Cohomological transfer and equivariant anticanonical sections

1. Build a relative section from the common ratio field

Write $N_i=M+iP$. For $p<q<b$, the identity $(b-p)N_q=(b-q)N_p+(q-p)N_b$ preserves linearizations. Multiplication by the same fixed sections in numerator and denominator identifies the invariant ratio fields associated with interior indices. Independent ratios define a projective base; Stein factorization replaces its field by its relative algebraic closure in $\mathbb C(X)$. The torus acts trivially on this base.

Take a normal equidimensional model $\mu:W\to X$, $f_W:W\to Y$, and a smooth resolution $\pi:V\to X$, $f:V\to Y$. Since the ratios come from the base, horizontal section divisors are affine in the exponent; unboundedness makes their slope effective. Subtract the minimum order divided by pullback multiplicity over each base prime and clear denominators, obtaining $$\gamma\in H^0(W,k\mu^*P-f_W^*(kB))^T,\qquad k>0.$$ At least one component over every base prime is absent from the zero divisor of $\gamma$. If the ratio field has transcendence degree zero, its effective slope already gives a regular invariant anticanonical section.

§3, common ratio field and relative section · pp. 8–9

2. Obtain invariant rank one, including degree zero

The effective divisor $E=K_{V/X}$ has its natural invariant Jacobian section. A character shift in $P'=\pi^*P$ does not change this natural action on the Jacobian. For every $j\ge0$, $$\operatorname{rank}\bigl(f_*\mathcal O_V(E+jkP')\bigr)^T=1.$$ The lower bound comes from $1_E\gamma^j$. Two independent invariant sections could be globalized after an ample base twist, then multiplied by input sections and powers of $\gamma$ to yield an invariant ratio of sections of an equal multiple of $N_b$. That ratio would belong to the common base field, contradicting independence on the generic fiber. Degree zero is included.

§3, equation (6) · p. 9

3. Separate the metric for varying twists from the degree-zero metric

Proposition 2.2 puts a continuous semipositive metric on $kB$. Separately, pseudoeffectivity of $-M$ and ample domination on the base give a singular metric on $L'=\pi^*L$ satisfying $$i\Theta_h\ge\delta f^*\omega_Y,\qquad\mathcal I(h)=\mathcal O_V\qquad(\delta>0).$$ A small mixture with the smooth metric and Skoda integrability give the trivial ideal.

Project the degree-zero adjoint integral to the invariant holomorphic orthogonal quotient, giving a density $F_\xi$ on $-K_Y$ that is locally integrable throughout the base. At a boundary prime with pullback multiplicity $\ell$ and Jacobian order $a$, the local estimate is $F_\xi(y)\ge c|y_1|^{2((a+1)/\ell-1)}$. It determines an effective correction $D_*$ allowing extension of the metric on $-K_Y+D_*$. Its support is restricted to base primes whose dominating components are all exceptional. A dimension estimate on the equidimensional model also treats components mapping into codimension at least two. Consequently the entire $f^*D_*$ is exceptional over $X$.

§3, equations (7)–(8) · pp. 9–10 [PT14, Theorems 3.3.4–3.3.5, p. 24]

4. Keep one possibly nontrivial multiplier ideal

Let $\mathcal J$ be the extended metric's multiplier ideal. Local integrability of the density yields $$\mathcal O_Y(-D_*)\subseteq\mathcal J,$$ without asserting $\mathcal J=\mathcal O_Y$. Tensoring with the bounded metric on $jkB$ leaves this ideal unchanged. For all $j\ge0$, Nadel vanishing gives $$H^v(Y,\mathcal O_Y(D_*+jkB)\otimes\mathcal J)=0\quad(v>0).$$ The sheaf $\mathcal F=\mathcal O_Y(D_*)\otimes\mathcal J$ contains $\mathcal O_Y$, so $\chi(Y,\mathcal F)=h^0(Y,\mathcal F)>0$. Its Euler polynomial is not zero and gives a section for some $j>0$. Pull back, multiply by $\gamma^j$, remove the exceptional correction $f^*D_*$, and push to $X$. The resulting section of $jkP$ is invariant for the specified linearization.

