}, ( L|_{S_0} )^2 = ( K_{S_0} + B_{S_0} )^2,
L|_{S_a} ~ K_{S_a} + B_{S_a}, ( L|_{S_a} )^2 = ( K_{S_a} + B_{S_a} )^2.
Since the S_a squares all vanish by hypothesis, the middle terms drop
out and we get
( K_{Y_c} + B_{Y_c} )^2 = ( K_{S_0} + B_{S_0} )^2. (+)
Now apply Lemma 3.0.2 (the projective formula for beta) to the model
S = Y_c -> X_c (for c != 0) and to S = S_0 -> X_0 = X_0. Both meet the
hypotheses:
* X_c is integral normal projective, smooth outside Gamma(c)
(from Lemma 3.2.1);
* Y_c is a projective birational morphism over X_c that is an
isomorphism outside Gamma(c);
* B_c := B_{Y_c} is the reduced inverse image of Gamma(c) (Lemma
3.4.2(e));
* (Y_c, B_c) is log canonical (Lemma 3.4.1);
* K_{Y_c} + B_c is nef over X_c (since L is h-nef).
So
beta( X_c, Gamma(c) ) = K_{X_c}^2 - ( K_{Y_c} + B_c )^2,
beta( X_0, Gamma(0) ) = K_{X_0}^2 - ( K_{S_0} + B_{S_0} )^2.
The canonical squares K_{X_c}^2 and K_{X_0}^2 are equal: they both
equal d(d-4)^2 from Lemma 3.2.1(iv). Subtracting:
beta( X_c, Gamma(c) ) - beta( X_0, Gamma(0) )
= [ ( K_{S_0} + B_{S_0} )^2 - ( K_{Y_c} + B_c )^2 ]
(by distribution of the minus sign)
= 0
(by identity (+) above).
Thus beta is the same on all nearby fibers. This finishes Proposition
3.7.2. []
Remark. The finite base change constructed in Proposition 3.3.1
repeats the individual surface germs: the base change C' -> C is
locally of the form s^b = tau, so the fiber over s and the fiber over
zeta s (for zeta an e-th root of unity) are the same scheme-theoretic
surface. So the beta equality on the base-changed curve implies the
beta equality on the original curve after a Zariski shrinking. This
finishes the reduction of the algebraic assertion to the hypothesis
that all extra squares (K_{S_a}+B_{S_a})^2 vanish.
-----------------------------------------------------------------------
3.8 Assembling the numerical theorem
-----------------------------------------------------------------------
Putting together the pieces from 3.1-3.7 yields the main theorem of the
chapter.
Theorem 3.8.1 (beta is constant in the non-log-canonical case). Let
F(x, t) be a holomorphic family of isolated hypersurface singularities
in C^3 with f_t(0) = 0 and constant positive Milnor number mu. Suppose
that the central normal surface germ X_0 = { f_0 = 0 } is not log
canonical. Then beta(X_t, 0) = beta(X_0, 0) for every sufficiently
small t.
PROOF. By Proposition 3.1.2 (analytic -> algebraic curve), it suffices
to prove the algebraic assertion AB from section 3.1. So consider a
polynomial family g_c over a smooth algebraic curve C, with constants
Milnor number mu and non-log-canonical central fiber.
Step 1 (reduce to the compactified family). Apply Lemma 3.2.1 to
compactify: we get a projective hypersurface family X -> C in P^3 x C,
with the singularities preserved at the marked point and with
K_{X_c}^2 = d(d-4)^2 and chi_c(punctured fiber) = d(d^2-4d+6) - mu - 1
for all c. Log canonicity is constant by Lemma 3.0.3, so all germs
X_c are non-log-canonical.
Step 2 (build the log model). Apply Proposition 3.3.1 (which includes
a finite base change). This gives a projective birational morphism
h : Y -> X, a horizontal boundary H, a reduced fiber
Y_0 = S_0 + sum_a S_a, and K_Y + H nef over X.
Step 3 (full boundary). Lemma 3.4.1 and Proposition 3.4.2 verify all
adjunction and full-boundary requirements:
S_0° ≈ X_0 minus Gamma,
Y_c minus Supp(B_{Y_c}) ≈ X_c minus Gamma (c != 0),
and B_{S_a} is the full reduced exceptional preimage on each S_a.
Step 4 (interior correction and Langer). Lemma 3.6.1 gives interior
correction chi(M_p) - 1 >= 0 for isolated interior singularities of
each S_a. Apply Proposition 3.5.3 (Euler comparison) and use the
constancy of the punctured-fiber Euler characteristic from Step 1.
After cancellation of the S_0° contribution, we get
0 = sum_a chi_c(S_a°) + delta, delta >= 0,
which combined with Lemma 3.6.2 (surface inequality):
0 <= sum_a (K_{S_a} + B_{S_a})^2 <= 3 sum_a chi_c(S_a°) = -3 delta <= 0,
forces all (K_{S_a} + B_{S_a})^2 to vanish.
Step 5 (specialize). Apply Proposition 3.7.2 with the vanished extra
squares: beta is constant along the (base-changed) algebraic curve.
Step 6 (return to the analytic family). Unwind the base change (the
individual surface germs are repeated by the finite cover), and use
Proposition 3.1.2 again to pass from the algebraic curve back to the
analytic disk family. So beta is constant in t near 0. []
This is the end of Chapter 3. The numerical theorem 3.8.1 is the entire
content of the "non-log-canonical" branch of the main theorem's proof.
The next chapters do the rest: construct the resolution, lift to an
ambient homeomorphism, handle the log-canonical and double-point cases.
==========================================================================
CHAPTER 4. FROM THE NUMERICAL INVARIANT TO A RESOLVED FAMILY
==========================================================================
We now leave the numerical/geometric world of Chapter 3 and enter a more
topological world. Our input is: a mu-constant family
F(x, t) = f_t(x) of isolated hypersurface singularities in C^3 with
central surface X_0 non-log-canonical, AND the fact (Theorem 3.8.1)
that beta is constant along the family.
Our output is a smooth family N -> Delta of surfaces, an honest
simultaneous resolution of all X_t at once, with a reduced exceptional
divisor having relative normal crossings. That smooth resolved family
is what Chapter 5 will lift to an ambient homeomorphism.
There are two key intermediate theorems. The first is a classical
result of Okuma (2004) that gets us most of the way, but only
*after a finite base change* t = u^e (say e disjoint sheets glued at
u^e = t). The second is a new argument using the fixed-ball Euler
characteristic identity from Lemma 2.2, which upgrades Okuma's
simultaneous resolution to have relative normal crossings (not
just fiberwise normal crossings). Without this second argument, the
exceptional divisor could have nodal singularities in the total space
of the resolved family; with it, every exceptional component is smooth
over the parameter.
Definitions. A surface resolution is *semigood* when its reduced
exceptional divisor has normal crossings and each irreducible component
is smooth but may self-intersect in a node. For a divisor in a smooth
family of surfaces parametrized by u, *relative normal crossings*
means that near every point of the divisor there are coordinates
(z1, z2, u) with the family map equal to u and the divisor locally
given by z1 = 0 or by z1 * z2 = 0. This is the strongest reasonable
statement: every fiber is SNc, and the SNc structure varies smoothly
in u.
Proposition 4.1 (resolved family, relative NC). Let
F(x, t) = f_t(x) satisfy the hypotheses of the main theorem (Chapter 7
for the exact statement): isolated critical point at 0 of every fiber,
f_t(0) = 0, constant Milnor number mu > 0. Assume:
(a) the central surface X_0 = { f_0 = 0 } is not log canonical at 0;
(b) beta(X_t, 0) is constant in t (proved in Theorem 3.8.1).
