diff --git a/cohomological_transfer_equivariant_v2.txt b/cohomological_transfer_equivariant_v2.txt
index 790b049..625694a 100644
--- a/cohomological_transfer_equivariant_v2.txt
+++ b/cohomological_transfer_equivariant_v2.txt
@@ -44,7 +44,7 @@ title[Cohomological transfer and anticanonical sections]{Cohomological transfer
 
 
 Abstract: 
-Let $X$ be a smooth projective complex variety, and suppose its anticanonical line bundle $L=-K_X$ admits a Hermitian metric whose Chern curvature is nonnegative: in a word, $L$ is smoothly semipositive. We prove that $X$ then has a nonzero section of some positive multiple $mL$. We prove more. If an algebraic torus $T$ acts on $X$ and a section of $M+iL$ exists for unbounded positive integers $i$, invariant under the torus and with one fixed error $M$ whose inverse $-M$ is pseudoeffective, then an invariant section of a positive untwisted power $mL$ exists. The number $m$ may depend on the data; the paper does not produce a uniform $m$. The engine is a field of ratios of invariant sections. The engine has three consequences: natural invariant nonvanishing under a nonzero Euler characteristic; ordinary nonvanishing without that hypothesis, via compact monodromy and a finite norm; and exact source characters from a one-parameter action. Several independent routes to the same conclusion (Iitaka, harmonic restriction, alteration, multiplier ideal, stabilized direct image) are given so that the reader can see how the same obstructions are handled in different regularity regimes. 
+Let X be a smooth projective complex variety, and suppose its anticanonical line bundle L=-K_X admits a Hermitian metric whose Chern curvature is nonnegative: in a word, L is smoothly semipositive. We prove that X then has a nonzero section of some positive multiple mL. We prove more. If an algebraic torus T acts on X and a section of M+iL exists for unbounded positive integers i, invariant under the torus and with one fixed error M whose inverse -M is pseudoeffective, then an invariant section of a positive untwisted power mL exists. The number m may depend on the data; the paper does not produce a uniform m. The engine is a field of ratios of invariant sections. The engine has three consequences: natural invariant nonvanishing under a nonzero Euler characteristic; ordinary nonvanishing without that hypothesis, via compact monodromy and a finite norm; and exact source characters from a one-parameter action. Several independent routes to the same conclusion (Iitaka, harmonic restriction, alteration, multiplier ideal, stabilized direct image) are given so that the reader can see how the same obstructions are handled in different regularity regimes. 
 
 maketitle
 tableofcontents
@@ -94,20 +94,18 @@ The rest of the introduction to this paper has three parts. Section~sec:vocab is
 # Vocabulary
 
