Invariant anticanonical indices and conversion of twisted differentials

1. From invariant Euler index to invariant forms of one degree

Write the universal cover of a finite cover as $\widetilde S=\mathbf C^b\times C\times F$, with $C$ Ricci-flat and $F$ a compact factor without positive-degree holomorphic forms. Residual monodromy acts through a compact torus $T$. Hodge symmetry gives $H^q(F,\mathcal O_F)=0$ for $q>0$, hence $I_T(0)=1$. The polynomial in Theorem 1.1 is nonzero, so for one fixed $q$ there are invariant classes in $H^q(F,L_F^m)^T$ at unbounded exponents. Hard Lefschetz with coefficient $L_F^{m+1}$, followed by compact averaging, gives $H^0(F,\Omega_F^p\otimes L_F^{m+1})^T\ne0$ for one fixed $p$.

[OpenAI26a, Theorem 2.1; Lemma 2.2, pp. 5–8] Theorem 2.4; Corollary 2.5 · pp. 8–9 [DPS01, Theorem 2.1.1, p. 8]

2. Descend forms before constructing the pseudoeffective error

Multiplication by invariant canonical frames in Proposition 5.1 injects these forms into $H^0(S,\Omega_S^p\otimes(-K_S)^r)$. If $p>0$, a fixed determinant of their generic span yields a line bundle $M$ saturated in a cotangent tensor power. Nefness makes $-M$ pseudoeffective. Taking $D=-M$ turns the forms into effective divisors $N_j\sim d_j(-K_S)-D$ for unbounded $d_j$. If $p=0$, the forms are already anticanonical sections.

Proposition 5.1 · p. 16 Theorem 4.1; Lemma 4.2; Corollary 4.3 · p. 15 [LMPTX23, Theorem 4.1, print p. 1409 / PDF p. 17]

3. Ordinary conversion and the finite étale norm

The anticanonical metric on $S$ is the smooth semipositive pullback metric, so Theorem 1.2 applies. The norm for $d=\deg(S/X)$ sends its ordinary section of $-kK_S$ to a nonzero section of $-dkK_X$. No $T$-invariance of this section is required. The finite cover is never treated as a global product.

Theorem 1.2 · p. 2 Corollary 5.2 · pp. 16–17 [OpenAI26a, Theorem 2.1; Lemma 2.2, pp. 5–8]

1. Fixed-component localization gives all divisible exponents

For a fixed component $Z\subset F^T$, let $\nu_i$ be the normal tangent characters and $x_i$ the formal Chern roots. The natural anticanonical character is $w=\sum_i\nu_i$, and the contribution is $$\int_Z\operatorname{td}(Z)e^{m c_1(L_F|_Z)}t^{mw}\prod_i(1-t^{-\nu_i}e^{-x_i})^{-1}.$$ Polarize all denominators in one integral direction. The trivial-character coefficient becomes a weighted lattice count with $\sum_i\alpha_i b_i=-mw$, $b_i>0$ for $i\in I$, and $b_i\ge0$ otherwise; all $\alpha_i$ are positive under the same linear functional. Lemma 2.1 triangulates closed rational polytopes and uses a fundamental parallelepiped with binomial identity (2.1). This gives the polynomial at every divisible exponent, including small ones. Truncated cohomological degree makes the weights polynomial. Only the complete symmetric sum is interpreted as characteristic classes and integrated.

§2.2, (2.2)–(2.4) · pp. 7–8 Lemmas 2.1–2.2 · pp. 4–6 [ASeg68, Theorem 2.12, print p. 540 / PDF p. 11] [ASin68, Theorems 3.9, 4.3, print pp. 562–565 / PDF pp. 18–21]

2. The natural character supplies a retraction onto deleted faces

Set $D=\{x_i\ge0:\sum\alpha_i x_i=-w\}$ and $A=\bigcup_{i\in I}(D\cap\{x_i=0\})$. If $I\ne\varnothing$ and $D\ne\varnothing$, the vector $u_i=-1$ for $i\in I$ and $u_i=1$ otherwise lies in the same affine equation space. This is where $w=\sum\nu_i$ enters. With $t(x)=\max_{i\in I}(x_i+1)^{-1}$, the map $r(x)=u+t(x)(x-u)$ retracts $D$ onto $A$; the segment homotopy fixes $A$. Thus $\chi(D)=\chi(A)=1$, and inclusion–exclusion makes the polynomial constant term $(\chi(D)-\chi(A))W(0,0)=0$.

For $I=\varnothing$, the closed count retains $W(0,0)$. At the actual exponent $m=0$, positive pairing forces $b=0$, allowed precisely when $I=\varnothing$. Hence $P(0)=I_T(0)$. If $F^T=\varnothing$, localization and character continuity give $I_T=0$ at every exponent, so take $P=0$. Trivial actions and zero-dimensional polytopes are included. The argument uses the difference of Euler characteristics of compact sets, not the ordinary Euler characteristic of the nonclosed complement $D\setminus A$.

