diff --git a/cohomological_transfer_equivariant_v2.txt b/cohomological_transfer_equivariant_v2.txt
index 625694a..489bd4d 100644
--- a/cohomological_transfer_equivariant_v2.txt
+++ b/cohomological_transfer_equivariant_v2.txt
@@ -37,7 +37,7 @@ makeatother
 
 
 
-newcommand{hol}[1]{mathcal O_calO_{#1}}
+newcommand{hol}[1]{O_{#1}}
 
 title[Cohomological transfer and anticanonical sections]{Cohomological transfer and equivariant anticanonical sections}
 
@@ -62,11 +62,11 @@ is a line bundle, the anticanonical line of (X). In coordinates, a section of (L
 
 For which smooth connected projective (X) over (C) is
 [
- H^0(X,mL)neq 0
- qquadfor some integer m>0?
+ H^0(X,mL)!= 0
+  for some integer m>0?
 
 
-Nonzero sections of an ample line bundle give maps of (X) into projective space; the spaces of sections grow polynomially in (m); the linear system of a big line eventually has positive dimension. The anticanonical line (L) need not be ample. It can be trivial, as on an abelian variety, and it can be a torsion point of the Picard group. Question~q:nonvanishing therefore asks whether some positive power of (L) is at least effective (that is, whether (H^0(X,mL)neq 0) for some (m)), a condition much weaker than amplitude. It is the first question one can ask about positivity of (L) and the natural warm-up for the harder question of semiampleness.
+Nonzero sections of an ample line bundle give maps of (X) into projective space; the spaces of sections grow polynomially in (m); the linear system of a big line eventually has positive dimension. The anticanonical line (L) need not be ample. It can be trivial, as on an abelian variety, and it can be a torsion point of the Picard group. Question~q:nonvanishing therefore asks whether some positive power of (L) is at least effective (that is, whether (H^0(X,mL)!= 0) for some (m)), a condition much weaker than amplitude. It is the first question one can ask about positivity of (L) and the natural warm-up for the harder question of semiampleness.
 
 The word nonvanishing in the literature refers to this question. The companion problem, abundance, asks for much more: whether the sections of (L) generate a morphism nice enough to be a fibration. We do not address abundance.
 
@@ -83,7 +83,7 @@ Smooth semipositivity of (L) does not imply smooth semipositivity of (L^{ ⊗  m
 Our main theorem, restated in the introduction, is the following.
 
 [Restatement of Corollary~thm:ordinary below]
-Every smooth connected projective complex variety (X) with smoothly semipositive (L=-K_X) has (H^0(X,mL)neq 0) for some integer (m>0).
+Every smooth connected projective complex variety (X) with smoothly semipositive (L=-K_X) has (H^0(X,mL)!= 0) for some integer (m>0).
 
 
 The power (m) in Corollary~cor:main-informal is not uniform. It depends on (X). A uniform bound cannot exist: on the projective variety (X=PP^1), (m=1); on an abelian surface, where (L) is trivial, (m) must equal (1); on some Fano manifolds, small powers are non-effective and large powers are effective; on a K3 surface with smooth semipositive anticanonical line (that is, a K3 with (L) trivial), the same (m=1) applies. These examples are not an obstruction to the theorem, they are its proof strategy: it takes some power, not a specific one, and it depends on the geometry of (X).
@@ -101,7 +101,7 @@ This section is a glossary. Each item is one sentence; a reader who already know
 [(V5)] A section of a line bundle (A) over an open set (U) is a holomorphic way to choose a point in each fibre of (A|_U). The symmetric power (H^0(X,mL)) is the vector space of sections of (mL); its elements vanish along effective divisors when nonzero.
 [(V6)] The cohomology group (H^q(X,A)) is the (q)-th derived functor of (H^0(X,-)), computed in v{C}ech or Dolbeault terms. On a smooth projective (X), (H^q(X,OO_X)) is a Hodge summand of (H^q(X,C)). The Euler characteristic of a vector space or sheaf is (chi(X,A)=sum_q(-1)^qdim H^q(X,A)).
 [(V7)] A Hermitian metric (h) on a line bundle (A) is a smooth family of inner products on the fibres. Its Chern curvature (iΘ_h(A)) is a closed smooth ((1,1))-form representing the first Chern class (c_1(A)). The metric is semipositive when (iΘ_h(A)ge0) as a ((1,1))-form. It is pseudoeffective when (c_1(A)) lies in the closure of the cone of effective divisor classes, equivalently when (A) carries a singular Hermitian metric with (iΘge0).
-[(V8)] An algebraic torus (T=(C^times)^k) acts algebraically on (X). A character (chi) of (T) is a holomorphic group homomorphism (TtoC^times), necessarily monomial and therefore of the form (tmapsto t^a) for (ainZ^k); we write such characters additively. A linearization of a (T)-action on a line bundle (A) is a lift of the action to (A) that is linear on fibres; if (A=mL), the natural linearization uses the natural action of (T) on (T_X^{ ⊗  m}).
+[(V8)] An algebraic torus (T=(C^times)^k) acts algebraically on (X). A character (chi) of (T) is a holomorphic group homomorphism (TtoC^times), necessarily monomial and therefore of the form (t|-> t^a) for (ainZ^k); we write such characters additively. A linearization of a (T)-action on a line bundle (A) is a lift of the action to (A) that is linear on fibres; if (A=mL), the natural linearization uses the natural action of (T) on (T_X^{ ⊗  m}).
 [(V9)] A section is invariant under (T) if the linearization sends it to itself. The invariant subspace (H^0(X,A)^T) is what we control. On a compact torus and a shift (P=L ⊗ character), invariants of the natural and shifted linearizations agree; on a non-compact torus they do not.
 [(V10)] An abelian variety is a complex torus that is projective. A Fano variety is smooth projective with (-K_X) ample. A K3 surface is a simply connected compact complex surface of complex dimension two with trivial canonical bundle.
 [(V11)] A normal variety is one whose local rings are normal domains; it is smooth outside a closed set of codimension at least two. A birational morphism is a proper morphism that is an isomorphism outside a proper subvariety. Resolving a variety means replacing it by a smooth variety that maps to it by a birational morphism and is isomorphic outside a proper subvariety; over (C), resolutions always exist (Hironaka).
@@ -179,7 +179,7 @@ Plain form. On the generic fiber, invariant sections of the adjoint line (E+jkP'
 
 Technical form. For every interior index (i) and every divisor (H') on (Y), the ratio construction gives (H^0(V,api^*N_i-f^*H')^Tne0) for some (a>0). Combined with Step 3, this proves
 [
- rank(f_*mathcal O_V(E+jkP'))^T=1calO_V(E+jkP'))^T=1
+ rank(f_*O_V(E+jkP'))^T=1calO_V(E+jkP'))^T=1
  (jge0).
 ]
 The section (1_Egamma^j) (where (1_E) is the canonical section of (E)) gives the lower bound.
@@ -200,7 +200,7 @@ Plain form. We need a bounded metric on the base and a positive-curvature metric
 
 Technical form. Average a smooth semipositive metric (h_0) on (L) over the maximal compact torus; it is invariant for both the natural and shifted linearizations, because a character is unitary on the compact torus. The ratio domination plus pseudoeffectivity of (-M) gives a singular invariant metric (h) on (L') with
 [
- iΘ_hgedelta f^*omega_Y, mathcal I(h)=calI(h)=OO_V, delta>0.
+ iΘ_hgedelta f^*omega_Y, I(h)=, delta>0.
 ]
 The first inequality uses ratio domination to make (L'-eta f^*H) pseudoeffective for some (eta>0), mixing the resulting current with (h_0).
 
@@ -208,12 +208,12 @@ The first inequality uses ratio domination to make (L'-eta f^*H) pseudoeffective
 noindentStep 7: Nadel vanishing and Euler interpolation.
 Plain form. For each (jge0), the base twists give a polynomial in (j). At (j=0) the polynomial is one; hence it is nonzero for some positive (j). The resulting section pulls back, multiplies, and pushes down to give the required invariant section.
 