§3, equation (9) and interpolation · p. 10 [Dem09, Theorem 5.11, p. 38]

1. Normalize divisors and use equidimensionality for continuity

Subtract the minimal normalized order of $s$ over each base prime, then clear denominators to construct $Q$ and its injection. Its zero divisor misses at least one component over every base prime. For the image $s_e$ of a local frame, $M_e(y)=\max_{W_y}|s_e|^2$ is upper semicontinuous by properness and lower semicontinuous by openness of the normal equidimensional morphism. Its zero locus is algebraic closed and has codimension at least two by the omitted-component property. The rank condition at $k=0$ also supplies connected generic fibers.

Proposition 2.2 · pp. 6–8

2. Treat the invariant part as a holomorphic orthogonal quotient

Apply Berndtsson to the smooth adjoint coefficient $(kb+1)D_V$. Positivity of the invariant line does not follow merely from being a subline. Compact averaging makes the action unitary; the nonzero-weight spaces form the orthogonal complement and define a holomorphic quotient. Normalized rank-one integrals satisfy $$-\frac1k\log\int_{V_y}|s_e|^{2k}d\mu_y\ \longrightarrow\ -\log M_e(y).$$ On smooth general fibers, the measure has full support and convergence is locally uniform. The negative logarithm of the maximum is therefore psh.

[Ber09, Theorem 1.2, print p. 533 / PDF p. 3] Proposition 2.2 · pp. 6–8

3. Exclude codimension-two zeros and retain equivariance

Continuity extends the psh weight wherever $M_e>0$. At the remaining codimension-two zero locus, use analytic discs and parallel translates whose boundaries stay off that locus. The maximum principle gives a uniform upper bound for $-\log M_e$, contradicting a zero of the continuous maximum. Hence $M_e$ is positive everywhere, and the frame transformation law defines the continuous semipositive metric on $Q$. Invariance of the normalized vertical divisors and relative section preserves the equivariant injection and the trivial base action.

Proposition 2.2 · pp. 6–8

1. Use the index polynomial with its actual value at zero

A connected torus acts by maps homotopic to the identity, hence trivially on $H^q(X,\mathcal O_X)$ through Hodge decomposition. The natural invariant index at zero is therefore $\chi(X,\mathcal O_X)\ne0$. The companion's index theorem identifies this actual value with the constant term of a polynomial on a divisible progression. Eventual polynomiality alone would not exclude the zero polynomial. Fix a cohomological degree along a subsequence, and apply equivariant hard Lefschetz to $mL=K_X+(m+1)L$, obtaining invariant twisted forms.

[OpenAI26b, Theorem 1.1, p. 1] Proposition 4.1; Lemma 4.2 · pp. 10–11 [DPS01, Corollary 2.1.2, p. 8]

2. Remove the fixed determinant error by Theorem 1.1

If $q=0$ or $q=n$, a positive anticanonical section already exists. Otherwise take the generic span of the forms and its saturated determinant line $M$. Wedge a form of unbounded exponent with a fixed complementary basis to obtain nonzero invariant sections of $M+a_iL$ with $a_i\to\infty$. The cotangent tensor theorem applies to the smooth zero-boundary pair with nef $L$, making $-M$ pseudoeffective. Theorem 1.1 then applies with naturally linearized $P=L$.

Corollary 4.3, proof · pp. 11–12 [LMPTX23, Theorem 4.1, print p. 1409 / PDF p. 17]

1. Descend the compact-factor section to the finite cover

The companion's structure theorem gives a finite étale cover $X'\to X$ whose deck action is by translations on the Euclidean factor, trivial on $V$, and through a compact torus $H$ on $F$. Positive-degree holomorphic forms vanish on $F$, so Hodge symmetry gives $\chi(F,\mathcal O_F)=1$. Since $\operatorname{Pic}^0(F)=0$, the compact action can be algebraized through a very ample line. Corollary 4.3 supplies an invariant section for its algebraic-torus closure. Tensoring with invariant canonical frames makes it deck invariant and it descends to $X'$. For a point factor $F$, use the constant section.