 This section is a glossary. Each item is one sentence; a reader who already knows the term can skip it. Everything later refers back to these items by the item number.
-
-
-item[(V1)] A smooth projective complex variety (X) is the common zero locus of homogeneous polynomials in some (PP^N) that is a complex manifold of pure dimension (n). Projective means embeddable in (PP^N). Smooth means no singularity. Connected means one piece.
-item[(V2)] The tangent bundle (T_X) is the complex vector bundle whose fibre at a point is the tangent space in the holomorphic category. The cotangent bundle (Ω_X^1=T_X^vee) is its dual. Higher exterior powers (Ω_X^p=∧^pT_X^vee) are the holomorphic (p)-forms.
-item[(V3)] The canonical line bundle is (K_X=detΩ_X^1=∧^nΩ_X^1); its inverse (L=-K_X=det T_X) is the anticanonical line bundle. On a point (K_X=OO) and (L) is trivial.
-item[(V4)] A line bundle on (X) is a complex vector bundle of rank one; equivalently an invertible sheaf of (OO_X)-modules. Two are linearly equivalent when isomorphic. Tensoring line bundles makes their set into a group, the Picard group (Pic(X)). We write (A+B) for the tensor product (A ⊗  B) and (mL) for (L^{ ⊗  m}).
-item[(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.
-item[(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)).
-item[(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).
-item[(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}).
-item[(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.
-item[(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.
-item[(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).
-item[(V12)] A flat morphism (f:Vto Y) between algebraic varieties is one whose fibres deform without jumping in dimension or in arithmetic genus. An equidimensional morphism is one whose fibres all have dimension exactly (dim V-dim Y). A flat projective morphism with integral target is equidimensional if and only if every irreducible component of every fibre has the expected dimension.
+[(V1)] A smooth projective complex variety (X) is the common zero locus of homogeneous polynomials in some (PP^N) that is a complex manifold of pure dimension (n). Projective means embeddable in (PP^N). Smooth means no singularity. Connected means one piece.
+[(V2)] The tangent bundle (T_X) is the complex vector bundle whose fibre at a point is the tangent space in the holomorphic category. The cotangent bundle (Ω_X^1=T_X^vee) is its dual. Higher exterior powers (Ω_X^p=∧^pT_X^vee) are the holomorphic (p)-forms.
+[(V3)] The canonical line bundle is (K_X=detΩ_X^1=∧^nΩ_X^1); its inverse (L=-K_X=det T_X) is the anticanonical line bundle. On a point (K_X=OO) and (L) is trivial.
+[(V4)] A line bundle on (X) is a complex vector bundle of rank one; equivalently an invertible sheaf of (OO_X)-modules. Two are linearly equivalent when isomorphic. Tensoring line bundles makes their set into a group, the Picard group (Pic(X)). We write (A+B) for the tensor product (A ⊗  B) and (mL) for (L^{ ⊗  m}).
+[(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}).
+[(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).
+[(V12)] A flat morphism (f:Vto Y) between algebraic varieties is one whose fibres deform without jumping in dimension or in arithmetic genus. An equidimensional morphism is one whose fibres all have dimension exactly (dim V-dim Y). A flat projective morphism with integral target is equidimensional if and only if every irreducible component of every fibre has the expected dimension.
 
 
 A note on ample and nef. A line bundle is ample when some positive power admits a projective embedding; nef when its degree on every curve is nonnegative. Ample implies nef implies pseudoeffective; none of the converses hold. Smoothly semipositive anticanonical (V7) is a differential-geometric strengthening of nef. It rules out all of the pathological curvature-zero cases only when the curvature form is zero on a large locus; on those cases the analysis in later sections becomes essential.
@@ -132,11 +130,11 @@ An analogy.:  Imagine (s_i) is a polynomial of degree (2i-1) on the circle (S^1)
 [Invariant transfer]
 Let (X) be smooth connected projective and let (L=-K_X) carry a smooth semipositive metric. Let an algebraic torus (T) act on (X), and let (P,M) be linearized line bundles with
 [
- P=L as ordinary line bundles,qquad -M pseudoeffective.
+ P=L as ordinary line bundles, -M pseudoeffective.
 ]
 The linearization of (P) may differ from the natural linearization of (L) by a character. Suppose that for an unbounded subset (IsubsetZ_{>0}) there are nonzero invariant sections
 [
- s_iin H^0(X,M+iP)^T,qquad iin I.
+ s_iin H^0(X,M+iP)^T, iin I.
 ]
 Then (H^0(X,mP)^Tne0) for some integer (m>0).
 
@@ -153,7 +151,7 @@ Proof. [Proof of Theorem~thm:transfer]
 
 The proof is long and technical; it splits into seven steps. The plain-language form of each step is stated first; the technical form follows.
 
-medskip
+
 noindentStep 1: the common ratio field.
 Plain form. Ratios of invariant sections of the same power behave like rational functions and form a field. As (i) varies, these fields are all the same, eventually.
 
@@ -163,30 +161,30 @@ Technical form. Put (n=dim X), (N_i=M+iP). For each (i), let (Q_i) be the field
 ]
 holds. Given any ratio in degree (aN_p) or (aN_b), multiply numerator and denominator by appropriate powers of the fixed (s_p,s_b) to put the ratio in the field for (N_q). Taking (q) sufficiently large shows that all (Q_i) with (i>min I) coincide; call the common field (Q).
 
-medskip
+
 noindentStep 2: if the field is trivial, we are done.
 Plain form. If the ratios are all constant, the section divisors are linearly related, and two ratios of sections give a section of a positive power of (P) directly.
 