Lemmas 2.1–2.2 · pp. 4–6 §2.2, (2.2)–(2.4) · pp. 7–8

1. A maximal dominated base and normalized vertical orders

Choose a rational map with largest base dimension among those satisfying $c\pi^*L-f^*H$ pseudoeffective on a resolution $\pi:V\to S$, with $H$ ample on the base $Y$. Make its function field relatively algebraically closed as well. If $c'L-A$ is pseudoeffective, any ratio of two sections of $A$ lies in $\mathbf C(Y)$: otherwise its pencil together with the chosen map would give a larger dominated base.

Equation (3.3) for $N_j\sim m_jL-D$ makes horizontal orders affine functions of $m_j$. Their nonnegativity and $m_j\to\infty$ imply horizontal effectivity of $N_2-N_1$. On a normal model, subtract the minimum normalized vertical order above each base prime and clear denominators. This gives $\sigma:f^*B\to\ell\pi^*L$, with $\ell>0$, nonvanishing on at least one component above every base prime. A point base already produces a section at this stage.

§3.1, (3.2)–(3.7) · pp. 9–12

2. Rank-one integrals converge to a bounded maximum metric

The anticanonical identity gives an effective exceptional divisor $E=K_V-\pi^*K_S=K_V+\pi^*L$ and its canonical section $e$. Maximality forces $f_*\mathcal O_V(E+k\ell\pi^*L)$ to have generic rank one for every $k\ge0$. A larger rank would produce, after a sufficiently positive base twist, two sections with a ratio outside the base field, contradicting maximality after pushdown.

For a local frame $\beta$ of $B$, define on the smooth-fiber locus $$\varphi_\beta=-\log\max_{V_y}|\sigma_\beta|^2_{h_L^\ell}.$$ Berndtsson's theorem gives psh weights from $-1/k$ times the logarithm of the $L^2$ integral of $e\sigma_\beta^k/f^*\tau$. High moments against a positive full-support measure, together with Dini's theorem, converge locally uniformly to $\varphi_\beta$. Properness gives a lower bound for the weight. A local power-map chart at a component where $\sigma$ is nonzero gives an upper bound. Extension across divisors and then codimension two produces a locally bounded semipositive metric $h_B$.

§3.2, (3.8)–(3.10) · pp. 12–13 [Ber09, Theorem 1.2, p. 3]

3. Vanishing including k=0 forces a nonzero polynomial

Pseudoeffective domination supplies a singular metric positive in base directions. Mix a small proportion of it with smooth $h_L$. Small-exponent integrability of psh weights gives $\Theta_h(\pi^*L)\ge\varepsilon f^*\omega_H$ for some $\varepsilon>0$, with $\mathcal I(h)=\mathcal O_V$. Boundedness of $h_B$ means multiplication by $f^*h_B^k$ preserves that ideal for each fixed $k\ge0$. Fujino's theorem with direct-image degree $q=0$ yields $$H^i(Y,\mathcal G\otimes\mathcal O_Y(kB))=0\quad(i>0,k\ge0),\qquad\mathcal G=f_*\mathcal O_V(E).$$ Consequently $Q(k)=\chi(Y,\mathcal G(kB))=h^0(Y,\mathcal G(kB))$ at every nonnegative exponent, including zero. The section $e$ gives $Q(0)>0$, so this polynomial is nonzero and gives sections at positive exponents. Multiply one by $\sigma^k$ and use $\pi_*\mathcal O_V(E)=\mathcal O_S$ to obtain a section of $k\ell L$. The identity $L=-K_S$ is used both in the rank-one adjoint expression and in the fixed sheaf at exponent zero.

§3.3, (3.11)–(3.12) · pp. 13–14 [DK01, §1.4(8), p. 7] [Fuj16, Theorem 1.3, p. 2]

1. Natural actions agree on both tensor factors

Let $\eta$ be the product of the invariant canonical frames. Send an invariant form $u$ to $\eta^{-r}\otimes\operatorname{pr}_F^*u$. The natural actions on the differential and anticanonical factors agree with the deck action, so this tensor descends. Pullback of forms under the projection is injective, tensoring with the frame is invertible, and local covering charts preserve nonvanishing. Projectivity of $S$ makes the descended holomorphic section algebraic.

Proposition 5.1 · p. 16 [OpenAI26a, Theorem 2.1; Lemma 2.2, pp. 5–8]

2. Keep the exponent shift and determinant saturation

Corollary 2.5, with $I_T(0)=1$, uses exponent $m+1$, so apply descent with $r=m+1$. A point factor gives the same conclusion with $p=0$. After descent, if $p>0$, fix a basis of the generic span and replace one basis vector with forms at infinitely many exponents. Their exterior products have exponents $m_j$ plus a fixed sum, still unbounded. The saturated line $M$ in $\bigwedge^{r'}\Omega_S^p$ is reflexive and hence invertible. Its inclusion in a cotangent tensor power preserves saturation because the exterior power is a direct summand. LMPTX23 makes $-M$ pseudoeffective. Ordinary Theorem 1.2 now applies; it is not claimed to convert invariant forms directly into invariant anticanonical sections.

Theorem 2.4; Corollary 2.5 · pp. 8–9 Theorem 4.1; Lemma 4.2; Corollary 4.3 · p. 15 [LMPTX23, Theorem 4.1, print p. 1409 / PDF p. 17]