Then, after shrinking the parameter disk and making a finite base
change t = u^e, there is a proper modification
b : N -> X' = { F(x, u^e) = 0 }
such that:
(i) N -> Delta_u is a smooth family of complex surfaces;
(ii) b is an isomorphism off the origin section;
(iii) the reduced exceptional divisor E = b^{-1}({0} x Delta_u)
has relative normal crossings over Delta_u;
(iv) for each u, the fiber b_u : N_u -> X_{u^e} is the minimal
semigood resolution of X_{u^e} at 0;
(v) E|_{N_u} is the full reduced exceptional divisor of b_u.
PROOF.
First, note a small reduction. The zero-fiber family X -> Delta is a
flat deformation of a normal Gorenstein surface singularity. Each
fiber (by Lemma 2.1) has only the one singular point at the origin in
the chosen representative. So the beta invariant at the origin section
is the same as the sum of the beta invariants over all the singular
points of the fiber. This is exactly the form in which Okuma's
hypotheses are stated.
Step 1: Set up the small-deformation hypotheses of Okuma's paper.
Okuma (2004, Sections 3-4) requires (i) an analytic total space X,
(ii) a resolution of X, and (iii) the ideal sheaf of the singular
section principalized on the resolution. Let us produce this.
Choose a projective resolution of singularities of the analytic total
space X (say r : X~ -> X smooth, proper, iso over X minus Gamma, and
with r^{-1}(Gamma) a divisor with SNc support on X~). This provides
the ambient smooth space X~.
Now choose a principalization of the ideal sheaf of the section
Gamma on X~. If the section ideal has already been processed by r, we
may need to further resolve; the procedure is the same as in
Wlodarczyk 2009 Theorems 2.0.1-2.0.3, Lemma 4.0.3 (a combination of
Hironaka-style resolution of the ideal and its support). The result is
a proper map (still call it r) X~ -> X with X~ smooth, r iso off
Gamma, and r^{-1}(Gamma) a simple normal crossing divisor.
The composite is therefore a proper map from a smooth threefold X~ to
X, isomorphic off Gamma, and r^{-1}(Gamma) a full SNc divisor over
the section. This is exactly Okuma's local setup.
Step 2: Smoothness over the punctured parameter disk. We restrict to
the parameter disk minus the origin. Over Delta minus {0}, X minus
Gamma is smooth; after base change by u delta -> t = u^e, over c
parameter Delta_u minus {0}, X_u is a smooth surface with
r_u : X~_u -> X_u a birational contraction.
Actually we need to be more careful: r is constructed on the *total*
space X, which is three-dimensional. The family X_u -> u is a surface
family; the total X~ with the map X~ -> Delta_u is a threefold over a
disk. Is it smooth over Delta_u minus 0?
To check: near the sphere boundary of the fixed ball (Lemma 2.2), r
is an isomorphism and the original family X -> Delta is smooth there.
So r|_{boundary} is an iso and X~_u is smooth over the ball boundary.
Interior of X~ minus r^{-1}(Gamma): r is an iso, X is smooth over
Delta minus {0} there. Over r^{-1}(Gamma) with u != 0: the fiber is
r^{-1}(Gamma)_u which is the exceptional divisor of the fiber
resolution r_u. The critical locus of the parameter map on X~ and on
closed intersections of exceptional components lies in r^{-1}(Gamma),
proper over the disk. On each such component a critical point of the
parameter map has constant parameter value (each component maps
constantly onto the disk near u=0, since it lies over Gamma = {0}xDelta,
so any critical locus sits over u = 0 exactly). Thus over Delta_u minus 0
the parameter map is smooth with relative NC exceptional divisor.
Consequence: over Delta_u minus {0}, X_u -> Delta_u is a locally
trivial family of surfaces (or a locally trivial deformation of the
exceptional divisor E_u). This is the "punctured-disk weak simultaneous
resolution" that Okuma's setup requires.
Step 3: Apply Okuma's theorem. Okuma 2004 (Theorem 1.2): suppose
X -> Delta is a flat family of normal Gorenstein surface singularities
with constant Wahl invariant and non-log-canonical central fiber. After
a finite base change t = u^e, there is a simultaneous resolution
N -> X' = { F(x, u^e) = 0 }
over Delta_u such that:
* N is smooth over Delta_u;
* N -> X' is proper, birational, iso off the section;
* each fiber N_u -> X_{u^e} is the minimal semigood resolution;
* the reduced exceptional divisor E has every fiber E_u equal to the
full reduced exceptional divisor of the fiber resolution.
Take the branch of the finite base change (Delta_u -> Delta, u -> u^e);
normalize if necessary. A local parameter u at 0 puts the map in the
form t = u^e. Flatness and the smooth-fiber property propagate through
the base change: X' is still flat over Delta_u, and the smooth fibers
of N -> X' over Delta_u minus 0 push through. Also the "reducedness of
E" holds at u = 0 (by Okuma's theorem) and away from u = 0 (where the
fiber N_u is a resolution of the smooth surface X_{u^e}, and the
exceptional divisor E_u is by definition reduced). Hence E is a
relative effective Cartier divisor (its restrictions are Cartier, no
fiber contains E, and E's support is proper over the base).
The support of E is exactly b^{-1}({0} x Delta_u): that is, the
preimage of the origin section. So (i), (ii), (iv), (v) are achieved.
Step 4: The critical remaining point, relative normal crossings.
Okuma's theorem gives fiberwise normal crossings of E_u. But this
does NOT imply that E is a relative normal crossing divisor in N over
Delta_u: fiberwise nodes could smooth (or vice versa) as u moves. We
now prove that they do not, using Euler characteristics.
Choose the fixed closed Milnor ball B from Lemma 2.2 and set
N_B = b^{-1}( X' cap (B x Delta_u) ) subset N.
Its projection to the disk is a proper smooth submersion with boundary
(near the sphere boundary dB, b is an isomorphism and the family
X' -> Delta_u is a fibration, so N_B -> disk is a smooth fibration with
boundary by Ehresmann's theorem). Therefore chi((N_B)_u) is constant in
u.
The complement of E_u in this fiber is isomorphic to
(X_{u^e} cap B) minus {0}, by construction of the modification
b (it's an iso off the origin section). By Lemma 2.2(c),
chi(X_t cap B) = 1 and chi_c((X_t cap B) minus {0}) = 0. Using
additivity of chi_c on complements:
chi_c( (X_{u^e} cap B) minus {0} ) = 0.
So chi((N_B)_u) = chi(E_u) + chi of the complement
= chi(E_u) + 0.
Therefore chi(E_u) = chi((N_B)_u), which is constant in u.
The compact reduced nodal curves E_u have constant chi.
Step 5: A node cannot smooth. Let us examine a node of E_u near u = 0.
The divisor E is Cartier in the smooth total space N. At a smooth
point of E_0, E is already smooth over the parameter (by the fiber-is-
SNc property of the fiber E_u near the smooth point). At a node of
E_0, the holomorphic Morse lemma with parameter gives relative
coordinates (z1, z2, u) in which
E = { z1 * z2 = a(u) }, a(0) = 0, a(u) holomorphic,
where a(0) = 0 because E_0 has a node at that point. This is the
local model of a nodal curve smoothing.