 Technical form. Fix (p<q) in (I). For (b>q), the ratio (s_q^{b-p}/(s_p^{b-q}s_b^{q-p})) belongs to (Q). If (trdeg_C Q=0), this ratio is constant. The section divisors are affine in the index (i), and their slope is effective because (I) is unbounded; thus (s_q/s_p) extends to a regular invariant section of ((q-p)P).
 
-medskip
+
 noindentStep 3: if the field is nontrivial, it defines a base.
 Plain form. A nontrivial field of ratios gives a fibration (f:Vto Y), and the torus acts trivially on (Y). On each fiber, the sections become unique up to scalar.
 
 Technical form. Assume (trdeg_C Q=e>0). Choose (e) independent ratios to define a rational map to (PP^e), resolve its graph, take Stein factorization. The Stein base (Y) is smooth projective after resolution; its field is relatively algebraically closed in (C(X)) and contains (Q); its torus action is trivial. Choose a normal equidimensional model (Wto Y) and an equivariant resolution (Vto W) as in Lemma~lem:flat-model (Section~sec:models). Write (pi:Vto X), (f:Vto Y), (P'=pi^*P), (L'=pi^*L), (E=K_{V/X}).
 
-medskip
+
 noindentStep 4: the fiberwise adjoint rank is one.
 Plain form. On the generic fiber, invariant sections of the adjoint line (E+jkP') are unique up to scalar, for every (j).
 
 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
- qquad(jge0).
+ (jge0).
 ]
 The section (1_Egamma^j) (where (1_E) is the canonical section of (E)) gives the lower bound.
 
-medskip
+
 noindentStep 5: a regular relative section.
 Plain form. Remove the vertical poles by subtracting, over each base prime, the minimal relative pole order. The result is a regular section of (kP_W-f_W^*(kB)) for a base divisor (B).
 
@@ -196,24 +194,24 @@ Technical form. Let (D_i) be the zero divisor of (s_i) on (W). By Step 3 its hor
 ]
 is regular and nonzero at the general point of at least one component above every base prime.
 
-medskip
+
 noindentStep 6: two metrics and the fixed multiplier ideal.
 Plain form. We need a bounded metric on the base and a positive-curvature metric at twist zero. The bounded metric comes from the maximum construction (Section~sec:models); the positive-curvature metric uses pseudoeffectivity of (-M). Its multiplier ideal is not assumed trivial; it is fixed while the exponent varies.
 
 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,qquad mathcal I(h)=calI(h)=OO_V,qquad delta>0.
+ iΘ_hgedelta f^*omega_Y, mathcal I(h)=calI(h)=OO_V, 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).
 
-medskip
+
 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
 [
  H^v(Y,OO_Y(D_*+jkB) ⊗ mathcal J)=0calJ)=0
- qquad(v>0, jge0).
+ (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).
 
@@ -234,7 +232,7 @@ The following is an input theorem, imported from the companion paper [cite]; we
 [Invariant anticanonical index]
 Let (F) be a compact complex manifold with a holomorphic action of a compact real torus (T). Give (D=-K_F) its natural linearization. There are an integer (a>0) and a polynomial (PinQ[t]) such that
 [
- sum_q(-1)^qdim H^q(F,maD)^T=P(m)qquad(m>0),qquad P(0)=sum_q(-1)^qdim H^q(F,OO_F)^T.
+ sum_q(-1)^qdim H^q(F,maD)^T=P(m)(m>0), P(0)=sum_q(-1)^qdim H^q(F,OO_F)^T.
 ]
 The statement admits a trivial torus, an empty fixed locus, and zero-dimensional fixed components; it assumes neither positivity, projectivity, nor K"ahlerity.
 