-Technical form. Apply Proposition~prop:maximum to obtain a continuous semipositive metric on (kB). The singular coefficient metric at (j=0) gives a metric on the base with positive curvature and multiplier ideal (mathcal JcalJ), not assumed trivial. Tensor with the bounded metric on (jkB); the ideal remains (mathcal JcalJ), and Nadel vanishing gives
+Technical form. Apply Proposition~prop:maximum to obtain a continuous semipositive metric on (kB). The singular coefficient metric at (j=0) gives a metric on the base with positive curvature and multiplier ideal (J, not assumed trivial. Tensor with the bounded metric on (jkB); the ideal remains (J, and Nadel vanishing gives
 [
- H^v(Y,OO_Y(D_*+jkB) ⊗ mathcal J)=0calJ)=0
+ H^v(Y,OO_Y(D_*+jkB) ⊗ J)=0
  (v>0, jge0).
 ]
-The Euler characteristic of (OO_Y(D_*) ⊗ mathcal JcalJ) twisted by (jkB) is a polynomial in (j). At (j=0) its value is (h^0(Y,mathcal J)calJ)ge1), because (OO_Y(D_*) ⊗ mathcal J⊇calJ⊇OO_Y) at the level of the multiplier ideal. A nonzero section therefore exists for some positive (j). Pull back, multiply by (gamma^j), push down to (X), and remove the exceptional twist (E).
+The Euler characteristic of (OO_Y(D_*) ⊗ J twisted by (jkB) is a polynomial in (j). At (j=0) its value is (h^0(Y,J), because (OO_Y(D_*) ⊗ J⊇calJ⊇OO_Y) at the level of the multiplier ideal. A nonzero section therefore exists for some positive (j). Pull back, multiply by (gamma^j), push down to (X), and remove the exceptional twist (E).
 
 
 Two technical ingredients are used without proof inside Theorem~thm:transfer: the maximum metric of Proposition~prop:maximum (built in Section~sec:models), and the extension of the base singular metric across boundary divisors (built in Section~sec:local-ideal). If the reader prefers to see the proof of the fixed multiplier ideal first, they can read Section~sec:local-ideal before returning to this proof.
@@ -354,7 +354,7 @@ A second factor argument, stronger than vanishing of holomorphic forms, is avail
 [Covariant-tensor holonomy dichotomy]
 Let (P) be a compact simply connected nonflat K"ahler factor, irreducible in the Riemannian de Rham sense, with nonnegative Ricci curvature. Either (K_P) has a nonzero parallel holomorphic volume form, or
 [
- H^0(P,(Ω_P^1)^{ ⊗  m})=0qquadfor every integer m>0.
+ H^0(P,(Ω_P^1)^{ ⊗  m})=0 for every integer m>0.
 ]
 
 
@@ -441,7 +441,7 @@ Let (X,Y) be smooth connected projective, let (L=-K_X) have a smooth semipositiv
 ]
 where (W) is normal, (g) is projective surjective equidimensional, (mu) is birational, and (V) is a smooth projective resolution. Set (D_V=(munu)^*L) and (E=K_{V/X}), and let (s_E) be the Jacobian section of (E). Suppose a nonzero rational section (s) of (bmu^*L), (b>0), has no horizontal poles and on the generic fibre (V_eta)
 [
- H^0(V_eta,E+kbD_V)=C(Y),s_Es^kqquad(kge0).
+ H^0(V_eta,E+kbD_V)=C(Y),s_Es^k (kge0).
 ]
 In the invariant case, the same condition with (H^0(-)^T) and (P) in place of (L), (s) invariant, (Y) trivially acted on. Then after replacing (b,s) by a common multiple and a power, there are a line (Q) on (Y) and a regular injection
 [
@@ -457,7 +457,7 @@ The rank one at (k=0) forces generic geometric connectivity. The constant functi
 
 For every prime divisor (Bsubset Y), put
 [
- c_B=min_{Gammamapsto B}(ord_Gamma(s))/(ord_Gamma(g^*B)).
+ c_B=min_{Gamma|-> B}(ord_Gamma(s))/(ord_Gamma(g^*B)).
 ]
 Equidimensionality supplies divisorial lifts of (B), so the minimum is over a nonempty set; finiteness and integrality after clearing denominators give the base divisor. Sending the canonical section of the resulting base line to the powered (s) gives a regular bundle map by normality. The effective Cartier zero divisor (G) omits at least one component above every base prime.
 
@@ -480,9 +480,9 @@ In words.:  The transfer theorem of Section~sec:transfer was written for invaria
 ## Injectivity and base cohomology
 
 [Injectivity and base cohomology]
-Let (g:Vto Y) be a morphism of smooth projective complex varieties, let (A) be very ample on (Y) with smooth metric of positive curvature (omega_A). Suppose (N) on (V) has a singular Hermitian metric (h) with (mathcal I(h)=calI(h)=OO_V) and (iΘ_h(N)geε g^*omega_A) for some (ε>0). Then
+Let (g:Vto Y) be a morphism of smooth projective complex varieties, let (A) be very ample on (Y) with smooth metric of positive curvature (omega_A). Suppose (N) on (V) has a singular Hermitian metric (h) with (I(h)= and (iΘ_h(N)geε g^*omega_A) for some (ε>0). Then
 [
- H^i(Y,R^jg_*OO_V(K_V+N))=0qquad(i>0, jge0).
+ H^i(Y,R^jg_*OO_V(K_V+N))=0 (i>0, jge0).
 ]
 
 
@@ -491,9 +491,9 @@ Fujino--Matsumura injectivity [cite] says that on a compact K"ahler manifold, mu
 
 We reduce to base cohomology by induction on (dim Y). Choose (m) large for Serre vanishing on all direct images (R^jg_*OO_V(K_V+N)). A general smooth divisor (Cin|mA|) contains no associated point of these sheaves; its restricted coefficient metric has trivial multiplier ideal. The long direct-image sequence of
 [
- 0tomathcal FcalFtomathcal FcalF ⊗  g^*OO_Y(mA)toOO_{V_C}(K_{V_C}+N|_{V_C})to0
+ 0toF FcalF ⊗  g^*OO_Y(mA)toOO_{V_C}(K_{V_C}+N|_{V_C})to0
 ]
-breaks into short exact sequences, since multiplication by the cutting equation is injective on each (R^jg_*mathcal FcalF). The induction hypothesis and Serre vanishing give (H^i(Y,R^jg_*mathcal F)=0calF)=0) for (ige2). In the Leray spectral sequence, only columns zero and one survive. For each (j), the group (H^1(Y,R^jg_*mathcal F)calF)) is a subspace of (H^{j+1}(V,mathcal F)calF)); after twisting by (g^*(mA)) this subspace maps to zero. Functoriality plus injectivity then makes the original subspace zero, and the case (i=1) follows.
+breaks into short exact sequences, since multiplication by the cutting equation is injective on each (R^jg_*F The induction hypothesis and Serre vanishing give (H^i(Y,R^jg_*F)=0 for (ige2). In the Leray spectral sequence, only columns zero and one survive. For each (j), the group (H^1(Y,R^jg_*F) is a subspace of (H^{j+1}(V,F); after twisting by (g^*(mA)) this subspace maps to zero. Functoriality plus injectivity then makes the original subspace zero, and the case (i=1) follows.
 
 
 ## Conversion on an Iitaka base
@@ -531,7 +531,7 @@ Proposition~prop:maximum then gives a continuous semipositive metric on (B), bou
 
 At zero exponent, pseudoeffectivity of (F_0) gives (nL'-f^*A) pseudoeffective for some (n>0); a current from (F_0) gives a singular metric on (L') with curvature dominating a positive multiple of (f^*omega_A). Mix with the smooth semipositive part using a small coefficient, Skoda integrability, and finite cover. The multiplier ideal becomes trivial, and curvature dominates a positive multiple of (f^*omega_A). For (kge0), put (C_k=L'+kf^*B) and add (kf^*phi_B) to the coefficient weights. Local boundedness keeps the multiplier ideal trivial and curvature positive. Lemma~lem:leray and the projection formula give
 [
- H^i(Y,f_*OO_V(E) ⊗ OO_Y(kB))=0qquad(i>0, kge0).
+ H^i(Y,f_*OO_V(E) ⊗ OO_Y(kB))=0 (i>0, kge0).
 ]
 At (k=0) the Euler characteristic is (h^0(V,E)=1). It is a polynomial in (k); hence it is nonzero at some positive integer (k). A section of (E+kf^*B), multiplied by (s_H^k), pushes down to a nonzero section of (kdL) on (X).
 