There are finitely many nodes of E_0. Choose small disjoint neighborhoods
of them with transverse boundaries. Smoothness of N over Delta_u near
the remaining points of E_0, together with properness of E -> Delta_u,
excludes new nodes outside these neighborhoods after shrinking the
disk (compactness + the fact that no new singularities appear at
u = 0).
The remainder of E is a proper smooth family with boundary, so its
topology is unchanged with u.
Now consider a single node. Inside its neighborhood:
* At u = 0: a single crossing point, Euler characteristic 1.
* At u != 0 with a(u) != 0: an annulus (or disk pair), whose
Euler characteristic is 0 (a connected annulus, b_0 = 1, b_1 = 1,
chi = 0).
* At u != 0 with a(u) = 0 identically on that neighborhood: a
crossing persists, chi = 1.
So for general small u,
chi(E_u) = chi(E_0)
- (number of nodes whose smoothing function a is
not identically zero).
No global labeling of components is needed; the calculation works
pointwise (in disjoint neighborhoods) and also includes
self-intersection nodes.
Since by Step 4 chi(E_u) is constant, the number of nodes smoothing is
zero. Every smoothing function a(u) is identically zero. This
identifies a node neighborhood that persists into the total space: the
equation z1 z2 = 0 in relative coordinates makes E locally
SNc in N over Delta_u. Together with the smooth-point charts, this
proves relative normal crossings of E near E_0.
Properness of E -> Delta_u then extends the conclusion over a smaller
disk. This finishes Proposition 4.1. []
The output of Chapter 4 is a smooth family N -> Delta_u together with a
reduced relative-NC divisor E, after a finite base change t = u^e. This
is the "clean" input for Chapter 5's construction of the ambient
homeomorphism. Getting rid of the base change will be the very last
step of Chapter 5.
==========================================================================
CHAPTER 5. FROM A RESOLVED FAMILY TO AN AMBIENT HOMEOMORPHISM
==========================================================================
The goal of this chapter: given the resolved family N -> Delta_u of
Chapter 4, plus the defining function G(x,u) = F(x, u^e) of the
zero-fiber family X', construct the ambient coordinate changes
H_u : T_0 -> T_u, G(H_u(x), u) = G(x, 0),
as in the statement of the main theorem, jointly continuous in u with
jointly continuous inverse. Then remove the finite base change
t = u^e to get the trivialization of the original family F(x,t).
Throughout: B is a closed Euclidean ball around 0 in C^3 (from
Lemma 2.2), D is a smaller parameter disk, and
T_u = { x in B : |G(x,u)| < eta }
is the ambient tube at parameter u. The tube T_0 is the source; T_u is
the target. The origin 0 sits in each T_u; the section 0 x D is the
union of the origins.
Proposition 5.1 (ambient tube lemma). Let G(x,u) be holomorphic on a
neighborhood of B x D, with G(0,u) = 0 and d_x G(x,u) != 0 for
x != 0. Assume that, for some eta > 0, the levels
G(.,u) = c of absolute value |c| < eta meet the sphere dB transversely
(for every u in D).
Define the punctured zero family
Z* = { (x, u) in (B minus {0}) x D : G(x,u) = 0 }.
Suppose there exist smooth real vector fields v_1, v_i on Z*, tangent to
the sphere boundary dB, satisfying:
(a) v_1, v_i project to the vector fields ∂/∂Re(u) and ∂/∂Im(u) on D
respectively;
(b) for every u in D, the ODE transport of Re(u) v_1 + Im(u) v_i for
time one starting from (x, 0) exists for all x on the zero level
(G(x,0) = 0), x != 0, and the reverse transport starting from
(y, u) exists for all y on the target zero level.
Then there are homeomorphisms
H_u : T_0 -> T_u,
satisfying
(i) G( H_u(x), u ) = G( x, 0 );
(ii) H_0 = identity;
(iii) H_u(0) = 0;
(iv) the combined map (x,u) |-> (H_u(x), u) and its inverse are
jointly continuous.
PROOF. First we extend the vector fields. On the ambient tube minus
the origin-axis, the pair map (G, u) : (B x D) minus (0 x D) -> C x D
is a smooth submersion (dG is surjective because d_xG != 0 off the
axis; du is trivially surjective). Let K = ker(dG, du) be the
associated real 2-plane bundle over (B x D) minus (0 x D).
The given v_a are sections of K over Z*, tangent to dB. We want smooth
extensions V_a to the whole ambient tube minus the axis, still sections
of K, still tangent to dB (where defined), and agreeing with v_a on Z*.
Construction: fix the prescribed projection du(V_a) = a (a = 1, i).
This reduces the constraint to: V_a agrees with v_a on Z* (a
codimension-1 submanifold of the ambient tube minus the axis), is
tangent to dB, and is in K. All these are affine-linear conditions on
V_a. In submersion coordinates (G-coordinate, u-coordinate, and
2 transverse coordinates) the kernel bundle K is trivial, and the
values v_a|Z* extend from the closed submanifold Z* (which is the
preimage {G = 0} intersected with the ambient tube).
Boundary tangency to dB is a closed condition. A partition of unity
patches local extensions to a global V_a. Notice: we do NOT extend V_a
across the axis (x = 0); the axis is a codimension-2 "hole" where the
submersion (G,u) has singular behavior.
Now for u in D, set
W_u = Re(u) V_1 + Im(u) V_i on (B x D) minus (0 x D).
W_u has du(W_u) = u (parameter-direction 1) and dG(W_u) = 0.
Integrate W_u for time one, starting from (x, 0), for any x in T_0.
Existence of the flow for the whole time:
* For x = 0: the trajectory is the axis (0, s u) with s in [0, 1];
the axis is preserved by W_u because G(0, s) = 0 for all s.
* For x != 0 with G(x, 0) = 0: existence is the hypothesis (b).
* For x != 0 with G(x, 0) = c != 0: the trajectory stays in the
compact set
{ (y, s) : y in B, s in [0, u], G(y, s) = c }.
This set misses the origin-axis because c != 0. Smooth continuation
of W_u on the complement of the axis plus boundary tangency to dB
give existence for the whole time interval.
* Similarly for G(x, 0) = c with 0 < |c| < eta.
Define H_u(x) = the spatial part of the endpoint of W_u integrated for
time one from (x, 0). Since dG(W_u) = 0, we have
G(H_u(x), u) = G(x, 0). H_0 = identity (since W_0 = 0).
Reverse transport: integrate W_u in reverse time from (y, u). This
exists by hypothesis for y on the zero level, and by compactness in
the other levels. It defines an inverse map
H_u^{-1} : T_u -> T_0.
The joint continuity of H_u and its inverse near the axis is where the
real subtlety is. Here's the argument.
Let x_j -> 0 in T_0 and u_j -> u_0 in D. Since the trajectory of
(x_j, 0) is contained in a compact set (as in the boundedness argument
above, uniformly in j for u_j in a small disc), the H_{u_j}(x_j) lie
in B. Pass to a subsequential limit y in B. Then G(y, u_0) = 0 by
continuity. If y != 0, then y is a nonzero point over u_0 on the zero
level, and the reverse flow H_{u_0}^{-1} exists in a neighborhood of y
(smooth dependence of reverse transport near y). That neighborhood
contains H_{u_j}(x_j) for j large, so H_{u_j}^{-1}(H_{u_j}(x_j)) = x_j
converges to H_{u_0}^{-1}(y) = 0. But that would require x_j to
converge to a point over u_0 rather than to 0. Contradiction (unless
H_{u_0}(0) = y = 0). Hence any subsequential limit is 0, so
H_{u_j}(x_j) -> 0 = H_{u_0}(0).