@@ -252,7 +250,7 @@ The bridge between cohomology classes and sections of a line bundle is hard Lefs
 [Forms and determinant sections]
 Let (X) be smooth connected projective, let (L=-K_X) have a smooth semipositive metric, and let an algebraic torus (T) act on (X). If (chi^T(X,OO_X)ne0), then for some fixed (q) there are nonzero invariant sections of (Ω_X^{n-q} ⊗  L^{m+1}) for unbounded positive integers (m). If these do not already give a nonzero invariant section of a positive power of (L), then there are a linearized line bundle (M) and a nonzero map (Mto∧^eΩ_X^{n-q}) for some (e>0), together with invariant sections
 [
- 0ne s_iin H^0(X,M+a_iL)^T,qquad a_ito∞.
+ 0ne s_iin H^0(X,M+a_iL)^T, a_ito∞.
 ]
 
 
@@ -337,7 +335,7 @@ Descending a cohomological class.:  The invariant Euler characteristic of (F) is
 
 Descending a twisted form.:  The third descent map descends forms before converting them. On (F), Lemma~lem:forms produces invariant sections
 [
- 0nealpha_min H^0(F,Ω_F^{d-q} ⊗ (m+1)P)^H,qquad P=-K_F, d=dim F,
+ 0nealpha_min H^0(F,Ω_F^{d-q} ⊗ (m+1)P)^H, P=-K_F, d=dim F,
 ]
 for unbounded (m) and a fixed (q). Projecting to (F) and embedding the pulled-back cotangent bundle into the product cotangent bundle gives a twisted form
 [
@@ -402,7 +400,7 @@ Take a generic integral cocharacter (lambda) with (X^lambda=X^T), and let (S) be
 
 By Lemma~lem:source-character the polynomial in (m) (a specialization of the full character identity) has value one at zero. Some fixed cohomological degree thus has nonzero weight-(mmu) classes for unbounded (m). Equivariant hard Lefschetz produces nonzero eigenforms
 [
- alpha_min H^0(X,Ω_X^p ⊗ (m+1)L),qquad wt(alpha_m)=mmu,
+ alpha_min H^0(X,Ω_X^p ⊗ (m+1)L), wt(alpha_m)=mmu,
 ]
 for a fixed (p) and unbounded (m). Determinant extraction (as in Lemma~lem:forms) gives a linearized line (M_0subset∧^rΩ_X^p) and invariant sections of (M_0+iL) of weight ((i-r)mu) for unbounded (i). Shift the linearizations of (L) by (-mu) and of (M_0) by (rmu); the shift cancels in the weight, and the resulting sections are invariant sections of (M+iP) with (P=L) shifted by (-mu) and (M=M_0) shifted by (rmu). Cotangent-tensor theorem [cite] makes (-M) pseudoeffective, so Theorem~thm:transfer applies. It gives a section of (k_mu L) of weight (k_mumu).
 
@@ -439,7 +437,7 @@ The following proposition is the reusable engine that turns a rational relative
 [Divisorial normalization and maximum metric]
 Let (X,Y) be smooth connected projective, let (L=-K_X) have a smooth semipositive metric, and suppose
 [
- Vxrightarrow{nu}Wxrightarrow{mu}X,qquad g:Wto Y,qquad f=gnu,
+ Vxrightarrow{nu}Wxrightarrow{mu}X, g:Wto Y, f=gnu,
 ]
 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)
 [
@@ -519,13 +517,13 @@ For a geometric generic fiber (F), the Iitaka dimension of (pi^*N_b+E) restricte
 
 Restriction of the affine relation to (F) gives equality of effective divisors: ((pi^*(N_b-N_a))|_Fge0). So the ratio of (z=s_b/s_a) of two representative sections is a regular section of ((b-a)L') on the generic fiber. For every (kge0),
 [
- h^0(F,(E+kdL')|_F)=1,qquad d=b-a.
+ h^0(F,(E+kdL')|_F)=1, d=b-a.
 ]
 Lower bound: the indicated effective section. Upper bound: for (kge1), multiply by the section of (kpi^*N_a+(k-1)E) to inject into the sections of (k(pi^*N_b+E)|_F); for (k=0), multiply by the section of (pi^*N_b|_F).
 
 Now extract the base divisor. On the normal equidimensional (W), take for each prime (Psubset Y) the minimum coefficient (c_P=min_Qord_Q(z)/ord_Q(f_W^*P)), clear denominators, and obtain
 [
- dL'sim H+f^*B,qquad Hge0,
+ dL'sim H+f^*B, Hge0,
 ]
 with (H) pulled back from (W) and (B) integral on (Y). Above each base prime at least one component has coefficient zero in (H).
 