@@ -599,7 +599,7 @@ a distributional statement obtained by uniform convergence of (h_j) and weak (L^
 
 On the open set where the kernel condition holds, every pulled-back base covector annihilates (kertheta), so its wedge with (v) vanishes. The holomorphic bundle map
 [
- f^*Ω_Y^1toΩ_V^{n-ell+1} ⊗  A,qquadalphamapsto(df)^*alphawedge v,
+ f^*Ω_Y^1toΩ_V^{n-ell+1} ⊗  A, alpha|->(df)^*alphawedge v,
 ]
 therefore vanishes everywhere. Over the smooth fibration locus, this gives the base factor decomposition
 [
@@ -615,7 +615,7 @@ is injective.
 
 Finally apply Fujisawa's derived decomposition [cite] to the smooth coefficient (D_V):
 [
- Rf_*(K_V+D_V)simeq⊕_q R^qf_*(K_V+D_V)[-q].
+ Rf_*(K_V+D_V)~=⊕_q R^qf_*(K_V+D_V)[-q].
 ]
 Tensor by the line (Q^k) on (Y) and use the projection formula, to get
 [
@@ -649,7 +649,7 @@ Let (pi:Vto X) be birational between smooth connected projective varieties, let
 
 
 Proof. 
-Let (mathcal G=f_*calG=f_*OO_V(E)) or its invariant summand. Vanishing gives (h^0(Y,mathcal GcalG ⊗  Q^k)=chi(Y,mathcal GcalG ⊗  Q^k)) for every (kge0). The right side is a polynomial in (k) by Riemann--Roch for the coherent sheaf (mathcal GcalG). At zero it equals one: (pi_*OO_V(E)=OO_X), and the exceptional canonical section is invariant. The polynomial is nonzero at some positive integer (k). Multiply the resulting section of (E+kf^*Q) by the injection's (k)-th power; the effective exceptional divisor (E) changes neither sections nor characters, by normality of (X).
+Let (G=f_*calG=f_*OO_V(E)) or its invariant summand. Vanishing gives (h^0(Y,G ⊗  Q^k)=chi(Y,G ⊗  Q^k)) for every (kge0). The right side is a polynomial in (k) by Riemann--Roch for the coherent sheaf (G At zero it equals one: (pi_*OO_V(E)=OO_X), and the exceptional canonical section is invariant. The polynomial is nonzero at some positive integer (k). Multiply the resulting section of (E+kf^*Q) by the injection's (k)-th power; the effective exceptional divisor (E) changes neither sections nor characters, by normality of (X).
 
 
 ## Why this was a different proof
@@ -673,7 +673,7 @@ where (H_0) is ample, the torus acts trivially on (Y_0,H_0), and the defining se
 
 Maximality forces the invariant adjoint rank to be one on each generic fibre. Suppose two independent invariant sections existed. Their ratio would be transcendental over (C(Y_0)) by relative algebraic closedness of the Stein field. Multiplying by a high power of (H_0) would clear the vertical poles and combine with the original image coordinates, increasing the image dimension beyond the maximum. Consequently
 [
- dim_{C(Y_0)}H^0((X_0)_eta,E_0+kbpi_0^*L)^T=1qquad(kge0),
+ dim_{C(Y_0)}H^0((X_0)_eta,E_0+kbpi_0^*L)^T=1 (kge0),
 ]
 where (E_0=K_{X_0/X}) and (b=a_j-a_i>0) is a positive index difference.
 
@@ -694,19 +694,19 @@ The preceding proof uses Proposition~prop:maximum to build the base metric. A di
 [Pole-sheaf stabilization]
 In the setting above, with (D_V=pi^*(-K_X)), (E=K_{V/X}), and (bD_V=G+f^*Q) the divisorial normalization, the increasing sheaves
 [
- mathcal P_k=(f_*calP_k=(f_*OO_V(E+kG))^Tqquad(kge0)
+ P_k=(f_*calP_k=(f_*OO_V(E+kG))^T (kge0)
 ]
 viewed as sheaves of rational functions on (Y), stabilize. The line (Q) is nef, and (G) contains no entire fibre. These conclusions give a second proof that the fibrewise maximum is positive.
 
 
 Proof. 
-On (V), form the sheaves (mathcal P_k=(f_*calP_k=(f_*OO_V(E+kG))^T) and view them as sheaves of rational functions on (Y) by dividing by (s_Es_G^k). They have rank one, contain (OO_Y), and increase with (k). For each prime divisor (P) of (Y), take a lift (Gamma) omitted by (G); regularity of (f^*a,s_Es_G^k) bounds the possible pole order independently of (k) by one divisor (C). Normality embeds all (mathcal P_kcalP_k) in (OO_Y(C)), so Noetherianity gives a stable value (mathcal P_∞calP_∞).
+On (V), form the sheaves (P_k=(f_*calP_k=(f_*OO_V(E+kG))^T) and view them as sheaves of rational functions on (Y) by dividing by (s_Es_G^k). They have rank one, contain (OO_Y), and increase with (k). For each prime divisor (P) of (Y), take a lift (Gamma) omitted by (G); regularity of (f^*a,s_Es_G^k) bounds the possible pole order independently of (k) by one divisor (C). Normality embeds all (P_k in (OO_Y(C)), so Noetherianity gives a stable value (P_∞calP_∞).
 
 Take a very ample (H) on (Y). Apply Theorem~thm:kernel with trivial base twist to ((1+kb)D_V+lf^*H), (k,lge0). Its curvature dominates the original, so its kernel is still vertical. Projection to invariants and the projection formula give
 [
- H^i(Y,mathcal P_kcalP_k ⊗ (kQ+lH))=0qquad(i>0).
+ H^i(Y,P_k ⊗ (kQ+lH))=0 (i>0).
 ]
-For large (k), Castelnuovo--Mumford regularity makes (mathcal P_∞calP_∞ ⊗ (kQ+(dim Y)H)) globally generated. Pulling back to a curve and removing torsion gives that the fixed sheaf has positive rank, so its degree plus (k) times the rank times (deg Q) is nonnegative for unbounded (k). Thus (Q) is nef. If (G) contained an entire fibre (Y_y), blow up (y) and recompute; the new rational base divisor is the pullback of (Q) plus a strictly positive multiple of the exceptional divisor, and its degree on a line in the exceptional projective space is negative, a contradiction. Hence no entire fibre lies in (G), and the fibrewise maximum is positive.
+For large (k), Castelnuovo--Mumford regularity makes (P_∞calP_∞ ⊗ (kQ+(dim Y)H)) globally generated. Pulling back to a curve and removing torsion gives that the fixed sheaf has positive rank, so its degree plus (k) times the rank times (deg Q) is nonnegative for unbounded (k). Thus (Q) is nef. If (G) contained an entire fibre (Y_y), blow up (y) and recompute; the new rational base divisor is the pullback of (Q) plus a strictly positive multiple of the exceptional divisor, and its degree on a line in the exceptional projective space is negative, a contradiction. Hence no entire fibre lies in (G), and the fibrewise maximum is positive.
 
 
 ## What the harmonic proof did
@@ -724,17 +724,17 @@ Throughout this section, assume (H^i(X,OO_X)=0) for (i>0), and let (T) act on th
 
 A prime divisor is small if (theta^r) vanishes on its smooth locus. An effective divisor has small support exactly when its intersection with ([theta]^r[omega]^{n-r-1}) vanishes. Smallness persists under pullback to smooth birational models and through exceptional divisors centered in small supports: resolve the original prime, take a dominating component of the fiber product with the proposed source, and test the pulled-back form on a smooth dominating space; continuity extends the zero identity.
 