Same argument for the inverse: if y_j in T_{u_j} -> 0, then a nonzero
subsequential limit of H_{u_j}^{-1}(y_j) would contradict smoothness
of the forward transport. So the inverse map is jointly continuous at
the axis as well.
Therefore: H_u extends continuously to 0 with H_u(0) = 0, and the
combined map (x,u) |-> (H_u(x), u) and its inverse are jointly
continuous everywhere on the tube union and on the ball minus origin,
respectively.
The restriction to the interiors of the tubes T_0, T_u is the desired
ambient trivialization. This finishes Proposition 5.1. []
Corollary 5.2 (Whitney regular zero fibers). Let G(x,u) be a holomorphic
family with G(0,u) = 0, no spatial critical points off the origin
section near (0,0), and let
( X* := {G = 0} minus {x = 0}, A := {x = 0} )
satisfy Whitney conditions (a) and (b). Then G has the trivialization
of Proposition 5.1.
PROOF. Write ρ(x,u) = ||x||^2. Claim: (u, ρ) is a real submersion on
X* sufficiently near (0,0). Suppose otherwise. Choose points
(x_j, u_j) in X* -> (0,0) witnessing failure of submersivity.
Move to a subsequence where the unit spatial vectors x_j/||x_j||
converge to some unit v.
Whitney (a) says the tangent spaces of X* converge to the tangent
spaces of the fiber A, so the limiting tangent space L of X* contains
the two real base directions. The fiber tangent spaces Q_j of
{G(., u_j) = 0} lie in ker du and have real dimension 2(n-1). So
Q := lim Q_j = L intersect ker du (both inclusions, with equal
dimension). Whitney (b) applied to secant lines (x_j, u_j) to (0, u_j)
gives (v, 0) in L. So (v, 0) in Q. But submersivity of (u,ρ) fails at
these points: dρ vanishes on the fiber tangent space. So (v, 0) is
orthogonal to Q. Contradiction.
So (u, ρ) is submersive near (0, 0). Choose a sufficiently small
closed ball and parameter disk inside this submersion neighborhood.
Smooth local right inverses to d(u, ρ), patched by partition of unity,
produce lifts V_1, V_i of the parameter directions with dρ(V_a) = 0.
Their trajectories preserve the positive value of ρ = ||x||^2, hence
stay in a compact set disjoint from the axis. So complete transport
holds in both directions.
These fields are tangent to the sphere boundary (ρ = const). The zero
levels are transverse to it. Lemma 5.1 applies. []
Proposition 5.3 (lifts supplied by a resolution). Let G(x,u) satisfy
the analytic and transversality hypotheses of Proposition 5.1. Suppose
that the zero family {G = 0} admits a proper holomorphic map
b : M -> {G = 0} over the parameter disk D, such that:
(a) M -> D is a smooth family of complex surfaces;
(b) b is an isomorphism away from the origin section;
(c) E = b^{-1}({x = 0}) is a relative normal-crossing divisor.
Then G has the trivialization of Proposition 5.1.
PROOF. Choose the ball B and a smaller parameter disk within
representatives of b. Above the closed ball, M -> D is a smooth
submersion with boundary: near the sphere dB, b is an isomorphism and
the fibers meet dB transversely (Lemma 2.2).
At each point of E, relative normal-crossing coordinates give local
lifts of the two real parameter directions ∂/∂Re(u), ∂/∂Im(u) tangent
to every branch of E. Concretely, at a crossing point in coordinates
(z_1, z_2, u), the branches of E are the coordinate sources; and the
parameter-direction fields ∂/∂Re(u), ∂/∂Im(u) are tangent to each
z_1 = 0 and z_2 = 0.
Away from E, use submersion coordinates (M -> D is smooth off E);
near the sphere boundary, use boundary submersion coordinates. A
partition of unity patches these local lifts to smooth global lifts
Ṽ_1, Ṽ_i on M, tangent to E and to the sphere boundary.
Properness makes the inverse image of the closed ball over any compact
parameter segment compact. So the smooth lifts are complete along that
segment in both directions. Their flows preserve E and its complement:
tangency preserves each branch, and uniqueness in both time directions
prevents a trajectory in the complement from reaching E.
Push the fields down through the isomorphism b to fields on
X* = {G = 0} minus {x = 0}. Then all the hypotheses of Proposition 5.1
are met: complete transport on the punctured zero family, jointly
continuous lift of parameter directions. Lemma 5.1 gives the desired
trivialization. []
Lemma 5.4 (removal of a finite disk base change). Let F(x,t) be a
holomorphic family with F(0,t) = 0. Suppose that, after the base change
t = u^e (e >= 1), the family G(x,u) = F(x, u^e) has the tube
trivialization H_u of Proposition 5.1. Then F is ambiently topologically
right trivial over the original disk, by a section-preserving
homeomorphism whose central map is the identity.
PROOF. Let ζ = exp(2πi / e), the primitive e-th root of unity. For
real s >= 0 sufficiently small, define
J_s : T_0 -> T_0,
J_s := H_{ζ s}^{-1} o H_s.
Both H_s and H_{ζ s} are available (both are defined for s in a small
interval [0, ε) with ε > 0), and their targets coincide because
s^e = (ζ s)^e. So J_s is a homeomorphism from T_0 to T_0. It preserves
the defining function:
H_{ζ s} restricts to the target at parameter ζ s, so H_s maps
T_0 to T_s, and H_{ζ s}^{-1} maps T_s to T_0, and both preserve
G-values (i.e. G(H_u(x),u) = G(x,0)). So the composition preserves
f_0 = F(., 0).
At s = 0, both H_0 = id, so J_0 = id. The families J_s and
J_s^{-1} = H_s^{-1} o H_{ζ s} are jointly continuous in s by continuity
of H and its inverse.
Now consider the polar description t = r^e exp(i θ), r >= 0,
0 <= θ <= 2π. Define, for t != 0,
φ_t : T_0 -> T_t,
φ_t := H_{ r exp(i θ / e) } o J_{ r θ / (2π) }.
Note that the base-change target at parameter u = r exp(i θ / e) is
t = u^e = r^e exp(i θ). And J_{r θ/(2π)} maps T_0 to itself,
preserving f_0. So the composite maps T_0 to T_t = T_u via two steps
of the tube trivialization.
Check the seam: at θ = 2π, we have
r exp(i θ / e) = r exp(2πi / e) = ζ r,
r θ / (2π) = r.
So the target-side factor is H_{ζ r} o J_r. And
φ_t at θ = 2π = H_{ζ r} o J_r.
Meanwhile at θ = 0 (i.e. wrapped around once), we have
r exp(0) = r,
r * 0 / (2π) = 0,
so
φ_t at θ = 0 = H_r o J_0 = H_r o id = H_r.
These must agree:
H_{ζ r} o J_r = H_r,
which is exactly the defining property of J_r:
J_r = H_{ζ r}^{-1} o H_r => H_{ζ r} o J_r = H_r.
So the two endpoint formulas coincide. The map φ_t glues.
Inverse: φ_t^{-1} = J_{r θ/(2π)}^{-1} o H_{r exp(i θ / e)}^{-1}. Its
seam identity is J_r^{-1} o H_{ζ r}^{-1} = H_r^{-1}, which also
holds from the definition of J_r.
