diff --git a/cohomological_transfer_equivariant_v2.txt b/cohomological_transfer_equivariant_v2.txt
index 489bd4d..9574d1b 100644
--- a/cohomological_transfer_equivariant_v2.txt
+++ b/cohomological_transfer_equivariant_v2.txt
@@ -202,7 +202,7 @@ Technical form. Average a smooth semipositive metric (h_0) on (L) over the maxim
 [
  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).
+The first ineuality uses ratio domination to make (L'-eta f^*H) pseudoeffective for some (eta>0), mixing the resulting current with (h_0).
 
 
 noindentStep 7: Nadel vanishing and Euler interpolation.
@@ -213,7 +213,7 @@ Technical form. Apply Proposition~prop:maximum to obtain a continuous semipositi
  H^v(Y,OO_Y(D_*+jkB) ⊗ J)=0
  (v>0, jge0).
 ]
-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).
+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⊇J⊇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.
@@ -274,7 +274,7 @@ The hypothesis (chi(X,OO_X)ne0) holds in a large class of examples: Fano varieti
 
 ## A degree trick that removes the pseudoeffectivity hypothesis
 
-The proof of Corollary~thm:main uses the cotangent-tensor theorem to get (-M) pseudoeffective. The following lemma obtains the degree inequality directly from a prescribed-Ricci-curvature argument, without invoking the pseudoeffectivity theorem. It is needed in the harmonic constructions of Sections~sec:harmonic and~sec:fields to control the zero divisors of (M).
+The proof of Corollary~thm:main uses the cotangent-tensor theorem to get (-M) pseudoeffective. The following lemma obtains the degree ineuality directly from a prescribed-Ricci-curvature argument, without invoking the pseudoeffectivity theorem. It is needed in the harmonic constructions of Sections~sec:harmonic and~sec:fields to control the zero divisors of (M).
 
 [Curvature-null determinant divisors]
 Let (X) be smooth connected projective of dimension (n), let (L=-K_X) have smooth semipositive curvature (theta) of maximal rank (r<n). Suppose a line bundle (M) admits a nonzero map (Mto∧^eΩ_X^p) for some (e,p>0). Then (c_1(M)G^{n-1}le0) for every K"ahler class (G). If (0ne sin H^0(X,M+aL)) for an integer (a), its zero divisor (D) satisfies
@@ -649,7 +649,7 @@ Let (pi:Vto X) be birational between smooth connected projective varieties, let
 
 
 Proof. 
-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).
+Let (G=f_*G=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
@@ -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
 [
- P_k=(f_*calP_k=(f_*OO_V(E+kG))^T (kge0)
+ P_k=(f_*P_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 (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_∞).
+On (V), form the sheaves (P_k=(f_*P_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_∞P_∞).
 
 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,P_k ⊗ (kQ+lH))=0 (i>0).
 ]
-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.
+For large (k), Castelnuovo--Mumford regularity makes (P_∞P_∞ ⊗ (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
@@ -768,7 +768,7 @@ In words.:  Section~sec:extremal obtained a character at a source using fixed-po
 Let (S) be a smooth connected projective rationally connected variety with smoothly semipositive (L=-K_S) and a (C^*)-action with the natural anticanonical linearization. Let (S_0) be the source component for (tto0). Write (w_C(U)) for the fiber weight of a linearized line (U) on the fixed component (C).
 
 [Source weights and retraction]
-Let (S) be as above, allowing the trivial action. Let (S_0) be the attracting component of the action. A nonzero eigensection of a linearized line (U) has weight at most (w_{S_0}(U)). If (U) is nef, then (w_C(U)le w_{S_0}(U)) for every fixed component, and the multiplicity of weight (m w_{S_0}(U)) in the virtual character (chi(S,mU)) is (chi(S_0,mU|_{S_0})), for (m>0). If (U) is nef, on a smooth equivariant resolution (d:widehatwidehat SSto S) of the source retraction (p:widehatwidehat SSto S_0) there is an equivariant nonzero bundle map (p^*(U|_{S_0})to d^*U).
+Let (S) be as above, allowing the trivial action. Let (S_0) be the attracting component of the action. A nonzero eigensection of a linearized line (U) has weight at most (w_{S_0}(U)). If (U) is nef, then (w_C(U)le w_{S_0}(U)) for every fixed component, and the multiplicity of weight (m w_{S_0}(U)) in the virtual character (chi(S,mU)) is (chi(S_0,mU|_{S_0})), for (m>0). If (U) is nef, on a smooth equivariant resolution (d:SStohat S) of the source retraction (p:SStohat S_0) there is an equivariant nonzero bundle map (p^*(U|_{S_0})to d^*U).
 
