Rationality of the Weil–Petersson Chern numbers of Weil–Petersson subvarieties of Calabi–Yau moduli (Lu–Sun conjecture)

Lu and Sun proved that the Weil–Petersson volume of the moduli space of polarized Calabi–Yau manifolds is rational, and that all integrals of powers of the Ricci form against powers of the Weil–Petersson form are rational over arbitrary subvarieties; they closed their paper by conjecturing the same for all Chern forms: for every Weil–Petersson subvariety $X$ of dimension $q$ and all $k+l=q$, the integral $\int_X c_k(\omega_{WP})\wedge\omega_{WP}^{l}$ is a rational number. Douglas and Lu's Gauss–Bonnet–Chern theorem for Chern–Weil forms on moduli spaces later settled the case $X=M$ of the full Calabi–Yau moduli in a stronger form (all rational invariant polynomials of the Weil–Petersson curvature, mixed with powers of the metric). The open core is: rationality of the intrinsic Chern numbers of the restricted Weil–Petersson metric for $k\ge 2$ on proper Weil–Petersson subvarieties, such as Hodge loci, together with the case $X=M$ for Weil–Petersson varieties other than the full Calabi–Yau moduli. The problem is to prove or refute the conjecture in this remaining generality.

Background

A polarized Calabi–Yau $n$-fold is a compact Kähler manifold with trivial canonical bundle together with an ample line bundle polarizing the Kähler class. Deformations of such manifolds are unobstructed, so complex structures vary in smooth families, and the moduli space $M$ of polarized deformation classes is a quasi-projective complex orbifold (locally the quotient of a smooth space by a finite group); it is typically noncompact. Infinitesimal deformations are governed by the Kodaira–Spencer map, which identifies the tangent space of $M$ at a point with $H^1(Z,T_Z)$, the first cohomology of the holomorphic tangent bundle of the fiber $Z$. The Weil–Petersson metric $\omega_{WP}$ on $M$ is the $L^2$ inner product on these deformation classes induced by the Ricci-flat Kähler metric representing the polarization; it is Kähler but generally incomplete, with unbounded curvature near the boundary.

For Calabi–Yau manifolds this metric is governed by Hodge theory. The middle cohomology of the fibers carries a polarized Hodge decomposition varying holomorphically over $M$ (a variation of Hodge structure), encoded in a holomorphic period map from the universal cover of $M$ into a classifying domain $D$ of polarized Hodge structures; the map is horizontal, meaning the Hodge filtration can move only one step at a time (transversality). The Hodge bundles $F^p$ are the holomorphic bundles whose fibers are the pieces of the Hodge filtration, pulled back along the period map. A theorem of Tian identifies the Weil–Petersson form with the first Chern form of the top Hodge line bundle $F^n$ (fiber $H^{n,0}(Z)$, spanned by the holomorphic volume form $\Omega$):

$\omega_{WP}=-\frac{\sqrt{-1}}{2\pi}\,\partial\bar{\partial}\log(\Omega,\Omega).$

Lu and Sun axiomatized this into the notion of a Weil–Petersson variety: a Kähler orbifold whose universal cover immerses horizontally into a classifying space with the tangent bundle identified with the quotient $F^{n-1}/F^n$ of consecutive Hodge bundles, whose Kähler form is the curvature of $F^n$, and whose Hodge bundles satisfy nilpotent-orbit type extension properties near the boundary (the local normal form of a degenerating variation of Hodge structure). Moduli spaces of polarized Calabi–Yau manifolds are the basic examples. A Weil–Petersson subvariety is a subvariety $X\subset M$ that is itself a Weil–Petersson variety for the restricted structures; in Calabi–Yau moduli the natural examples are the Hodge loci, also called Noether–Lefschetz loci, where the varying Hodge structure acquires extra Hodge classes.