At r = 0: every constituent map is the identity independently of θ
(H_0 = id, J_0 = id). So define φ_0 = id. Joint continuity of the
constituent families and their inverses passes through: the polar disk
with its collapsed center circle is homeomorphic to a standard disk,
and joint continuity on the polar disk (with each point in a
continuous system of radial parameter and θ) descends to joint
continuity on the disk.
For every t: source is the tube T_0, target is
{ x in B : |f_t(x)| < eta }. Additionally,
f_t o φ_t = f_0 o J_{r θ/(2π)} = f_0,
φ_t(0) = 0.
Restricting to the interiors of the tubes gives the asserted
homeomorphism germ (x,t) |-> (φ_t(x), t) over the original disk. []
Remark. A finite map between smooth pointed disks has the form t = u^e
after a change of coordinates: write it as u^e times a nonvanishing
holomorphic unit and absorb a holomorphic e-th root of that unit into u.
Accordingly Lemma 5.4 applies to a normalized disk branch of the finite
base change used for a simultaneous resolution. It requires no
deck-equivariance of that resolution or of its chosen lifts.
That concludes the topological core of the proof. The next two chapters
handle the second case (log canonical and double-point central fiber),
which uses a different but simpler invariant (sectional Milnor numbers),
and then assemble the main theorem from these pieces.
==========================================================================
CHAPTER 6. THE DOUBLE-POINT AND LOG-CANONICAL CASES
==========================================================================
Our treatment so far handles the case X_0 non-log-canonical. The
remaining cases split into two:
(Case D = "double point"): mult_0(f_0) = 2.
(Case L = "log canonical"): X_0 is log canonical at 0 (and
mult_0(f_0) >= 3).
In each, we have an alternative route to topological right triviality
that passes through the *sectional Milnor numbers* mu^(j) and the
Whitney regularity theorem of Teissier.
First, recall the sectional Milnor numbers. For a holomorphic germ f
in n variables with isolated critical point at 0, define for
1 <= j <= n:
mu^(j)(f) := Milnor number of f restricted to a general
j-dimensional linear subspace of C^n, at 0.
mu^(n) = mu (full Milnor number),
mu^(1) = mult_0(f) - 1,
mu^(j) = number of j-by-j minors in a general plane section.
Write mu^*(f) = (mu^(1), mu^(2), mu^(3)) in our case (n = 3).
We need two theorems from the literature:
Theorem A (equivmultiplicity, Fernandez de Bobadilla-Pelka 2024).
Let F(x, t) be a holomorphic family of isolated hypersurface
singularities with constant finite Milnor number mu(f_t) = mu > 0. Then
mult_0(f_t) is constant. In particular, in our notation,
mu^(1)(f_t) = mult_0(f_t) - 1 is constant. Moreover the same conclusion
holds when f_t are formal power series whose coefficients vary
continuously along a real interval.
The proof in Fernandez-Bobadilla-Pelka is a substantial theorem based
on symplectic geometry of A'Campo's radius-zero monodromy model and on
a Floer-theoretic spectral sequence. We take it as a black box.
Theorem B (Teissier's criterion, Teissier 1973, Chapter II Theorem 3.9).
Let F(x,t) be a deformation of an isolated hypersurface singularity
with a section over a smooth curve, and suppose the complement of the
section in the total space is smooth over the base. If mu^*(f_t) is
constant along the deformation, then the pair
(punctured zero set, section) satisfies Whitney conditions (a) and (b).
Combined with Corollary 5.2 (which takes Whitney regularity to
topological right triviality via the Whitney lift), the conclusion of
Theorem B gives us right-triviality in the Whitney regular cases.
Proposition 6.1 (the two easy cases). Let F(x,t) = f_t(x) be a
holomorphic family of isolated hypersurface singularities in three
variables with each f_t having an isolated critical point at 0 and
constant Milnor number mu. If either
(i) mult_0(f_0) = 2 (double point), or
(ii) X_0 = {f_0 = 0} is log canonical at 0,
then F is topologically right-trivial over a sufficiently small
parameter disk.
PROOF. Case (i): mult_0(f_0) = 2.
The multiplicity of f_0 at 0 is exactly 2, so the quadratic part of
f_0 is a nondegenerate quadratic form in some linear direction.
Choose a spatial direction in which the second partial of f_0 is
nonzero. After a linear change of coordinates, this is the direction z
(write x = (z, y1, y2)). The implicit function theorem solves the
equation f_{0,z} = 0 in a small neighborhood as
z = a(y1, y2, 0),
with a(y1, y2, 0) a holomorphic function vanishing at 0 and with the
solution being unique near 0 for small t as well:
z = a(y1, y2, t), a(0,0,t) = 0.
Translate z by a(y1, y2, t). Then the quadratic part of f_t in z is
c(y1, y2, t) * (z - a(y1, y2, t))^2 + (higher order in z)
+ g_t(y1, y2),
with c(0, t) != 0. Take a holomorphic square root of c near 0, well
defined for small t (choose the branch with positive value at t = 0).
Rescale z accordingly. This gives biholomorphic coordinates depending
holomorphically on t and preserving the origin, in which
f_t = z^2 + g_t(y1, y2).
Then the Jacobian quotient of f_t equals the Jacobian quotient of g_t:
the Jacobian of f_t is (2z, g_{t,1}, g_{t,2}), and mod z this is just
the Jacobian of g_t in the y's. So g_t has an isolated critical point
at 0 and mu(g_t) = mu(f_t).
In particular mu(g_t) is constant (equal to mu). So by Theorem A
Fernandez-Bobadilla-Pelka, the multiplicity mult_0(g_t) is constant,
and hence mu^(1)(g_t) = mult_0(g_t) - 1 is constant.
Now g_t is a family of plane curves: y = (y1, y2) are two variables.
The sectional Milnor numbers for g_t are mu^(1)(g_t) (multiplicity
minus one) and mu^(2)(g_t) = mu(g_t). Both are constant. So by
Theorem B (Teissier), the pair (punctured zero set of g_t, section) is
Whitney regular. By Corollary 5.2, g_t is right-trivializable.
Extend the trivialization of g_t by the identity in the z coordinate.
This gives a trivialization of f_t in the coordinates above.
Conjugate by the coordinate changes (a(y,t) shift in z, square-root
rescaling) at 0 and at t. The result is the required
right-trivialization of f_t, with central map the identity. []
Case (ii): X_0 log canonical.
Subcase mult_0(f_0) = 2 has already been settled in Case (i).
Subcase mult_0(f_0) >= 3. Then we need to show the sectional Milnor
numbers mu^*(f_t) are constant.
Setup. Inversion of adjunction (Kawakita 2007) identifies log canonicity
of the surface germ (X_0, 0) with log canonicity of the pair
(C^3, X_0) as a pair in the sense of the log MMP. The different on the
normal Gorenstein hypersurface X_0 inside the smooth threefold C^3 is
zero, so we can work entirely in the ambient pair. These single-germ
algebraic statements also apply to the analytic germ by its analytic
right-equivalence to a sufficiently high polynomial jet (Lemma 2.3).
Blowup. Blow up the origin in C^3. The exceptional divisor of the
blowup (the projective space P^2 of all complex directions) with the
strict transform X~_0 of X_0 gives the "log model" of the blown-up
pair (C^3', X_0') with boundary X~_0 + P. Since we assumed (C^3, X_0)
log canonical, the pair (X~_0 + P, C^3') is log canonical with
coefficient-one boundary.