-Let (mathcal KcalK) consist of zero and the invariant rational functions with small pole support. It is a field. Sums and products introduce no new pole support; a rational function's zeros and poles are linearly equivalent effective divisors, so the zero support is small whenever the pole support is. It is finitely generated: take a transcendence basis of (mathcal KcalK), extend it to one of (C(X)), and note that adjoining the complementary independent variables preserves degrees of finite algebraic subextensions. The finite degree of the full function field over the larger purely transcendental field bounds those degrees.
+Let (K consist of zero and the invariant rational functions with small pole support. It is a field. Sums and products introduce no new pole support; a rational function's zeros and poles are linearly equivalent effective divisors, so the zero support is small whenever the pole support is. It is finitely generated: take a transcendence basis of (K, extend it to one of (C(X)), and note that adjoining the complementary independent variables preserves degrees of finite algebraic subextensions. The finite degree of the full function field over the larger purely transcendental field bounds those degrees.
 
 Use finitely many generators to map to a product of projective lines, take the image (Y_0), and resolve equivariantly: (pi_0:X_0to X), (f_0:X_0to Y_0). The base action is trivial. Indeterminacy lies in the small pole supports, so the resolution may be taken to be an isomorphism off them. The exceptional canonical divisor (E_0) is small, and a pullback of an ample (H_0) on (Y_0) has a small effective representative.
 
-The choice of field now controls the horizontal divisors. There are only finitely many invariant small primes on (X_0) horizontal over (Y_0), because (Pic^0(X_0)=0) by birational invariance, the kernel of the linearized Picard group over the ordinary one is characters, and infinitely many such primes would give a nonzero integral relation among their natural linearized divisor lines. That relation would be an invariant rational function on (X_0) with vertical divisor, but its poles downstairs are small, so it belongs to (mathcal K=calK=C(Y_0)), a contradiction.
+The choice of field now controls the horizontal divisors. There are only finitely many invariant small primes on (X_0) horizontal over (Y_0), because (Pic^0(X_0)=0) by birational invariance, the kernel of the linearized Picard group over the ordinary one is characters, and infinitely many such primes would give a nonzero integral relation among their natural linearized divisor lines. That relation would be an invariant rational function on (X_0) with vertical divisor, but its poles downstairs are small, so it belongs to (K=, a contradiction.
 
 The determinant sections are small, so their horizontal supports belong to this finite set. Order the multiplicities and take a ratio along the unbounded subsequence to obtain an invariant rational section (s) of (bpi_0^*L), (b>0), regular on the generic fiber. For every (kge0) both
 [
  (f_{0*}OO_{X_0}(kbpi_0^*L))^T, (f_{0*}OO_{X_0}(E_0+kbpi_0^*L))^T
 ]
-have rank one, because they contain (s^k) and (s_{E_0}s^k), and any other invariant generic section has small effective zero divisor and a small-pole ratio in (mathcal KcalK).
+have rank one, because they contain (s^k) and (s_{E_0}s^k), and any other invariant generic section has small effective zero divisor and a small-pole ratio in (K
 
 The kernel condition follows from the same smallness. Every level divisor of a generating function is small; at a regular level point in the rank-(r) open its tangent hyperplane must contain (kertheta), because a hyperplane not containing that kernel would surject onto the rank-(r) quotient. Thus (kertheta⊆ker df_0) on a nonempty ordinary open. Proposition~prop:maximum and Proposition~prop:euler finish.
 
@@ -784,7 +784,7 @@ The source (S_0) is rationally connected, being a smooth rational image of (S),
 [
  v_min H^0(S,Ω_S^{dim S-q} ⊗ (m+1)L)=H^0(S,textstyle∧^qT_S ⊗  mL)
 ]
-of weight (mw). For (q=0) they are the desired sections. Otherwise saturate the generic span in (∧^qT_S) and call it (mathcal UcalU) of rank (e>0); the determinant extraction gives nonzero eigensections (d_iin H^0(S,detmathcal U+k_iL)calU+k_iL)) of weight (k_iw), (k_ito∞). Cotangent-tensor theorem [cite] makes (eL-c_1(mathcal U)calU)) pseudoeffective and ((e+k_i)L-D_i) pseudoeffective.
+of weight (mw). For (q=0) they are the desired sections. Otherwise saturate the generic span in (∧^qT_S) and call it (U of rank (e>0); the determinant extraction gives nonzero eigensections (d_iin H^0(S,detU+k_iL)calU+k_iL)) of weight (k_iw), (k_ito∞). Cotangent-tensor theorem [cite] makes (eL-c_1(U) pseudoeffective and ((e+k_i)L-D_i) pseudoeffective.
 
 Let a divisor class be dominated if a positive multiple of (L) minus it is pseudoeffective. Maximize the image dimension of linear systems of dominated effective integral divisors; choose a full system (|P_0|) realizing this maximum. Rational connectedness gives (Pic^0(S)=0), so the system linearizes. Resolve its graph, take Stein factorization, flatten by the Hilbert-family construction, and resolve the base. The result is a normal equidimensional model (Wto Y) with a smooth resolution (V), and the line (L_W) pulled back from (S) with the exceptional canonical divisor (E). The whole effective canonical discrepancy is dominated by (L_W).
 
@@ -814,7 +814,7 @@ In words.:  All previous ordinary conversions followed (L) by constructing a bas
 Let (X) be smooth connected projective with smoothly semipositive (L=-K_X). Suppose that for one fixed (qge0), (H^q(X,mL)ne0) for unbounded positive integers (m). Then (H^0(X,rL)ne0) for some (r>0).
 
 
-Construction overview.:  The proof has three parts. First, the harmonic-star construction of Lemma~lem:forms turns the cohomology into nonzero invariant forms of type (Ω_X^{n-q} ⊗ (m+1)L), and determinant extraction gives invariant sections (s_i) of (M+a_iL) with (a_ito∞). Each (s_i) has zero divisor null along the semipositive curvature. Second, the field (mathcal KcalK) of rational functions constant in curvature-null directions is relatively algebraically closed and finitely generated; its projective model (Y) has (dim Yle r), and the alteration (f:X_1to Y) has reduced flat fibres. Third, a Hom line on this alteration carries a continuous semipositive metric; after a finite Galois descent an injection survives to the original source and base fields, and Proposition~prop:euler finishes.
+Construction overview.:  The proof has three parts. First, the harmonic-star construction of Lemma~lem:forms turns the cohomology into nonzero invariant forms of type (Ω_X^{n-q} ⊗ (m+1)L), and determinant extraction gives invariant sections (s_i) of (M+a_iL) with (a_ito∞). Each (s_i) has zero divisor null along the semipositive curvature. Second, the field (K of rational functions constant in curvature-null directions is relatively algebraically closed and finitely generated; its projective model (Y) has (dim Yle r), and the alteration (f:X_1to Y) has reduced flat fibres. Third, a Hom line on this alteration carries a continuous semipositive metric; after a finite Galois descent an injection survives to the original source and base fields, and Proposition~prop:euler finishes.
 
 ## The field constant in curvature-null directions
 
@@ -826,27 +826,27 @@ The intersection equality is Lemma~lem:null-divisors. If (r=0), all (D_i) vanish
 
 On the ordinary open (U) of maximal curvature rank, define
 [
- mathcal K=calK={ginC(X): dg(kertheta)=0 on U off a proper algebraic subset}.
+ K={ginC(X): dg(kertheta)=0 on U off a proper algebraic subset}.
 ]
-It is a field. It is relatively algebraically closed: differentiate a minimal polynomial for an element algebraic over (mathcal KcalK) in a null direction; its separable derivative is generically nonzero, so the element has zero derivative there as well. The intermediate-field argument from Section~sec:fields proves finite generation.
+It is a field. It is relatively algebraically closed: differentiate a minimal polynomial for an element algebraic over (K in a null direction; its separable derivative is generically nonzero, so the element has zero derivative there as well. The intermediate-field argument from Section~sec:fields proves finite generation.
 
 If a rational function has null pole support, its level divisors have the same zero intersection as its poles, hence each of their components is null. At a regular level point in (U), the tangent hyperplane contains (kertheta) by the rank-quotient argument. So this proves the assertion where the differential is nonzero and then everywhere.
 
