Weil's 1947 question: must an integral Kähler class be homologous to a positive analytic cycle?

In a 1947 letter to G. de Rham, published in the Commentarii Mathematici Helvetici, André Weil raised four open problems around the harmonic theory of Kähler manifolds, ending with the sharp question recorded here: if a Kähler form $\Omega$ on a compact complex analytic manifold $V$ has all its periods integers, must $\Omega$ be homologous to a positive analytic cycle, that is, must there exist an effective divisor on $V$ whose cohomology class has the same periods as $\Omega$, equivalently equals $[\Omega]$ in real cohomology? Later literature settled the surrounding problems directly: Hopf's quotient manifolds and the Calabi-Eckmann manifolds show that not every compact complex manifold admits a Kähler metric, and Kodaira and Spencer derived the Lefschetz theorem on $(1,1)$-classes from harmonic theory, while Kodaira's embedding theorem proves that a compact Kähler manifold with an integral Kähler class is projective, so that a positive multiple of every integral Kähler class is effective. The exact effectivity question, however, remains open; its modern counterpart is actively studied on fake projective planes, where Keum proved that, on the planes with enough automorphisms, $H^0$ vanishes for the cubic root $L_0$ of the canonical bundle (an ample generator with $L_0^2=1$) and for $2L$ for every ample generator $L$, and where no example with an effective divisor of degree at most 2 was known. The problem therefore remains open, in its original generality over all compact Kähler manifolds and all integral Kähler classes.

Background

A complex analytic manifold is a manifold whose coordinate changes are holomorphic; all manifolds below are compact. A Hermitian metric on such a manifold is called a Kähler metric if its associated real 2-form $\omega=\frac{i}{2}\sum h_{\alpha\beta}\,dz^\alpha\wedge d\bar z^\beta$ (of type $(1,1)$: one holomorphic and one antiholomorphic differential in each term) is closed, $d\omega=0$; this closed positive $(1,1)$-form — positive in the sense that the Hermitian matrix $(h_{\alpha\beta})$ is positive definite — is the Kähler form, and a manifold carrying one is a Kähler manifold. The periods of a closed real 2-form are its integrals $\int_\gamma\omega$ over the integral 2-cycles $\gamma$ (cycles with integer coefficients); by de Rham's theorem, all periods being integers is equivalent to the cohomology class $[\omega]$ being an integral class, i.e. lying in the image of $H^2(V,\mathbb Z)$ in $H^2(V,\mathbb R)$. A closed form integrates to zero over every torsion cycle, so its periods determine an integral class only up to torsion in $H^2(V,\mathbb Z)$. An analytic cycle is a finite formal sum $\sum_i n_i C_i$ of irreducible closed analytic subvarieties with integer coefficients $n_i$; the cycle is positive (effective) when all $n_i\ge 0$, and a positive cycle of codimension one is an effective divisor. Every effective divisor determines an integral cohomology class, so it makes sense to ask which cohomology classes contain positive analytic cycles. On a complex torus, theta function theory shows that every integral polarization (integral Kähler class) is represented by effective divisors; this was the known case when Weil wrote.

Hodge's theorem decomposes the cohomology of a compact Kähler manifold into harmonic pieces of bidegree $(p,q)$ (spanned by forms with $p$ holomorphic and $q$ antiholomorphic differentials); the Lefschetz theorem on $(1,1)$-classes states, in its sheaf-cohomological form proved by Kodaira and Spencer in 1953, that the integral classes of type $(1,1)$ are exactly the characteristic classes (first Chern classes) of holomorphic line bundles, i.e. the divisor classes up to linear equivalence (two divisors are linearly equivalent when they differ by the divisor of zeros and poles of a meromorphic function). Being a divisor class is thus a Hodge-theoretic condition, but being an effective one is not: a holomorphic line bundle has a nonzero holomorphic section exactly when it is linearly equivalent to an effective divisor. The line bundles sharing one given characteristic class form a coset of the Picard variety $\mathrm{Pic}^0(V)$, the connected group which is the kernel of the characteristic-class map $c_1$ (it is trivial when the irregularity $q=\dim H^0(V,\Omega^1_V)$, the space of holomorphic 1-forms, vanishes), while the line bundles whose characteristic classes differ from a given one only by a torsion class are its twists by torsion line bundles; the distinction matters below, because a form with integral periods determines its class only up to torsion. A compact Kähler manifold whose Kähler form has integral periods was called a Kähler manifold of restricted type by Hodge (1951) and a Hodge manifold by Weil (1952); Kodaira's embedding theorem (1954) proves that such a manifold is projective algebraic, embedded by the sections of $hF_1$ for large integers $h$ ($hF_1$ denoting the $h$-fold tensor power of the line bundle $F_1$ with $c(F_1)=[\omega]$). Consequently some positive multiple of every integral Kähler class is the class of an effective, even base-point-free (the sections have no common zero), divisor; the embedding theorem does not assert that the class itself is effective, and that gap is exactly Weil's question.