Log discrepancy of the exceptional plane. On the blowup, the
exceptional divisor P has log discrepancy 2 - m, where
m = mult_0(f_0) (its self-intersection in the blown-up pair is -1
because it is a P^2 with normal O(-1), so the discrepancy is
2 - m). Log canonicity requires 2 - m >= 0, so m <= 3.
Since the origin is a critical point of f_0, m >= 2. Subcase m = 2 is
Case (i) already. So m = 3.
Adjunction to P. Restricting the boundary to P gives the pair
(P, C), where C = X~_0|_P is the tangent cubic of the exceptional
plane, i.e. the zero set in P^2 of the cubic initial form of f_0,
counted with scheme-theoretic multiplicities from f_0's Taylor
expansion. Adjunction to a boundary component of an lc pair preserves
log canonicity, so (P, C) is log canonical. A boundary component of an
lc pair has coefficient at most 1, so C is a reduced divisor in
P^2, i.e. C is a reduced (possibly reducible, possibly singular)
plane curve of degree 3.
Equimultiplicity. By Theorem A, mult_0(f_t) is constant, so
mult_0(f_t) = 3 for all t near 0. So the tangent cubic C_t of f_t at 0
is a cubic plane curve for all t.
Choice of a good transversal. Pick a projective line L subset P^2
that is transverse to C (reduced) at exactly three distinct smooth
points. This choice can be made because the degree is 3 and C is
reduced. This transversality condition persists as t varies near 0,
because the family of cubics C_t depends continuously on t.
For such a line, the plane section f_t|_{L^⊥} (where L^⊥ is the
2-plane in C^3 whose projectivization is L) has a singularity at 0
whose tangent cone is a reduced cubic in L = P^1 cut by three distinct
points. Concretely: in the coordinates adapted to L, the quadratic
form of the Jacobian ideal of f_t|_{L^⊥} at the origin is a "product of
three linear forms"; the Jacobian ideal's local intersection has
Milnor number 4.
Milnor number computation for an ordinary triple point. If
h(y1, y2) is a plane curve with an ordinary triple point at 0 (three
distinct tangents), then h generates a curve singularity of Milnor
number (3-1)^2 = 4. Reason: h is right-equivalent to
y1 y2 (y1 + y2) (or y1 y2 (y1 - y2), etc.) near 0, which has an
isolated singularity of multiplicity 3; the Jacobian quotient is
generated by the three pairwise products, giving complex dimension 4.
Therefore for all t near 0:
mu^(2)(f_t) = 4 (general plane section has ordinary triple point),
mu^(1)(f_t) = 3 - 1 = 2 (multiplicity 3),
mu^(3)(f_t) = mu(f_0) (constant by hypothesis).
So mu^*(f_t) is constant throughout. Teissier's Theorem B gives Whitney
regularity. Corollary 5.2 gives topological right-triviality. This
completes Case (ii) m = 3.
Putting (i) and (ii) together: Proposition 6.1 holds. []
==========================================================================
CHAPTER 7. THE MAIN THEOREM
==========================================================================
We now assemble the preceding chapters into the main theorem. Every
piece of the statement can be checked against one of the case splits
below.
Theorem 7.1 (topological right-triviality of mu-constant families of
surface singularities). Let Delta be a disk centered at 0 in C. Let
F : (C^3 x Delta, {0} x Delta) -> (C, 0)
be holomorphic. Assume that for every t in Delta:
* f_t(0) = 0,
* f_t has an isolated critical point at 0,
* mu(f_t, 0) = mu(f_0, 0) = mu > 0 (constant Milnor number).
Then after shrinking the disk and the germ representatives (in both the
spatial and the parameter directions), there exists a homeomorphism
germ
Phi(x, t) = ( phi_t(x), t ),
with phi_t : (C^3, 0) -> (C^3, 0) obeying
(i) phi_0 = identity,
(ii) phi_t(0) = 0,
(iii) f_t( phi_t(x) ) = f_0(x),
(iv) Phi and Phi^{-1} are both jointly continuous near the origin
section {0} x Delta (over the original disk Delta).
PROOF. We split into cases according to the local analytic type of the
central surface X_0 = {f_0 = 0} at 0.
Case 1: mult_0(f_0) = 2 (ordinary double point).
Then Proposition 6.1(i) directly provides the trivialization Phi over a
sufficiently small disk. Specifically: after the splitting
f_t = z^2 + g_t(y) of that proposition, the equimultiplicity theorem
gives constant sectional Milnor numbers of g_t, Teissier's criterion
gives Whitney regularity, and Corollary 5.2 (Whitney lift) gives the
trivialization of g_t which, extended by the identity in z and
conjugated back through the coordinate changes, gives the required
trivialization of f_t.
Case 2: X_0 is log canonical at the origin (and mult_0(f_0) >= 3).
Proposition 6.1(ii) handles this. The blowup gives the exceptional
plane and the tangent cubic; log canonicity forces the tangent cubic to
be reduced; equimultiplicity forces mult_0(f_t) = 3 for small t; and
the sectional Milnor numbers are mu^*(f_t) = (2, 4, mu(f_0)) for all
small t. Teissier's criterion gives Whitney regularity, and Corollary
5.2 gives the trivialization.
Case 3: X_0 is not log canonical at the origin (and mult_0(f_0) >= 3).
Step 3a (numerical): By Theorem 3.8.1, the Wahl invariant is constant
along the family: beta(X_t, 0) = beta(X_0, 0) for small t.
Step 3b (resolution): By Proposition 4.1, after a finite base change
t = u^e and shrinking Delta, there is a smooth family N -> Delta_u
with a proper birational map b : N -> X' = {F(x, u^e) = 0} over the
origin section, b is an isomorphism off the origin section, and its
reduced exceptional divisor E has relative normal crossings.
Step 3c (fixed-ball hypothesis): Lemmas 2.1 and 2.2 provide the
hypotheses of the ambient tube lemma with respect to the appropriate
Milnor ball B and small eta > 0: (a) the spatial critical locus is
exactly the origin section; (b) the levels meet the sphere dB
transversely; (c) chi(X_t cap B) = 1 for all t.
Step 3d (ambient tube): Proposition 5.3 (lifts supplied by a
resolution) gives smooth vector fields on the punctured zero family
with complete transport. Proposition 5.1 (ambient tube lemma) then
produces homeomorphisms
H_u : T_0 -> T_u, G(H_u(x), u) = G(x, 0),
jointly continuous in u with jointly continuous inverse, where
G(x, u) = F(x, u^e).
Step 3e (remove base change): Lemma 5.4 removes the finite base
change: from H_u we construct
phi_t(x) = H_{ r exp(i theta / e) } o J_{ r theta / (2 pi) } (x),
for t = r^e exp(i theta), with the seam correction
J_s = H_{zeta s}^{-1} o H_s. The seam identity
H_{zeta r} o J_r = H_r makes the formula agree at theta = 0 and
theta = 2 pi; at r = 0, phi_0 = identity. The resulting phi_t satisfy
f_t( phi_t(x) ) = f_0(x), phi_t(0) = 0,
and Phi = (x, t) -> (phi_t(x), t) is jointly continuous with jointly
continuous inverse.
Conclusion: In all cases, we obtain a trivialization Phi satisfying all
four properties (i)-(iv) of Theorem 7.1, over the original disk after
shrinking. The finite base change introduced by the simultaneous
resolution is removed in Step 3e; the parameter disk size shrinks
finitely many times (each argument introduces its own small disk or
ball), but any finite intersection of disks contains a smaller disk
around 0. This finishes the proof of Theorem 7.1. []
That completes the main theorem. The theorem positively answers the
surface case of the mu-constant problem: a holomorphic family of
isolated hypersurface singularities in C^3 with constant Milnor number
is topologically right-trivial, and the trivialization fixes the
parameter, the origin section, and the defining function, with the
identity at the central parameter.