@@ -589,7 +587,7 @@ The curvature commutator in bidegree ((n,ell)) is monotone in the curvature form
 
 Let (v_j=*u_j), absorbing a dimensional constant so that (u_j=omega^ellwedge v_j). Metric compatibility of the star operators gives
 [
- barbar∂∂ v_jto0 in L^2,qquad D'_jv_j=0.
+ barbar∂∂ v_jto0 in L^2, D'_jv_j=0.
 ]
 Passing to a weak limit gives a distributionally holomorphic (v), hence smooth holomorphic. Since (omega) is K"ahler, (u=omega^ellwedge v) is (barbar∂∂)-closed and represents (c). Projection to the harmonic forms for one fixed reference metric is a bounded finite-rank operator with the same value on each (u_j); the nonnegative square root of ([theta,Λ]) annihilates the limit, so every positive-curvature holomorphic covector is a wedge factor of (v).
 
@@ -635,7 +633,7 @@ In Theorem~thm:kernel, replace the kernel hypothesis by the assumption that (bD_
 Proof. 
 Keep the uniformly convergent metrics. For (ell>0), put
 [
- S=i^{(n-ell)^2}h,vwedgebarbar v,v,qquad T=Swedgeomega^{ell-1},
+ S=i^{(n-ell)^2}h,vwedgebarbar v,v, T=Swedgeomega^{ell-1},
 ]
 with coefficient pairing understood. The distributional Chern equation and holomorphy make (T) a continuous closed positive ((n-1,n-1))-form. The curvature-energy equality gives (thetawedge T=0). Choose a positive curvature form (eta) of (H). A positive current in the class of (bD_V-f^*H) pairs nonnegatively with (T), since positive currents have order zero. Regularise (T) by de Rham (which commutes with d) to approximate uniformly by smooth closed forms; the pairing with a cohomologous smooth form is approximated consistently, giving
 [
@@ -669,7 +667,7 @@ We start with the determinant sections of Lemma~lem:forms, keeping only the case
 
 Fix (a_-<a), keep unbounded indices (>a). Put (N=M+aL). Among all rational maps given by invariant sections of positive powers of (N), choose one with maximal image dimension. Resolve its base ideal equivariantly and take Stein factorization: this gives a smooth projective (X_0), a birational (pi_0:X_0to X), and (f_0:X_0to Y_0) with connected fibres and normal projective base, together with
 [
- dpi_0^*N=f_0^*H_0+G_0,qquad G_0ge0,
+ dpi_0^*N=f_0^*H_0+G_0, G_0ge0,
 ]
 where (H_0) is ample, the torus acts trivially on (Y_0,H_0), and the defining section of (G_0) is invariant.
 
@@ -734,7 +732,7 @@ The choice of field now controls the horizontal divisors. There are only finitel
 
 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,qquad (f_{0*}OO_{X_0}(E_0+kbpi_0^*L))^T
+ (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).
 
@@ -810,7 +808,7 @@ For a generic integral one-parameter subgroup (lambda), let (C) be its source, (
 
 # The curvature-kernel field and a reduced flat alteration
 
-In words.:  All previous ordinary conversions followed (L) by constructing a base and using a relative anticanonical section. This section instead constructs a field of functions constant in the curvature-null directions, takes a model of that field, and uses an alteration with reduced fibres to make a base line on it. The metric weight along this line makes the fibrewise maximum positive even on a non-smooth model. Only after this is done does the birational Euler argument come back to (X). The route does not use any Iitaka or multiplier-ideal setup, and it assumes nothing on $Pic^0(X)$.
+In words.:  All previous ordinary conversions followed (L) by constructing a base and using a relative anticanonical section. This section instead constructs a field of functions constant in the curvature-null directions, takes a model of that field, and uses an alteration with reduced fibres to make a base line on it. The metric weight along this line makes the fibrewise maximum positive even on a non-smooth model. Only after this is done does the birational Euler argument come back to (X). The route does not use any Iitaka or multiplier-ideal setup, and it assumes nothing on Pic^0(X).
 