 
 Proof. 
@@ -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 (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.
+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)U+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).
 
@@ -794,7 +794,7 @@ The pullbacks of (D_i) have horizontal supports in a fixed finite set. Order mul
 
 Source-weight comparison. Let (Y_0) be the source of the induced action on (Y). The unbounded eigensections force (w_{Y_0}(Q)ge aw). After fixing (a), take an infinite subsequence of (k_i) in one residue class modulo (a), divide by the first section of the subsequence, and clear the resulting vertical poles by an invariant effective Cartier divisor (C) on (Y). The resulting sections come from (jQ+C); the source upper bound gives (jawle jw_{Y_0}(Q)+w_{Y_0}(C)), and divide by (j) and let (jto∞).
 
-Euler interpolation on the base. The continuous semipositive metric makes (Q) nef by Lemma~lem:smoothing. Resolve the source retraction of (Y) and dominate this resolution by a smooth model (V'). Lemma~lem:source and the normalization give (p^*Q_0to aA'). Proposition~prop:dominationvanish applies to (jQ_0). The action on (Y_0) is trivial and (Q_0) has constant fiber weight (w_{Y_0}(Q)). Restrict to the weight-zero summand of (p_*OO(E')); sections of it twisted by (jQ_0) have precisely weight (jw_{Y_0}(Q)). Vanishing identifies their dimension with the Euler polynomial for every (jge0), which is positive at (j=0) because the canonical exceptional section is invariant. For some (j>0) a nonzero section of (E'+jp^*Q_0) of that weight appears. Map into (E'+jaA'), push down to (S); the weight is at least (jaw) by the source-weight inequality and at most (jaw) by the source bound on (S). Equality gives the required extremal section.
+Euler interpolation on the base. The continuous semipositive metric makes (Q) nef by Lemma~lem:smoothing. Resolve the source retraction of (Y) and dominate this resolution by a smooth model (V'). Lemma~lem:source and the normalization give (p^*Q_0to aA'). Proposition~prop:dominationvanish applies to (jQ_0). The action on (Y_0) is trivial and (Q_0) has constant fiber weight (w_{Y_0}(Q)). Restrict to the weight-zero summand of (p_*OO(E')); sections of it twisted by (jQ_0) have precisely weight (jw_{Y_0}(Q)). Vanishing identifies their dimension with the Euler polynomial for every (jge0), which is positive at (j=0) because the canonical exceptional section is invariant. For some (j>0) a nonzero section of (E'+jp^*Q_0) of that weight appears. Map into (E'+jaA'), push down to (S); the weight is at least (jaw) by the source-weight ineuality and at most (jaw) by the source bound on (S). Equality gives the required extremal section.
 
 
 [From extremal characters to invariants]
@@ -854,7 +854,7 @@ 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 (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.
+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')H')) 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
 [
@@ -868,15 +868,15 @@ Give (B_b) the maximum metric through the map and the pulled-back smooth metric
 
 Resolve the alteration and its section by first taking a finite Galois extension of (C(Y)) containing (C(Y_b)), of degree (ell) and group (G), and then normalizing the simultaneous graph of all conjugate rational maps to (Y_b). This gives a normal projective (W) with a regular (G)-action and a map (rho:Wto Y_b). The quotient (W_0=W/G) is normal projective birational to (Y). The invariant product (C_W=⊗_{gin G}g^*B_W) descends to a line (C_0) on (W_0): its power (C_W^ell) has trivial stabilizer actions on fibres, so it descends locally to a line by averaging equivariant values. The invariant metric descends continuously.
 
-Resolve (widehatwidehat YYto W_0) and let (B_0) be the pullback of (C_0); its metric is plurisubharmonic on the finite-étale locus and by continuity everywhere. Choose a smooth projective birational model (widehatwidehat XXto X_1) mapping to (widehatwidehat YY), with maps (widehatmu:widehatwidehat XXto X) and (widehatwidehat f:f:widehatwidehat XXtowidehatwidehat YY), and put (widehatwidehat A=A=widehatmu^*L). Normalize the main component of (widehatwidehat XXtimes_{W_0}W), calling it (R). It is finite over (widehatwidehat XX) with the same Galois group; geometric integrality of the generic fibre makes the function-field extension linearly disjoint from the finite Galois base extension.
+Resolve (YYtohat W_0) and let (B_0) be the pullback of (C_0); its metric is plurisubharmonic on the finite-étale locus and by continuity everywhere. Choose a smooth projective birational model (XXtohat X_1) mapping to (YYhat), with maps (muhat:XXtohat X) and (fhat:f:XXtohathathat YY), and put (Ahat=A=muhat^*L). Normalize the main component of (XXtimeshat_{W_0}W), calling it (R). It is finite over (XXhat) with the same Galois group; geometric integrality of the generic fibre makes the function-field extension linearly disjoint from the finite Galois base extension.
 