==========================================================================
A NOTE ON THE SHAPE OF THE PROOF
==========================================================================
The proof has three conceptually distinct pieces:
(1) A numerical-algebraic piece (Chapter 3): the Wahl invariant
beta is constant under the hypothesis of constant Milnor number,
using the log minimal model program, the nearby Euler
characteristic, and Langer's inequality. This is by far the
longest part of the argument. Its central idea is the "cancellation
of Euler characteristics", which forces the logarithmic canonical
squares of all the extra vertical surfaces S_a to vanish, and
thereby forces the log square on the whole special fiber to agree
with the log square on the general fiber.
(2) A numerical-to-topological piece (Chapter 4): constant beta, by
Okuma's theorem, lets us build a simultaneous resolution after a
finite base change. The auxiliary Euler-characteristic argument on
the exceptional divisor upgrades Okuma's resolution to one with
*relative* normal crossings: no nodes of the exceptional curve can
smooth as the parameter moves.
(3) A topological piece (Chapter 5): the resolved family lets us build
a family of vector fields on the punctured zero set that lift the
two real parameter directions. Flowing them along a parameter
segment gives the ambient homeomorphisms H_u. A careful seam
correction at the angular boundary removes the finite base change,
producing the original-disk trivialization.
The log-canonical and double-point cases (Chapter 6) use a wholly
different technology (sectional Milnor numbers, equimultiplicity,
Teissier's criterion for Whitney conditions), because the numerical
invariant approach requires non-log-canonicity to force a strictly
negative coefficient in the log discrepancy.
The next (final) appendices gather technicalities that were invoked in
the main argument but that deserve a self-contained treatment.
==========================================================================
APPENDIX A. TECHNICALITIES
==========================================================================
A.1. Marked monomial reduction in dimension three (Proposition 3.3.2)
The proof of Proposition 3.3.1 in section 3.3 invoked an auxiliary
fact: that a given simple-normal-crossings morphism
q : Z -> C from a smooth complex threefold to a smooth curve can be
turned, after a finite base change s^b = tau and a projective
birational modification, into one whose special fiber is reduced and
simple normal crossing and whose total space is smooth. This section
proves that fact, following the dimension-three monomial reduction of
Kollar-Mori 1998, Chapter 7, and making the corresponding toric
argument explicit.
Statement (repeated). Let q : Z -> C be a morphism from a smooth
quasi-projective complex threefold to a smooth curve, with marked
point 0 in C. Let Z_0 = sum_i a_i Z_i (its scheme-theoretic fiber),
and let H be a reduced divisor with horizontal components. Suppose:
(i) (Z_0)_red + H is simple normal crossing in Z;
(ii) away from 0, q is smooth with relative-snc divisor H.
Let tau be a uniformizer at 0 (a regular parameter on C whose only zero
is 0), and suppose C is shrunk so that tau has no other zero on it.
Choose any positive integer b divisible by every a_i. After the finite
base change s^b = tau and normalization, there is a projective
birational modification
psi : Z' -> Z x_C Delta_s
such that Z' is smooth, psi is an isomorphism over every open subset
where the original morphism was smooth, and the special fiber
(Z')_{s = 0} is reduced and simple normal crossing in Z', together with
the transformed horizontal marking.
The modification psi = psi_1 o psi_2 o psi_3 o psi_4 is a composition of
four steps. Step 1 is the finite base change. Step 2 is normalization.
Step 3 is a toric star-subdivision. Step 4 is a composition of blowups
in smooth centers. All steps are projective and all work in char 0.
STEP 1: Finite base change. Put Delta = the disk or affine line in s
parametrized by the base-change variable s, with relation s^b = tau.
The pullback of C to Delta is smooth away from the closed subscheme
s = 0 (which is a single point since C has marked point 0 only), so the
fiber product Z x_C Delta is smooth away from (q^{-1}(0)) x_C Delta.
But the relation s^b = tau has an ordinary ramification of order b over
0. If V is smooth at a generic point of q^{-1}(0), then the local
equation of q^{-1}(0) inside V is tau = 0, and in the base-changed
scheme V x_C Delta we have s^b = tau = 0, i.e. s = 0 with multiplicity
b in the naive fiber product. That is not reduced. Normalization will
fix this. So far, the total space is smooth, but the special fiber has
multiplicity b at each stratum.
STEP 2: Normalization. Take the normalization
Z~ := (normalization of Z x_C Delta).
The universal property of normalization in the total fraction ring
means: every stratum of q^{-1}(0) x Delta whose multiplicity b is
already coprime to the ramification order of the local structure
becomes reduced. In dimension three, a point of q^{-1}(0) has at most
three local coordinates contributing to the special fiber, and taking
the normalized base change in s^b = tau is equivalent to the usual
"toric normalization" of the local model
tau = u * z_1^{a_1} * z_2^{a_2} * z_3^{a_3}
where u is a unit (choose local coordinates adapted to the SNc fiber).
Specifically, after taking the b-fold base change and normalizing, the
result is again smooth locally near each point, provided b is a
multiple of each a_i. This follows from the standard toric computation
carried out below in Step 3, but here is the key computation: write
tau = (u) * z_1^{a_1} * z_2^{a_2} * z_3^{a_3},
b = a_i * d_i, and take one extra local coordinate w with
w^b = u (which is available after an etale extension of the base).
STEP 3: Toric star-subdivision. At a point z of the reduced fiber
(Z_0)_red + H, the SNc assumption means we can choose local coordinates
z_1, z_2, z_3 in which
tau = u * prod_{i in I} z_i^{a_i}, |I| <= 3,
where u is a unit and I is the subset of indices i <= 3 for which z_i
divides the fiber. The horizontal components of H present are the
remaining coordinate hyperplanes (those of indices not in I).
Apply the etale covering that extracts the b-th root of u. This
reduces the local calculation to the monomial case tau = prod z_i^{a_i}
over the etale base extension (with multiplication of the base by the
new variable w satisfying w^b = u, which is etale since b is coprime
to the residue characteristic in char 0). The horizontal factor is a
product over one common etale factor of the same size.
Now consider the local model via toric geometry. After the base change
s^b = tau and normalization, the local special fiber is the union of
the toric strata, i.e.
{ s = 0 } union { z_i = 0 } for i in I,
with multiplicities governed by the toric fan. The fan consists of
cones:
sigma_I = R_{>= 0}^I (a cone in R^I, |I| <= 3),
N_I = { (d_i)_{i in I} in Z^I : sum_{i in I} a_i d_i
= 0 modulo b },
i.e. N_I is the lattice of integer vectors whose weighted sum of
coordinates is 0 in Z/b. The order function (integral height of the
monomial divisor on the base) is
l(d) = ( sum_{i in I} a_i d_i ) / b, d in N_I.
The primitive generator of each original ray is (b/a_i) * e_i, which
lies in N_I because sum_{j} a_j (b/a_i) delta_{ij} = b = 0 mod b. Its
height is (a_i * b/a_i) / b = 1: the ray has height one, hence survives
with coefficient 1 in the base-changed model (no base-change
enlargement at that stratum).