 [Ordinary conversion through a curvature-null field]
 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).
@@ -822,7 +820,7 @@ Construction overview.:  The proof has three parts. First, the harmonic-star con
 
 Let (X) be as in the theorem, put (n=dim X), let (theta) be the smooth semipositive curvature of (L), and let (r) be its maximal rank. If (q=0), the conclusion is already given. Otherwise apply Lemma~lem:forms to get nonzero sections of (Ω_X^{n-q} ⊗ (m+1)L). If (r=n), the positive-curvature energy on an open set excludes (q>0). In the remaining cases the determinant extraction gives effective divisors
 [
- D_iin|M+a_iL|,qquad a_ito∞,qquad D_i[theta]^r[omega]^{n-r-1}=0.
+ D_iin|M+a_iL|, a_ito∞, D_i[theta]^r[omega]^{n-r-1}=0.
 ]
 The intersection equality is Lemma~lem:null-divisors. If (r=0), all (D_i) vanish, and two distinct (a_i) show that a positive power of (L) is trivial. Assume (1le r<n).
 
@@ -885,7 +883,7 @@ We are now on birational models of the original source and base fields. The base
 
 In words.:  The rank-one direct image of an adjoint line need not be locally free. If it has, say, a codimension-two defect, the Euler interpolation of Section~sec:harmonic that drops it can add a section count that is not there. The present section identifies the defect with the multiplier ideal of a fibre-integral metric, so nothing is dropped. This gives a route to the transfer theorem that uses the actual degree-zero line, not its hull.
 
-The three ingredients are: a localised multiplier-ideal vanishing theorem, a strictness lemma for the quotient metric on an invariant adjoint line, and an identification of the direct image with its multiplier ideal. The application is a new proof of Corollary~thm:main under the hypothesis $H^q(X,OO_X)=0$ for $q>0$.
+The three ingredients are: a localised multiplier-ideal vanishing theorem, a strictness lemma for the quotient metric on an invariant adjoint line, and an identification of the direct image with its multiplier ideal. The application is a new proof of Corollary~thm:main under the hypothesis H^q(X,OO_X)=0 for q>0.
 
 ## A localised vanishing theorem
 
@@ -999,16 +997,16 @@ satisfies (mathscr E_sscrE_s ⊗ OO_Y(K_Y)=g_*(omega_Z ⊗ pi_Z^*OO_X((1+sd)L)))
 ]
 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,qquad h^0(Y,K_Y+A_0)=1.
+ 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
 [
- u_s(B)=leftlfloormin_i(e_i+sc_i)/(b_i)rightrfloor,qquad K_Y+A_s=OO_Y(C_s),quad C_s=sum_Bu_s(B)B.
+ 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.
 ]
 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),
 [
- C_s=C_∞+sV,qquad C_∞ge C_0,
+ C_s=C_∞+sV, C_∞ge C_0,
 ]
 so with (A_∞=OO_Y(C_∞)-K_Y) there is an inclusion
 [
@@ -1031,7 +1029,7 @@ Its curvature is strictly positive for (s/u) small. Strong openness and compactn
 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).
 
 
-Where the hypothesis $H^q(X,OO_X)=0$ appears.:  Under that hypothesis, Proposition~prop:index and Lemma~lem:forms supply the unbounded invariant twisted forms required to apply the proposition, exactly as in Corollary~thm:forms-input. So the present section is a third proof of invariant nonvanishing under (H^q(X,OO_X)=0) for (q>0).
+Where the hypothesis H^q(X,OO_X)=0 appears.:  Under that hypothesis, Proposition~prop:index and Lemma~lem:forms supply the unbounded invariant twisted forms required to apply the proposition, exactly as in Corollary~thm:forms-input. So the present section is a third proof of invariant nonvanishing under (H^q(X,OO_X)=0) for (q>0).
 