[OpenAI26a, Theorem 2.1, p. 5] Corollary 4.3 · pp. 11–12 §14.1, proof of Corollary 1.2 · p. 39

2. Take a norm to the original variety

For $d=\deg(X'/X)$, the finite étale norm sends the section to a nonzero section of $-mdK_X$. Common powers and the cover degree may increase the exponent; no uniform exponent is asserted. The structural input does not require the finite cover itself to be a product, and it does not use the companion's nonvanishing theorem.

[OpenAI26a, Lemma 2.2, p. 8] §14.1, proof of Corollary 1.2 · p. 39

1. Uniform approximation retains the curvature-null part

Fix $k\geq0$ and put $A=D_V+kf^*Q$. The Richberg-type approximation in Lemma 7.1 gives smooth metrics $h_j$ on $A$ with $h_j\to h$ uniformly and $\Theta_{h_j}\geq\theta-o(1)\omega$. A small negative curvature allowance is essential; the approximants are not asserted to be semipositive. The harmonic representatives $u_j$ of a fixed class in $H^\ell(V,K_V+A)$ have uniformly bounded $L^2$ norms. Bochner–Kodaira gives $\|(D'_j)^*u_j\|^2+\langle[\theta,\Lambda]u_j,u_j\rangle\to0$. A weak limit $v$ of $v_j=*u_j$ is holomorphic, and $u=\omega^\ell\wedge v$ represents the original class. The finite-rank harmonic projection for a fixed reference metric preserves that class through the weak limit.

Lemma 7.1 · p. 17 [Dem12, Theorem I.5.21, pp. 43–44] Theorem 7.2, equation (18) · p. 18

2. Restrict the distributional Chern equation to a general fiber

Zero curvature energy makes every positive-curvature holomorphic covector a wedge factor of $v$. The kernel inclusion implies that the base covectors are factors on the specified open set; holomorphy extends this property over the smooth fibration locus. Thus $v=f^*(dy_1\wedge\cdots\wedge dy_b)\wedge w$. In a local frame, $D'_jv_j=0$ reads $\partial(h_jv_j)=0$. Uniform convergence of the metrics and weak $L^2$ convergence give $\partial(hv)=0$ in distributions, without convergence of derivatives of the weights.

The continuous base factor is constant under vertical differentiation. On each smooth fiber this yields $D'_{D_V|V_y}w_y=0$ for its smooth coefficient metric. Restriction here means restriction of a $(0,\ell)$-form valued in $K_V+A$. Removing the constant factor $K_Y|_y$ gives the fiber-harmonic representative $\omega_y^\ell\wedge w_y$. It is not ordinary pullback of an $(n,\ell)$-form to a smaller-dimensional fiber. A nonzero class is nonzero on some general fiber, making the Leray edge map injective.

Theorem 7.2, equations (19)–(21) · pp. 18–19

3. Smooth-coefficient splitting recovers every quantified case

Apply Fujisawa's theorem to the smooth semipositive line $D_V$: for a line its curvature condition is Nakano semipositivity, and the Kähler and proper-surjective hypotheses hold. The theorem is not applied directly to the continuous metric on $D_V+kf^*Q$. First obtain $Rf_*(K_V+D_V)\simeq\bigoplus_qR^qf_*(K_V+D_V)[-q]$, then tensor by $Q^k$ on the base. Projection formula gives

The injective edge map bounds the left side by the $p=0$ summand, forcing all remaining summands to vanish. For $\ell>\dim V$, the left side itself is zero. Since $k$ was any nonnegative integer, this proves all $k,q\geq0$, $p>0$, including the zero twist. That zero-twist case supplies the later Euler interpolation in Proposition 7.5.

[Fujisawa15, Theorem 1, p. 1] Theorem 7.2; Proposition 7.5 · pp. 19–20