Now apply a star subdivision: triangulate the cone sigma_I into
simplicial cones by inserting all height-one lattice rays and then
subdividing recursively within each face. All insertions are lattice
points on height-one slices, so every cone in the resulting fan is
regular. The toric variety obtained by this subdivision therefore
satisfies:
(a) Smooth total space (a toric variety with regular fan is smooth);
(b) Reduced special fiber (all primitive rays have height one);
(c) Simple normal crossings (regular simplicial cones = SNc strata);
(d) Horizontal markings preserved (they are product factors
unaffected by the vertical subdivision).
Carrying out subdivisions on a single orthant with one coordinate for
each global Z_i (rather than locally at each point) and with the same
congruence lattice N_I restricted to each face makes the choices
compatible across all charts. A point not on a given face does not
change that face under further subdivision, so consistency holds
automatically.
STEP 4: Global patching. The local toric modifications are
compatible on overlaps, so they patch (with a partition of unity
argument in the etale topology, using that the etale cover is finite
and its structure group acts compatibly). The result is a projective
birational modification psi : Z' -> Z x_C Delta with Z' smooth, the
special fiber reduced and snc, and the horizontal marking preserved.
Small-dimensional regularity check (needed for Step 3). In the vertical
cone sigma_I has dimension at most three. Its height-one slice
is a segment, triangle, or point.
* An empty lattice segment (no lattice points inside) is primitive:
its endpoints are a basis for the lattice on its line.
* An empty lattice triangle (no interior or edge lattice points)
is unimodular. Proof: translate one vertex to zero and denote
its edge vectors v, w. If the lattice index of {v, w} exceeded one,
a nonzero lattice point of the half-open parallelogram would have
the form alpha v + gamma w with 0 <= alpha, gamma < 1. If
alpha + gamma <= 1, this point lies in the triangle and is not a
vertex. If alpha + gamma > 1, the point (1 - alpha) v +
(1 - gamma) w lies in the triangle. Both contradict emptiness.
So {v, w} is a lattice basis for the cone's lattice.
* Adding a height-one vertex to the triangle gives a basis of the
full lattice N_I on the cone's span. Hence the cone is regular.
The same conclusion follows in dimension 2 (primitive segment = basis)
and is immediate in dimension 1.
This completes the proof of Proposition 3.3.2. []
A.2. Continuous gradient lifts (alternative to resolution)
Chapter 5 built ambient homeomorphisms via a resolution. There is a
second, more direct route that uses a Whitney-regularity condition or
a smallness condition on the parameter-derivative of the defining
function, instead of a full simultaneous resolution. We record it
because several older references (Parusinski 1999, Le-Saito 1973,
Teissier 1973) phrase the argument in this language, and because it
explains why the surface case works even without the Okuma route. We
only sketch this alternative.
Statement. Let G(x,u) be a holomorphic family with G(0,u) = 0,
d_x G(x,u) != 0 for x != 0. Let (X*, A) be the pair (punctured zero
set, its origin section). Suppose:
(Wh1) The pair (X*, A) satisfies Whitney condition (a): every
tangent hyperplane limit K at each point of A contains the
A-direction.
(Wh2) The pair (X*, A) satisfies Whitney condition (b): every
limiting tangent space of the fibers of the total zero set
at x -> (0,a) contains the secant directions to A.
Then the ambient tube lemma (Proposition 5.1) applies: there are
ambient homeomorphisms H_u as in that statement.
The proof constructs explicit vector fields by a projection trick.
Let K be the limiting hyperplane at (0, u) (a complex 2-dimensional
tangent space to the fiber at 0). Let pi_{K} be the Hermitian
orthogonal projection onto K, and set q(x, u) = pi_{K(x, u)}(x) as
before. Whitney (a) plus a small-o condition on the parameter
derivative forces
||q(x,u)|| / ||x|| -> 1 as (x,u) -> (0, u).
This is proved by a limiting argument: along any sequence, pass to a
subsequence where both x/||x|| and K(x,u) converge, and use Whitney
(a) to show the limit is the appropriate projection. Then define
w_a = k_a - (Re / ||q||^2) * q, for a = 1, i,
where k_a are the coefficient lifts of the two parameter directions.
The corrections are chosen so that
w_a is in K(x, u) [so dG(w_a) = 0],
d||x||^2 (w_a) = 0 [preserves distance to origin],
and the norm bound ||w_a|| <= ||k_a|| * ||x|| / ||q||, together with
the projection estimate above, gives w_a -> 0 at A. Every constant
real combination preserves the positive radius of a trajectory in X*,
so no such trajectory can reach A in finite time. Proposition 5.1
applies directly.
For our family, the L-S-T (Le-Saito-Teissier) criterion provides the
small-o condition in every mu-constant isolated family, so this
alternative is available in all the cases of the main theorem. The
argument we gave in Chapter 5 using a resolution is the version that
does not require any estimate beyond the resolution existence. Both
routes give the same final trivialization.
A.3. Alternative curve comparison inside an irreducible component of
the jet locus
Section 3.1 proves constancy of the Wahl invariant by reducing a
holomorphic family to an algebraic curve inside the jet locus U_mu,
using the constructibility of the invariant beta. Here is an
alternative that only uses constructibility of U_mu itself
(which is a much easier algebraic statement) and replaces the
constructibility of beta by a more elementary path-argument.
Let a_0 in U_mu be the jet of the central polynomial f_0 (of degree
<= N with N >= mu + 1). Let Z be an irreducible component of U_mu
containing a_0. Claim: beta(V(g), 0) is constant for all g in Z.
Proof. Take two points a_0 and a_1 in Z. If they coincide, trivially
beta is equal. Otherwise:
* If dim Z = 1, Z itself is a smooth complete-intersection curve
through a_0 and a_1 up to resolution.
* If dim Z >= 2, choose general sufficiently ample hypersurfaces
through a_0 and a_1 that cut out a smooth irreducible projective
curve C inside a projective closure of Z. Existence follows from
Bertini: the linear system of hypersurfaces through two points in
general position has no base locus at these points after choosing
sufficiently many such hypersurfaces to separate first jets there.
Iterating irreducibility and smoothness, we get a smooth complete-
intersection curve passing through both a_0 and a_1.
Now the curve C sits inside Z subset U_mu. So the family of jets
parameterized by C has constant Milnor number mu (they are all in
U_mu). The central surface germ of the marked point (say a_0) is
non-log-canonical (or log canonical, consistently, by Lemma 3.0.3).
Apply assertion AB (from section 3.1) to this jet family, receiving
beta(V(a_0), 0) = beta(V(a_1), 0). Since a_1 was arbitrary in Z, the
result follows.
Applying the conclusion to the specific holomorphic family: pick a_0
= j^N f_0 in U_mu. Let a(t) = j^N f_t. Near t = 0, a(t) lies in U_mu.
Take an analytic neighborhood of a_0 in U_mu. The finitely many
irreducible components of U_mu not containing a_0 form a closed subset
missing a_0; a sufficiently small neighborhood of a_0 avoids it.
Therefore every nearby jet a(t) shares an irreducible component with
a_0. So beta(V(a(t)), 0) = beta(V(a_0), 0) = beta_0 for all small t.
Finally, individual finite determinacy (Lemma 2.3) returns
beta(X_t, 0) = beta(V(a(t)), 0) = beta_0 for the original analytic
family. This proves the constancy of beta.
This alternative avoids the substantive constructibility of beta
(which requires resolution theory in a smooth family) by passing to a
curve inside the jet locus. The argument is "local on the jet locus"
and works uniformly in any dimension of the ambient space.
==========================================================================
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==========================================================================
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