 ## The reduced flat alteration and the base line
 
-Choose a smooth projective model (Y) of (mathcal KcalK) and resolve the map by (mu:X_1to X), (f:X_1to Y). Put (A=mu^*L). The generic fiber is geometrically integral, because the extension is regular in characteristic zero. On a nonempty ordinary open (kermu^*thetasubsetker df), and (dim Yle r). For every smooth ((1,1))-form (eta) on (Y),
+Choose a smooth projective model (Y) of (K and resolve the map by (mu:X_1to X), (f:X_1to Y). Put (A=mu^*L). The generic fiber is geometrically integral, because the extension is regular in characteristic zero. On a nonempty ordinary open (kermu^*thetasubsetker df), and (dim Yle r). For every smooth ((1,1))-form (eta) on (Y),
 [
  (mu^*theta)^rwedge f^*eta=0.
 ]
 On the rank-(r) open this follows because every horizontal covector lies in the positive-curvature span; elsewhere the rank is smaller, and continuity extends across the algebraic exceptional locus.
 
-Any integral relation (sum b_i=sum b_ia_i=0) makes (sum b_iD_i) principal with null pole support. Its function is in (mathcal KcalK), so the horizontal parts of (mu^*D_i) satisfy the same relation. Three-term interpolation proves that the horizontal divisors are affine in (a_i). Their slope is effective because the coefficients are nonnegative and unbounded. For two indices with (b=a_2-a_1>0) the quotient (s) is a rational section of (bA) with effective horizontal divisor, regular over a dense base open.
+Any integral relation (sum b_i=sum b_ia_i=0) makes (sum b_iD_i) principal with null pole support. Its function is in (K, so the horizontal parts of (mu^*D_i) satisfy the same relation. Three-term interpolation proves that the horizontal divisors are affine in (a_i). Their slope is effective because the coefficients are nonnegative and unbounded. For two indices with (b=a_2-a_1>0) the quotient (s) is a rational section of (bA) with effective horizontal divisor, regular over a dense base open.
 
 Let (E=K_{X_1/X}=K_{X_1}+A). For every (tge0),
 [
  rank f_*OO_{X_1}(E+tA)le1,quadwith equality for t=kb, kge0.
 ]
-Upper bound: two hypothetical independent generic sections could be twisted to make them global, and their zero divisors push to effective divisors on (X) with zero intersection with ([theta]^r[omega]^{n-r-1}), using ((mu^*theta)^rwedge f^*eta=0), exceptionality of (E), and (theta^{r+1}=0). Their quotient has null poles, hence lies in (mathcal KcalK), contrary to independence. Lower bound: (s_Es^k).
+Upper bound: two hypothetical independent generic sections could be twisted to make them global, and their zero divisors push to effective divisors on (X) with zero intersection with ([theta]^r[omega]^{n-r-1}), using ((mu^*theta)^rwedge f^*eta=0), exceptionality of (E), and (theta^{r+1}=0). Their quotient has null poles, hence lies in (K, contrary to independence. Lower bound: (s_Es^k).
 
 Over a smooth dense open with geometrically integral fibres, (-logmax_{X_{1,y}}|s|^2) is plurisubharmonic: the rank-one adjoint section ((s_E/f^*eta)s^k) has smooth coefficient ((kb+1)A), so Berndtsson's theorem applies, and the moment limit converges to the fibrewise maximum.
 
@@ -854,13 +854,13 @@ Over a smooth dense open with geometrically integral fibres, (-logmax_{X_{1,y}}|
 
 By weak semistable reduction cite[Theorem~0.3]{AK2000} the fibration may be replaced by an alteration (Y_ato Y) and a modification (Z_a) of the main component of (X_1times_Y Y_a) such that (p_a:Z_ato Y_a) is equidimensional, toroidal, and has reduced fibres. Toroidal varieties are Cohen--Macaulay, so the equidimensional flatness criterion over the smooth base makes (p_a) flat.
 
-Take a sufficiently high power (mathcal PcalP) of a relatively very ample line. Relative Serre vanishing and base change give locally free bundles (mathcal H=p_calH=p_{a*}mathcal PcalP) and (mathcal H'=p_calH'=p_{a*}(mathcal PcalP ⊗  q_a^*A^b)), with surjective evaluation for (mathcal PcalP). On the generic fibre, multiplication by (s) gives a nonzero map (mathcal H_calH_etatomathcal H'_calH'_eta). Resolve its rational point in the projective bundle of lines in (Hom(mathcal H,calH,mathcal H')calH')) on a smooth projective birational base model (Y_bto Y_a). Let (B_b) be the pulled-back tautological subline, and (Z_b=Z_atimes_{Y_a}Y_b), with maps (p_b,q_b). The total space is integral.
+Take a sufficiently high power (P of a relatively very ample line. Relative Serre vanishing and base change give locally free bundles (H=p_{a*}P and (H'=p_calH'=p_{a*}(P ⊗  q_a^*A^b)), with surjective evaluation for (P On the generic fibre, multiplication by (s) gives a nonzero map (H_ H'_calH'_eta). Resolve its rational point in the projective bundle of lines in (Hom(H,H')calH')) on a smooth projective birational base model (Y_bto Y_a). Let (B_b) be the pulled-back tautological subline, and (Z_b=Z_atimes_{Y_a}Y_b), with maps (p_b,q_b). The total space is integral.
 
 There is a regular line map
 [
  p_b^*B_bto q_b^*A^b
 ]
-nonzero on every fibre, because composing the tautological Hom map with evaluation into (mathcal PcalP ⊗  q_b^*A^b) annihilates the kernel of evaluation, and base change plus reducedness gives a nonzero value at a point of every fibre. Without reducedness this implication would fail.
+nonzero on every fibre, because composing the tautological Hom map with evaluation into (P ⊗  q_b^*A^b) annihilates the kernel of evaluation, and base change plus reducedness gives a nonzero value at a point of every fibre. Without reducedness this implication would fail.
 
 Give (B_b) the maximum metric through the map and the pulled-back smooth metric of (A^b). The maximum is positive and continuous; properness gives upper semicontinuity, fibre-smoothness at dense smooth points gives lower semicontinuity. On a dense open the map is the pullback of (s) times a rational base factor, and its maximum weight equals the pulled-back good-locus weight plus a pluriharmonic frame term. Continuity and removable singularities extend plurisubharmonicity to (Y_b).
 
@@ -890,20 +890,20 @@ The three ingredients are: a localised multiplier-ideal vanishing theorem, a str
 [Localised multiplier-ideal vanishing]
 Let (Y) be a smooth connected projective variety of dimension (d>0). Suppose (R) has a singular Hermitian metric with plurisubharmonic weights (psi), continuous and dominating a positive smooth ((1,1))-form on some coordinate ball. Then
 [
- H^q(Y,OO_Y(K_Y+R) ⊗ mathcal I(calI(psi))=0qquad(q>0).
+ H^q(Y,OO_Y(K_Y+R) ⊗ I( (q>0).
 ]
 
 
 Proof. 
-Construct a globally strictly positive metric with the same multiplier ideal. For sufficiently small (delta>0) and large (l), a section of (lR-A) in (mathcal I(l(1+calI(l(1+delta)psi)) exists: on (lR) use weights (l(1+delta)psi-ldeltapsi_0), add (j_lA-K_Y) with (j_l=lceil Cdelta l+C'rceil), and add (eta l) times a logarithmic pole with (eta) small, chosen independently of (delta,l). Nadel vanishing in the strict-current form cite[Theorem 5.11]{Demailly2011} then gives surjectivity to jets of order (lflooreta lrfloor-d) at the chosen point. Continuity of (psi) there makes (mathcal I_lcalI_l) trivial near the point, and polar-coordinate integration gives the ideal (mathfrak m^frakm^{lflooreta lrfloor-d+1}) for the pole. So (h^0(Y,(lR+j_lA) ⊗ mathcal I_l)calI_l)ge cl^d) for some (c>0).
+Construct a globally strictly positive metric with the same multiplier ideal. For sufficiently small (delta>0) and large (l), a section of (lR-A) in (I(l(1+calI(l(1+delta)psi)) exists: on (lR) use weights (l(1+delta)psi-ldeltapsi_0), add (j_lA-K_Y) with (j_l=lceil Cdelta l+C'rceil), and add (eta l) times a logarithmic pole with (eta) small, chosen independently of (delta,l). Nadel vanishing in the strict-current form cite[Theorem 5.11]{Demailly2011} then gives surjectivity to jets of order (floor(eta l)-d) at the chosen point. Continuity of (psi) there makes (I_l trivial near the point, and polar-coordinate integration gives the ideal (mathfrak m^frakm^{floor(eta l)-d+1}) for the pole. So (h^0(Y,(lR+j_lA) ⊗ I_l) cl^d) for some (c>0).
 