For a Hermitian holomorphic vector bundle with curvature endomorphism $R$, the Chern–Weil construction produces the Chern forms: $c_k$ is, up to the conventional factor, the $k$-th elementary symmetric polynomial of the eigenvalues of $R$; the notation $c_k(\omega_{WP})$ applies this to the tangent bundle with the Weil–Petersson metric, and $c_1(\omega_{WP})$ is the Ricci form of the metric. On a compact complex manifold, the Gauss–Bonnet–Chern theorem identifies integrals of products of Chern forms with intersection numbers of Chern classes, hence integers. On a noncompact moduli space this conclusion fails in general: one needs growth control of the Chern–Weil forms near the boundary. Mumford's notion of a good metric on a bundle over a quasi-projective variety — at worst logarithmic growth of the metric and Poincaré-type growth of the connection and curvature along a normal-crossing boundary divisor — guarantees that Chern numbers still compute intersection numbers on a compactification; but the Weil–Petersson Chern–Weil forms are known not to be good, even at the level of the first Chern form, so this route is unavailable.

Prior results delimit the question. Lu and Sun proved that the Weil–Petersson volume is rational (independently proved by Todorov) and, more generally, that for every subvariety $X\subset M$ of dimension $q$ and all $k+l=q$ the integrals $\int_X(\mathrm{Ric}(\omega_{WP}))^{k}\wedge\omega_{WP}^{l}$ are rational, identifying them in favorable cases with intersection numbers involving the extension of $X$'s own anti-canonical bundle, the Hodge line bundle, and rational boundary contributions. Douglas and Lu then proved a Gauss–Bonnet–Chern theorem for the Chern–Weil forms of Hodge bundles over quasi-projective subvarieties of any polarized Kähler moduli, and deduced that over the full Calabi–Yau moduli the integrals of arbitrary invariant polynomials with rational coefficients in the Weil–Petersson curvature, mixed with rational multiples of the metric form, are rational. The volume, Ricci-power, and full-moduli cases of the rationality question are therefore theorems. What remains unresolved is the same question for the higher Chern forms $c_k(\omega_{WP})$, $k\ge 2$, on proper Weil–Petersson subvarieties, together with the case $X=M$ for Weil–Petersson varieties other than the full Calabi–Yau moduli, which is precisely the conjecture stated at the end of Lu and Sun's paper.

Problem Statement

Let $M$ be a Weil–Petersson variety in the sense of Lu–Sun — the basic example being the moduli space of polarized Calabi–Yau $n$-folds with its Weil–Petersson metric $\omega_{WP}$ — and let $X\subset M$ be a Weil–Petersson subvariety of complex dimension $q$: a subvariety whose smooth part, carrying the Weil–Petersson metric obtained by restriction (equivalently, defined through the restricted period map), is itself a Weil–Petersson variety; the basic examples in Calabi–Yau moduli are the Hodge (Noether–Lefschetz) loci. For nonnegative integers $k,l$ with $k+l=q$, let $c_k(\omega_{WP})$ denote the $k$-th Chern form of the tangent bundle of the smooth part of $X$ equipped with this restricted Weil–Petersson metric ($c_1$ being the Ricci form). The problem is the conjecture posed at the end of Lu–Sun's paper:

$\int_X c_k(\omega_{WP})\wedge\omega_{WP}^{l}\in\mathbb{Q}$

for every Weil–Petersson subvariety $X$ and all $k+l=\dim X$. Determine whether this statement is true: prove it in the remaining cases, or refute it by exhibiting a Weil–Petersson subvariety $X$ and a $k\ge 2$ for which the integral is irrational or diverges. The integrand is intrinsic to $X$ — the curvature of the restricted metric on $X$, not the restriction of ambient curvature forms — which is the reading forced by Lu–Sun's own Theorem 1.2 and Corollary 1.1 for the analogous Ricci-power integrals. Cases already settled: $k=0$ (volume) and the replacement of $c_k$ by powers of the Ricci form are theorems for all subvarieties (Lu–Sun 2006), and the case $X=M$ of the full Calabi–Yau moduli is a theorem for all $k$ in a stronger form (Douglas–Lu 2013). The open content is proper subvarieties $X\subsetneq M$ with $k\ge 2$, together with the cases $X=M$ for Weil–Petersson varieties $M$ that are not the full Calabi–Yau moduli.

References

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