The modern theory measures this gap by the interplay of the Kähler cone (the open cone of Kähler classes in $H^2(V,\mathbb R)$) with the semigroup of effective classes: a line bundle is called ample, i.e. positive, when some positive tensor power of it has enough holomorphic sections to embed the manifold into a projective space; on a compact Kähler manifold a line bundle is ample exactly when its characteristic class is a Kähler class, so 'ample cone' and 'Kähler cone' coincide for projective manifolds. Demailly and Paun (2004) characterize Kähler classes as those numerically positive on analytic cycles (pairing positively with every analytic cycle), a condition of positivity on cycles rather than representability by cycles. Fake projective planes, where the surviving instance of the question is studied today, are smooth compact complex surfaces with the same Betti numbers as $\mathbb P^2$ (the complex projective plane) but of general type (the classification class of surfaces whose canonical bundle has maximal growth of sections under powers, the analogue of high-genus curves), with geometric genus $p_g=\dim H^0(K)$ and irregularity $q$ both zero and Chern numbers $c_1^2=3c_2=9$, the values for $\mathbb P^2$ ($c_1$, $c_2$ are the Chern classes of the tangent bundle, and $K$ is the canonical bundle, the top exterior power of the cotangent bundle, whose sections are the holomorphic 2-forms); they are quotients of the complex 2-ball (the unit ball in $\mathbb C^2$) by discrete arithmetic groups of its automorphisms, their canonical bundle $K$ is ample, and their Picard group is $\mathbb Z$ plus finite torsion, so that the ample generators (the line bundles with self-intersection $L^2=c_1(L)^2=1$) all differ by torsion.

Problem Statement

Let $V$ be a compact complex analytic manifold and let $\Omega$ be a Kähler form on $V$ all of whose periods are integers, that is, $\int_\gamma\Omega\in\mathbb Z$ for every integral 2-cycle $\gamma$ on $V$. By de Rham's theorem this says exactly that the class $[\Omega]\in H^2(V,\mathbb R)$ is integral: it is the image of an integral class $c\in H^2(V,\mathbb Z)$, and the integral lifts of $[\Omega]$ form a single coset modulo the torsion subgroup of $H^2(V,\mathbb Z)$ (a closed form integrates to zero over torsion cycles, so its periods do not see torsion). Weil's 1947 question: does it follow that $\Omega$ is homologous to a positive analytic cycle on $V$, i.e. that there exists an effective divisor $D=\sum_i n_i C_i$ (a finite sum of irreducible codimension-one analytic subvarieties with integers $n_i\ge 0$) whose associated integral cohomology class has the same periods as $\Omega$, equivalently whose image in $H^2(V,\mathbb R)$ equals $[\Omega]$ — that is, some integral lift of $[\Omega]$ is the class of an effective divisor? (Fixing a single integral lift $c$ and demanding $[D]=c$ in $H^2(V,\mathbb Z)$ exactly is a strictly stronger reading whenever $H^2(V,\mathbb Z)$ has torsion; the question as posed through the form $\Omega$ and its periods is the reading above, and the two agree when $H^2(V,\mathbb Z)$ is torsion-free. In particular, a negative answer must exclude effective divisors arising from every integral lift.) The question appears at the end of Weil's letter to de Rham as the sharp instance of the general problem of which homology classes contain positive analytic cycles, the case of complex tori being settled classically by theta function theory. Of the surrounding issues raised in the same letter, two are settled in later literature (the existence of non-Kähler compact complex manifolds, and the derivation of the Lefschetz $(1,1)$ theorem from harmonic theory), and the up-to-multiple version of the present question is settled affirmatively by Kodaira's embedding theorem; the operative open target is therefore the exact effectivity question above, retained in its original generality over all compact Kähler manifolds and all integral Kähler classes, with as its documented actively-studied instance the question of which small ample integral classes, e.g. the ample generators of fake projective planes, are effective.

References

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