-Remove (j_l+1) copies of a very ample (A). At each step choose a smooth member of (|A|) containing no associated centre of (OO_Y/mathcal I_lcalI_l); division by its equation preserves membership in the ideal because the equation is a nonzerodivisor. The loss at a step is at most (C''l^{d-1}) uniformly for (0<deltale1), since (pA-R) can be assumed globally generated for some fixed (p), and Hilbert-polynomial bounds on the restriction control the drop. The total loss is (O(delta l^d)+O(l^{d-1})). Taking (delta) small and then (l) large proves the initial section existence.
+Remove (j_l+1) copies of a very ample (A). At each step choose a smooth member of (|A|) containing no associated centre of (OO_Y/I_l; division by its equation preserves membership in the ideal because the equation is a nonzerodivisor. The loss at a step is at most (C''l^{d-1}) uniformly for (0<deltale1), since (pA-R) can be assumed globally generated for some fixed (p), and Hilbert-polynomial bounds on the restriction control the drop. The total loss is (O(delta l^d)+O(l^{d-1})). Taking (delta) small and then (l) large proves the initial section existence.
 
 Let (sigma) be a nonzero section of the resulting line. Its divisor metric (psi') on (R) satisfies (i∂barbar∂∂psi'geomega_A/l), and the section's membership in the multiplier ideal says exactly that (exp(l(psi'-(1+delta)psi))) is locally integrable. Take (lge1/delta), set (rho=(1-ε)psi+εpsi'), and rewrite
 [
  -psi=-(rho)/(1+εdelta)+(ε V)/(1+εdelta).
 ]
-H"older with the integrability of (exp(V/delta)) gives (mathcal I(calI(rho)⊆mathcal I(calI(psi)). For the converse, strong openness [cite] and compactness give (gamma>0) with (mathcal I((1+calI((1+gamma)psi)=mathcal I(calI(psi)) on a finite cover, and small-exponent integrability of the divisor metric gives (gamma'>0) with (exp(-gamma'psi')) locally integrable. H"older with (s=(1-ε)/(1+gamma)) and (1-s), choosing (ε) small so that (ε/(1-s)<gamma'), proves (mathcal I(calI(psi)⊆mathcal I(calI(rho)). The metric (rho) has a global positive curvature lower bound and the same ideal, and Nadel vanishing finishes.
+H"older with the integrability of (exp(V/delta)) gives (I(⊆I( For the converse, strong openness [cite] and compactness give (gamma>0) with (I((1+calI((1+gamma)psi)=I( on a finite cover, and small-exponent integrability of the divisor metric gives (gamma'>0) with (exp(-gamma'psi')) locally integrable. H"older with (s=(1-ε)/(1+gamma)) and (1-s), choosing (ε) small so that (ε/(1-s)<gamma'), proves (I(⊆I( The metric (rho) has a global positive curvature lower bound and the same ideal, and Nadel vanishing finishes.
 
 
 ## Strictness on the invariant quotient
@@ -925,19 +925,19 @@ The second term is nonnegative by the primitive middle-degree Hodge--Riemann sig
 ## The exact ideal of the invariant adjoint line
 
 [Exact ideal]
-Let (X,V,Y) be smooth connected projective, (dim Y>0), (pi:Vto X) birational, (f:Vto Y) surjective, and an algebraic torus acting equivariantly on the maps and trivially on (Y). Give (L=-K_X) its natural linearization and an invariant smooth semipositive metric. Put (L'=pi^*L), (J=K_{V/X}), with (t) its natural invariant Jacobian section. Assume (mathcal B=(f_*calB=(f_*OO_V(J))^T) has rank one. Write (mathcal B^calB^{veevee}=OO_Y(B_0)) and (R=B_0-K_Y). Then the invariant adjoint (L^2)-metric extends to a singular semipositive metric (psi_R) on (R), and
+Let (X,V,Y) be smooth connected projective, (dim Y>0), (pi:Vto X) birational, (f:Vto Y) surjective, and an algebraic torus acting equivariantly on the maps and trivially on (Y). Give (L=-K_X) its natural linearization and an invariant smooth semipositive metric. Put (L'=pi^*L), (J=K_{V/X}), with (t) its natural invariant Jacobian section. Assume (B=(f_*calB=(f_*OO_V(J))^T) has rank one. Write (B^{veevee}=OO_Y(B_0)) and (R=B_0-K_Y). Then the invariant adjoint (L^2)-metric extends to a singular semipositive metric (psi_R) on (R), and
 [
- OO_Y(B_0) ⊗ mathcal I(calI(psi_R)=mathcal B.calB.
+ OO_Y(B_0) ⊗ I( B.calB.
 ]
 If on a nonempty patch in the smooth base-change locus, with (t) nonzero, the curvature of (L') dominates the pullback of a positive base form, then (psi_R) is smooth and strictly positive on a base coordinate ball.
 
 
 Proof. 
-The candidate for the divisor (B_0) is determined by regularity over the generic point of a base prime: (b_P=min_{Qmapsto P}lfloorord_Q(t)/ord_Q(f^*P)rfloor). This is the regularity test for ((f^*a)t), and on the smooth base-change locus the invariant summand is a line whose fibre-integral metric is semipositive by the compact-torus orthogonal-quotient argument.
+The candidate for the divisor (B_0) is determined by regularity over the generic point of a base prime: (b_P=min_{Q|-> P}floor(ord_Q(t)/ord_Q(f^*P))). This is the regularity test for ((f^*a)t), and on the smooth base-change locus the invariant summand is a line whose fibre-integral metric is semipositive by the compact-torus orthogonal-quotient argument.
 
 The metric extends across every base prime. At a general point of a component attaining (b_P), write the map as ((z_1^e,z_2,...,z_k)), where (e>0), and let (a) be (ord(t)). The relative canonical coefficient is a nonvanishing smooth factor times (z_1^{a-b_Pe-(e-1)}), whose exponent is nonpositive because (0le a-b_Pe<e). Integration on a small vertical polydisk bounds the squared norm below. So the weight is locally bounded above and extends by first crossing the divisorial neighbourhoods and then codimension at least two.
 
-To identify the ideal with (mathcal BcalB), choose local frames (g) of (OO_Y(B_0)) and (eta) of (K_Y). The frame (g/eta) corresponds on the smooth locus to ((f^*g)t/f^*eta). For a holomorphic function (a), Fubini identifies integrability of (a) with the squared norm of this frame, using the base volume from (eta), with square integrability of the absolute top form ((f^*(ag))t) with the smooth (L')-metric upstairs. The excluded analytic sets have measure zero. An integrable meromorphic top form has no divisorial pole, and smoothness upstairs extends it across codimension two. Conversely a holomorphic lift is integrable by properness. Both inclusions follow; the codimension-two difference between (mathcal BcalB) and its hull is preserved.
+To identify the ideal with (B, choose local frames (g) of (OO_Y(B_0)) and (eta) of (K_Y). The frame (g/eta) corresponds on the smooth locus to ((f^*g)t/f^*eta). For a holomorphic function (a), Fubini identifies integrability of (a) with the squared norm of this frame, using the base volume from (eta), with square integrability of the absolute top form ((f^*(ag))t) with the smooth (L')-metric upstairs. The excluded analytic sets have measure zero. An integrable meromorphic top form has no divisorial pole, and smoothness upstairs extends it across codimension two. Conversely a holomorphic lift is integrable by properness. Both inclusions follow; the codimension-two difference between (B and its hull is preserved.
 