 
 # The map of the whole argument
@@ -1041,23 +1039,22 @@ In words.:  The paper has one main theorem, one main corollary, and six complete
 ## The shortest path to the strongest conclusion
 
 Read Sections~sec:transfer, sec:index, sec:descent in this order. They prove:
-
-item Theorem~thm:transfer: unbounded invariant sections of (M+iP) with (-M) pseudoeffective give an invariant section of a positive power of (P). (Section~sec:transfer.)
-item Corollary~thm:main: natural invariant nonvanishing under (chi(X,OO_X)ne0), via the invariant anticanonical index. (Section~sec:index.)
-item Corollary~thm:ordinary: ordinary nonvanishing without the Euler hypothesis, via compact monodromy and a finite norm. (Section~sec:descent.)
+ Theorem~thm:transfer: unbounded invariant sections of (M+iP) with (-M) pseudoeffective give an invariant section of a positive power of (P). (Section~sec:transfer.)
+ Corollary~thm:main: natural invariant nonvanishing under (chi(X,OO_X)ne0), via the invariant anticanonical index. (Section~sec:index.)
+ Corollary~thm:ordinary: ordinary nonvanishing without the Euler hypothesis, via compact monodromy and a finite norm. (Section~sec:descent.)
 
 A reader who wants the strongest short answer can stop after Section~sec:descent. A reader who wants all the machinery should keep reading.
 
-subsection*{The six routes to Corollary~thm:ordinary}
+The six routes to Corollary~thm:ordinary
 
 The following table compares them. The common step is always ``produce a base, an injection, and rank-one adjoints, then apply Proposition~prop:euler or an equivalent''. The differences are the choice of base and the analytic input that makes the metric work.
 
 
 small
 {@{}p{.14textwidth}p{.28textwidth}p{.22textwidth}p{.26textwidth}@{}}
-toprule
+
 Route & Input & Base construction & Conversion 
-midrule
+
 Transfer (Sect.~sec:transfer) & Unbounded invariant sections of (M+iP), (-M) pseudoeffective & Common ratio field (Q_i); equivariant normal model & Fixed multiplier ideal, Nadel vanishing 
 Indexed (Sect.~sec:index) & (chi(X,OO_X)ne0) & Invariant anticanonical index; determinant sections of (M) & Transfer theorem applied with natural linearization 
 Descent (Sect.~sec:descent) & Structure of the universal cover & Compact factor (F); Zariski closure of the torus; canonical frames & Finite 'etale norm; or cohomological or twisted-form descent 
@@ -1068,7 +1065,7 @@ Fields (Sect.~sec:fields) & (H^q(X,OO_X)=0) for (q>0) and determinant sections &
 Alteration (Sect.~sec:alteration) & Unbounded (H^q(X,mL)) in one degree & Curvature-null field (K), reduced flat alteration & Galois descent of a Hom line and a finite norm 
 Degree-zero (Sect.~sec:local-ideal) & (H^q(X,OO_X)=0) and smooth semipositive (L) & Degree-zero pole field with (L^2)-metrics & Exact multiplier ideal, Lemma~lem:local-nadel 
 Stabilized (Sect.~sec:stabilized) & Same as transfer, but no ratio field & Toroidal equidimensional model with rational singularities & Affine boundary orders, actual direct-image line, strong openness 
-bottomrule
+
 
 
 
@@ -1095,9 +1092,9 @@ The following table collects symbols used in more than one section. An index in
 
 small
 {@{}p{.12textwidth}p{.50textwidth}p{.24textwidth}@{}}
-toprule
+
 Symbol & Meaning & Section 
-midrule
+
 (X) & Smooth connected projective complex variety & sec:question 
 (n) & (dim X) & sec:vocab 
 (T_X, Ω_X^p) & Tangent and cotangent bundles, exterior powers & sec:vocab 
@@ -1139,7 +1136,7 @@ midrule
 (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 
-bottomrule
+
 
 
 A note on notation. Tensor products of line bundles are written additively: (A+B) means (A ⊗  B), (mA) means (A^{ ⊗  m}), (-A) means the inverse line. The symbol (s_E) is a specific rational section of (E=K_{V/X}); (1_E) is its local trivialization. The symbol (P) is used both for a linearized line and for the transfer theorem's base twist; context resolves which is meant. The base line is called (Q) in Sections~sec:transfer--sec:harmonic-iitaka and (B) in Sections~sec:local-ideal--sec:stabilized; the two are related by normalization and by the identification (B_0=B+K_Y).