-On (R) the difference of the pullbacks (bwidehatwidehat A-B_WA-B_W) has a nonzero section: resolve the rational map from (R) to (Z_b) over (X_1) and (Y_b), and equation ((p_b^*B_bto q_b^*A^b)) gives a regular section there. Both line bundles come from (R); normality descends the section to (R). Multiply its Galois conjugates and raise to the (ell)-th power to obtain a section of the pullback of (bell^2widehatwidehat A-A-widehatwidehat f^*B_0f^*B_0). A further field norm for the finite surjection onto the normal target cite[Lemma~31.18.7]{StacksNormNormal}, of degree (ell), relative to a rational line frame, gives a regular nonzero section downstairs of
+On (R) the difference of the pullbacks (bAhat-B_WA-B_W) has a nonzero section: resolve the rational map from (R) to (Z_b) over (X_1) and (Y_b), and equation ((p_b^*B_bto q_b^*A^b)) gives a regular section there. Both line bundles come from (R); normality descends the section to (R). Multiply its Galois conjugates and raise to the (ell)-th power to obtain a section of the pullback of (bell^2Ahat-A-fhat^*B_0f^*B_0). A further field norm for the finite surjection onto the normal target cite[Lemma~31.18.7]{StacksNormNormal}, of degree (ell), relative to a rational line frame, gives a regular nonzero section downstairs of
 [
- bell^3widehatwidehat A-A-widehatwidehat f^*(f^*(ell B_0).
+ bell^3Ahat-A-fhat^*(f^*(ell B_0).
 ]
-Regularity is checked at every prime: all orders upstairs are nonnegative, hence so are the norm orders. The powers are not meant to be minimal. Put (widehatwidehat b=bb=bell^3) and (widehatwidehat Q=Q=ell B_0).
+Regularity is checked at every prime: all orders upstairs are nonnegative, hence so are the norm orders. The powers are not meant to be minimal. Put (bhat=bb=bell^3) and (Qhat=Q=ell B_0).
 
-We are now on birational models of the original source and base fields. The base line (widehatwidehat QQ) carries a continuous semipositive metric and an injection (widehatwidehat f^*f^*widehatwidehat QQtowidehatwidehat bbwidehatwidehat AA). The original kernel condition persists on a nonempty open of these models. Proposition~prop:euler then gives a section of a positive power of (L) on (X).
+We are now on birational models of the original source and base fields. The base line (QQhat) carries a continuous semipositive metric and an injection (fhat^*f^*QQtohathathat AAhat). The original kernel condition persists on a nonempty open of these models. Proposition~prop:euler then gives a section of a positive power of (L) on (X).
 
 
 # Local strictness and an exact multiplier ideal
@@ -895,7 +895,7 @@ Let (Y) be a smooth connected projective variety of dimension (d>0). Suppose (R)
 
 
 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 (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).
+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+I(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 (m^m^{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/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.
 
@@ -903,7 +903,7 @@ Let (sigma) be a nonzero section of the resulting line. Its divisor metric (psi'
 [
  -psi=-(rho)/(1+εdelta)+(ε V)/(1+εdelta).
 ]
-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.
+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+I((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
@@ -925,9 +925,9 @@ 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 (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
+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_*B=(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) ⊗ I( B.calB.
+ OO_Y(B_0) ⊗ I( B.B.
 ]
 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.
 
@@ -984,7 +984,7 @@ For every fixed (sge0), pick (h) large enough that (hD_{j_1}^Fgediv(e_0|_F)+sdiv
 
 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
 [
- E_s=f_*scrE_s=f_*OO_W(K_{W/Y}+(1+sd)pi^*L)
+ E_s=f_*E_s=f_*OO_W(K_{W/Y}+(1+sd)pi^*L)
 ]
 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
 [
@@ -1011,7 +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 (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 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_{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).
 
@@ -1019,7 +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)·(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_{loc})-integrability of (e^{-psi_u}), proves (e^{-rho_s}in L^1_{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).
 
@@ -1127,10 +1127,10 @@ Symbol & Meaning & Section
 (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 
 (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 
+(H, H'H') & 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 
+(Xhat,X, YYhat) & Resolved models after Galois base change & sec:alteration 
+(W, G, R) & Galois cover, its group, and the finite pullback to (XXhat) & sec:alteration