 The strictness assertion follows by applying Lemma~lem:strict to the invariant section determined by (t) on the patch. The fibre-integral metric is smooth on the base-change open, so its strictness at one point gives a positive curvature lower bound on a smaller coordinate ball.
 
@@ -957,12 +957,7 @@ The field (K) of invariant rational functions with zero-degree polar divisors is
 
 The invariant adjoint rank is one for (mge0) (Section~sec:fields). On the maximal-rank open, the differential of every rational function in (K) kills (keralpha); the differential semipositive linear algebra gives (pi^*alphage cf^*omega_Y) on a horizontal patch. The rank (k) of (Y) satisfies (kle r).
 
-Proposition~prop:maximum builds a continuous semipositive metric on (B). Proposition~prop:exact-adjoint-ideal at (m=0) gives (R=B_0-K_Y) with strictly positive curvature on a base ball, and the exact identity (OO_Y(B_0) ⊗ mathcal I(calI(psi_R)=mathcal BcalB). Adding (m) times the bounded metric on (B) preserves the ideal and the strictness. Lemma~lem:local-nadel gives (H^q(Y,mathcal BcalB ⊗ OO_Y(mB))=0) for (q>0), (mge0). At zero (h^0(Y,mathcal B)=h^0(V,calB)=h^0(V,OO_V(J))^T=1), so the Euler polynomial has value one. For some (m>0) lift a nonzero section, multiply by (s_D^m), and push down removing the exceptional Jacobian divisor.
-
-
-## What this route changed
-
-The proof of Proposition~prop:degree-zero-route falls into the same seven-step template as Theorem~thm:transfer, but replaces the singular positive base metric with the exact multiplier-ideal identification of (mathcal BcalB). The result is not a new theorem; it is a different way of seeing why the fixed multiplier ideal in the transfer proof can be dropped from the input and recovered from the direct image at degree zero.
+Proposition~prop:maximum builds a continuous semipositive metric on (B). Proposition~prop:exact-adjoint-ideal at (m=0) gives (R=B_0-K_Y) with strictly positive curvature on a base ball, and the exact identity (OO_Y(B_0) ⊗ I( BcalB). Adding (m) times the bounded metric on (B) preserves the ideal and the strictness. Lemma~lem:local-nadel gives (H^q(Y,B ⊗ OO_Y(mB))=0) for (q>0), (mge0). At zero (h^0(Y,B)=h^0(V,OO_V(J))^T=1), so the Euler polynomial has value one. For some (m>0) lift a nonzero section, multiply by (s_D^m), and push down removing the exceptional Jacobian divisor.
 
 
 # Stabilized orders on the actual direct images
@@ -978,31 +973,31 @@ Let (X) be smooth connected projective and let an algebraic torus (T) act on (X)
 Proof. 
 Generic adjoint rigidity. Fix (j_0<j_1) in (J) and choose (k>0) so that the invariant system (H^0(X,kD(j_1))^T) has maximal image dimension; write (D(j)=M+jL). Equivariant principalisation of its base ideal gives a smooth projective (pi_0:W_0to X) isomorphic off the base locus, and a connected-fibre Stein map (f_0:W_0to Y_0) such that (pi_0^*kD(j_1)=f_0^*H+G), with (H) ample and (G) effective, both fixed by the connected torus. The Stein base is trivially acted on. Maximality of image dimension forces
 [
- dim_K H^0(F,hpi_0^*D(j_1)|_F)^T=1qquad(h>0)
+ dim_K H^0(F,hpi_0^*D(j_1)|_F)^T=1 (h>0)
 ]
 for the generic fibre (F) over (K=C(Y_0)). Products of the invariant sections of the same linearized degree are proportional; hence the fibre divisor (D_{j_1}^F-D_{j_0}^F) is effective. So (z=sigma_{j_1}/sigma_{j_0}) is a rational invariant section of (dL), (d=j_1-j_0>0), regular on (F).
 
 For every fixed (sge0), pick (h) large enough that (hD_{j_1}^Fgediv(e_0|_F)+sdiv(z|_F)), where (e_0) is the Jacobian section of (K_{W_0}+pi_0^*L). The invariant quotient ((sigma_{j_1}|_F)^h/(e_0z^s|_F)) is regular, so multiplication by it injects the whole invariant adjoint space into a one-dimensional one:
 [
- H^0(F,(K_{W_0}+(1+sd)pi_0^*L)|_F)^T=Kcdot(e_0z^s)|_F.
+ H^0(F,(K_{W_0}+(1+sd)pi_0^*L)|_F)^T=K·(e_0z^s)|_F.
 ]
 
 Actual lines. Apply birational weak toroidalization cite[Theorem 1.1]{ADK2013}, then equidimensionalize by toroidal subdivisions cite[Proposition 4.4]{AK2000}, obtaining a normal toroidal (Zto Y) equidimensional and birational over (W_0). Take a smooth resolution (Wto Z), with (f:Wto Y), (pi:Wto X). Toroidal singularities are rational, so
 [
- mathscr E_s=f_*scrE_s=f_*OO_W(K_{W/Y}+(1+sd)pi^*L)
+ E_s=f_*scrE_s=f_*OO_W(K_{W/Y}+(1+sd)pi^*L)
 ]
-satisfies (mathscr E_sscrE_s ⊗ OO_Y(K_Y)=g_*(omega_Z ⊗ pi_Z^*OO_X((1+sd)L))), and is reflexive, since equidimensionality pushes codimension two down to codimension two and the reflexive canonical sheaf extends sections. A common smooth resolution of (W) and an equivariant model identifies the canonical direct images, and the invariant part
+satisfies (E_s ⊗ OO_Y(K_Y)=g_*(omega_Z ⊗ pi_Z^*OO_X((1+sd)L))), and is reflexive, since equidimensionality pushes codimension two down to codimension two and the reflexive canonical sheaf extends sections. A common smooth resolution of (W) and an equivariant model identifies the canonical direct images, and the invariant part
 [
- A_s=(mathscr E_s)^TscrE_s)^T
+ A_s=(E_s)^T
 ]
 is a genuine rank-one reflexive summand, hence a line bundle on smooth (Y). Birational canonical pushforward gives
 [
  H^0(Y,K_Y+A_s)=H^0(X,sdL)^T, h^0(Y,K_Y+A_0)=1.
 ]
 
-Affine boundary orders. Use the rational generator (e_Zz^s) of (K_Y+A_s) on the regular locus of (Z). For a base prime (Bsubset Y) with primes (Q_imapsto B), put (b_i=ord_{Q_i}(g^*B)>0), (e_i=ord_{Q_i}(E_Z)), (c_i=ord_{Q_i}(z)). The allowed pole order is
+Affine boundary orders. Use the rational generator (e_Zz^s) of (K_Y+A_s) on the regular locus of (Z). For a base prime (Bsubset Y) with primes (Q_i|-> B), put (b_i=ord_{Q_i}(g^*B)>0), (e_i=ord_{Q_i}(E_Z)), (c_i=ord_{Q_i}(z)). The allowed pole order is
 [
- u_s(B)=leftlfloormin_i(e_i+sc_i)/(b_i)rightrfloor, K_Y+A_s=OO_Y(C_s), C_s=sum_Bu_s(B)B.
+ u_s(B)=leftfloor(min_i(e_i+sc_i)/(b_i)right), K_Y+A_s=OO_Y(C_s), C_s=sum_Bu_s(B)B.
 ]
 Only finitely many primes occur with a nonzero slope or intercept. Replace (d,z) by a common multiple and power so that (v_B=min_i(c_i/b_i)inZ), and put (V=sum_Bv_BB). Then for all sufficiently large integer (s),
 [
@@ -1016,7 +1011,7 @@ for all large (s). This is the inclusion that lets us interpolate from exponent
 
 Integrable metrics and interpolation. The original moving part and (-M) pseudoeffective give that (pi^*L-ε f^*H_1) is pseudoeffective for some ample (H_1). Mix a metric from this with the smooth semipositive metric on (pi^*L): the result at (s=0) is singular with trivial multiplier ideal and curvature (gedelta f^*omega_Y), (delta>0). For (s>0) use the smooth semipositive metrics on ((1+sd)pi^*L).
 
-The singular adjoint metric and its extension cite[Theorems 3.3.4--3.3.5]{PT2018} make (A_s) carry a quotient metric (psi_s) on its locally free locus, which extends plurisubharmonically as a line bundle. Crucially, the local frames of (A_s) lift to holomorphic adjoint sections of (mathscr E_sscrE_s), so their squared norms bound the quotient norms, giving (e^{-psi_s}in L^1_{rm loc}(Y)). This is exactly what a frame of a reflexive hull would miss.
+The singular adjoint metric and its extension cite[Theorems 3.3.4--3.3.5]{PT2018} make (A_s) carry a quotient metric (psi_s) on its locally free locus, which extends plurisubharmonically as a line bundle. Crucially, the local frames of (A_s) lift to holomorphic adjoint sections of (E_s, so their squared norms bound the quotient norms, giving (e^{-psi_s}in L^1_{rm loc}(Y)). This is exactly what a frame of a reflexive hull would miss.
 
 The weight (psi_0) is strictly positive in curvature on all of (Y). To see it across the boundary, choose a smooth (p) with Hessian dominating (omega_Y) near a point and bounded above globally, multiply the upstairs (L^2)-metric by (e^{t f^*p}) for small (t>0), and use the extension theorem. A finite cover gives (i∂barbar∂∂psi_0gedelta'omega_Y) for some (delta'>0).
 
@@ -1024,7 +1019,7 @@ Now give (A_0+sV) a strict positive metric with trivial multiplier ideal for eve
 [
  rho_s=psi_0+(s)/(u)(psi_u-psi_∞).
 ]
-Its curvature is strictly positive for (s/u) small. Strong openness and compactness give (eta>0) with (e^{-(1+eta)psi_0}) locally integrable on a finite cover. H"older, together with (L^1_{rm loc})-integrability of (e^{-psi_u}), proves (e^{-rho_s}in L^1_{rm loc}) if ((s)/(u)cdot(1+eta)/(eta)le1). No exponent improvement uniform in (u) is needed.
+Its curvature is strictly positive for (s/u) small. Strong openness and compactness give (eta>0) with (e^{-(1+eta)psi_0}) locally integrable on a finite cover. H"older, together with (L^1_{rm loc})-integrability of (e^{-psi_u}), proves (e^{-rho_s}in L^1_{rm loc}) if ((s)/(u)·(1+eta)/(eta)le1). No exponent improvement uniform in (u) is needed.
 
 Nadel vanishing gives (H^i(Y,K_Y+A_0+sV)=0) for (i>0), (sge0). The Euler polynomial at zero is one. So it is nonzero for large integer (s); the inclusion (A_0+sV↪ A_s) and the identification at zero give a nonzero invariant section of (sdL).
 
@@ -1119,20 +1114,20 @@ Symbol & Meaning & Section
 (s_E, 1_E) & Jacobian section of (E), its trivialization & sec:models 
 (B, kB) & Integral rational base divisor and its clearing exponent & sec:models 
 (Q) (on base) & Base line with continuous semipositive metric & sec:models 
-(mathcal I(calI(psi)) & Multiplier ideal of weight (psi) & sec:local-ideal 
+(I( & Multiplier ideal of weight (psi) & sec:local-ideal 
 (psi_R) & Semipositive metric on (R=B_0-K_Y) from the direct image & sec:local-ideal 
-(mathcal BcalB) & Rank-one direct image ((f_*OO_V(J))^T) & sec:local-ideal 
+(B & Rank-one direct image ((f_*OO_V(J))^T) & sec:local-ideal 
 (u_s(B)) & Allowed pole order in base prime (B) at exponent (s) & sec:stabilized 
 (v_B) & Slope of (u_s(B)) in (s) & sec:stabilized 
 (A_s) & Actual direct-image line (K_Y+A_s) & sec:stabilized 
 (C_s) & Effective divisor with (K_Y+A_s=OO_Y(C_s)) & sec:stabilized 
-(mathcal KcalK) & Curvature-null field of rational functions & sec:fields, sec:alteration 
+(K & Curvature-null field of rational functions & sec:fields, sec:alteration 
 (S) (as variety) & Smooth projective rationally connected with (L=-K_S) & sec:extremal 
 (S_0) (as source) & Source component of a (C^*)-action & sec:extremal 
 (mu) & Full (T)-character of (U|_S), a source weight & sec:extremal 
 (gamma_C) & Fiber character of (L) on a fixed component (C) & sec:extremal 
-(mathcal PcalP) & Relatively very ample line on an alteration & sec:alteration 
-(mathcal H,calH, mathcal H'calH') & Direct images (p_{a*}mathcal PcalP) and (p_{a*}(mathcal PcalP ⊗  q_a^*A^b)) & sec:alteration 
+(P & Relatively very ample line on an alteration & sec:alteration 
+(H, H'calH') & Direct images (p_{a*}P and (p_{a*}(P ⊗  q_a^*A^b)) & sec:alteration 
 (B_b) & Tautological subline on the birational base (Y_b) & sec:alteration 
 (widehatwidehat X,X, widehatwidehat YY) & Resolved models after Galois base change & sec:alteration 
 (W, G, R) & Galois cover, its group, and the finite pullback to (widehatwidehat XX) & sec:alteration 
@@ -1145,9 +1140,9 @@ A note on notation. Tensor products of line bundles are written additively: (A+B
 
 We write (sim), (sim_{Q}), and (equiv) for linear, rational linear, and numerical equivalence. A local frame of squared norm (e^{-phi}) has curvature (i∂barbar∂∂phi); factors of (2pi) are included when viewing curvature as a first Chern form. A continuous semipositive metric has continuous plurisubharmonic weights; a locally bounded metric has both upper and lower bounds on compact charts and need not be continuous.
 
-The action on sections is ((gcdot s)(gx)=g(s(x))). Algebraic-torus invariants agree with invariants under the maximal compact torus: averaging a section over the compact torus is a holomorphic idempotent, hence a projection onto invariants, and it is the identity on the algebraic invariants. Averaging smooth metric weights and K"ahler forms over the compact torus preserves semipositivity. This compactness is used only through averaging, but it is essential: on a non-compact torus there is no averaging, and algebraically invariant sections of (mL) need not be the same as sections invariant for a shifted linearization (P). That distinction is precisely the content of Theorem~thm:transfer.
+The action on sections is ((g· s)(gx)=g(s(x))). Algebraic-torus invariants agree with invariants under the maximal compact torus: averaging a section over the compact torus is a holomorphic idempotent, hence a projection onto invariants, and it is the identity on the algebraic invariants. Averaging smooth metric weights and K"ahler forms over the compact torus preserves semipositivity. This compactness is used only through averaging, but it is essential: on a non-compact torus there is no averaging, and algebraically invariant sections of (mL) need not be the same as sections invariant for a shifted linearization (P). That distinction is precisely the content of Theorem~thm:transfer.
 
-For readers comparing this edition with the algebraic original, three translations are worth spelling out. The ratio field (Q_i) of Section~sec:transfer replaces the algebraic ratio field of the classical transfer proof. The singular positive metric coming from (F_0) replaces the effective (Q)-divisor representing (-M). The multiplier ideal (mathcal JcalJ) of the transfer proof is the exact analogue of the base ideal in the algebraicity-based proofs. Each pair is the analytic and the algebraic version of the same step.
+For readers comparing this edition with the algebraic original, three translations are worth spelling out. The ratio field (Q_i) of Section~sec:transfer replaces the algebraic ratio field of the classical transfer proof. The singular positive metric coming from (F_0) replaces the effective (Q)-divisor representing (-M). The multiplier ideal (J of the transfer proof is the exact analogue of the base ideal in the algebraicity-based proofs. Each pair is the analytic and the algebraic version of the same step.
 
 
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