Hodge degeneracy of simply connected Calabi–Yau manifolds in dimension greater than two

Kamenova and Vafa proved that a Calabi–Yau manifold $X$ whose mirror dual $\check{X}$ exists and is not Hodge degenerate contain possibly singular rational or elliptic curves, and hence is Kobayashi non-hyperbolic, i.e. carries an entire curve; here Hodge degeneracy is an equality among first Chern classes of the Hodge bundles over the deformation space, equivalent in dimension three to a proportionality between the Weil–Petersson Ricci form and the Weil–Petersson form. Beyond K3 surfaces and complex tori, no simply connected Calabi–Yau manifold of dimension greater than two is known to satisfy this degeneracy, and no nonexistence theorem is known. The problem asks whether a non-vacuously Hodge degenerate simply connected Calabi–Yau $n$-fold with maximal holonomy $\mathrm{SU}(n)$, $n>2$, exists, and, if so, whether Hodge non-degeneracy of the mirror is necessary for the mirror-symmetric non-hyperbolicity criterion or merely sufficient.

Background

A Calabi–Yau manifold is a compact Kähler manifold with trivial canonical bundle and trivial fundamental group; it has maximal holonomy $\mathrm{SU}(n)$, $n=\dim_{\mathbb{C}}X$, precisely when $h^{p,0}(X)=0$ for all $0<p<n$, which excludes products, complex tori, and hyperkähler (irreducible holomorphic symplectic) manifolds. The Kobayashi pseudometric $d_{X}$ is the largest pseudometric on $X$ that is distance-decreasing for all holomorphic maps from the Poincaré disk; $X$ is Kobayashi hyperbolic when $d_{X}$ is nondegenerate, and Kobayashi non-hyperbolic when it admits an entire curve, i.e. a non-constant holomorphic map $\mathbb{C}\to X$. Kobayashi conjectured in 1976 that every compact Calabi–Yau manifold has vanishing Kobayashi pseudometric; the weaker statement that every Calabi–Yau manifold is Kobayashi non-hyperbolic would follow from the existence of a possibly singular rational or elliptic curve on $X$. Kamenova–Lu–Verbitsky settled the hyperkähler case (K3 surfaces — the simply connected complex surfaces with trivial canonical bundle — completely, and large classes by deformation), and Verbitsky later proved non-hyperbolicity for all compact hyperkähler manifolds using ergodicity of complex structures; Heath-Brown and Wilson obtained rational curves, hence non-hyperbolicity, for Calabi–Yau threefolds of Picard number $\rho>13$ (the rank of the group of divisor classes on $X$ modulo numerical equivalence). The Calabi–Yau case with small Picard number remains open in general.

Kamenova and Vafa introduced a mirror-symmetric route. Mirror symmetry is the conjectural duality from string theory predicting that a Calabi–Yau manifold $X$ has a mirror partner $\check{X}$ whose complex geometry encodes the enumerative geometry of curves on $X$. For a polarized Calabi–Yau $n$-fold with universal family $\pi\colon\mathcal{X}\to\mathrm{Def}(X)$ over its deformation space $\mathrm{Def}(X)$ of complex structures, the Hodge bundles are the bundles $R^{q}\pi_{*}\Omega^{p}_{\mathcal{X}/\mathrm{Def}}$ of relative primitive $(p,q)$-cohomology (the primitive part is the $L^{2}$-orthogonal of the polarization class); these bundles carry the flat Gauss–Manin connection of the family, described in coordinates by Picard–Fuchs differential equations. The Weil–Petersson form, the natural Kähler form on $\mathrm{Def}(X)$ induced by the $L^{2}$-metric on holomorphic $n$-forms, is $\omega_{WP}=c_{1}(R^{0}\pi_{*}\Omega^{n}_{\mathcal{X}/\mathrm{Def}})$, and the generalized Hodge metrics $\omega_{H}^{i}$ are Hodge metrics built from these bundles. The BCOV torsion $T$ (holomorphic Ray–Singer torsion, the alternating product of determinants of the non-singular parts of the $\bar\partial$-Laplacians on $(p,q)$-forms, introduced by Bershadsky–Cecotti–Ooguri–Vafa) satisfies the curvature formula of Fang–Lu: $\sum_{i=1}^{n}(-1)^{i}\omega_{H}^{i}-\frac{\sqrt{-1}}{2\pi}\partial\bar\partial\log T=\frac{\chi(X)}{12}\omega_{WP}$, where $\chi(X)$ is the topological Euler characteristic. Kamenova and Vafa call $X$ Hodge degenerate when the torsion term vanishes, i.e. when

$\sum_{i=1}^{n}\sum_{p=1}^{i}(-1)^{i}\,p\,c_{1}\bigl(R^{i-p}\pi_{*}\Omega^{p}_{\mathcal{X}/\mathrm{Def}}\bigr)=\frac{\chi(X)}{12}\,c_{1}\bigl(R^{0}\pi_{*}\Omega^{n}_{\mathcal{X}/\mathrm{Def}}\bigr),$

an identity of classes on $\mathrm{Def}(X)$. For Calabi–Yau threefolds, using the Fang–Lu relation $\omega_{H}=(m+3)\omega_{WP}+\mathrm{Ric}(\omega_{WP})$ with $m=\dim\mathrm{Def}(X)=h^{1}(X,T_{X})$, this is equivalent (Lemma 2.8 of their paper) to $\mathrm{Ric}(\omega_{WP})=-\bigl(m+3+\tfrac{\chi(X)}{12}\bigr)\omega_{WP}$; for $n>3$ no comparable simplification is known and the condition must be read from the full Chern-class identity. Their main theorem states that a Calabi–Yau $n$-fold $X$, $n>2$, whose mirror dual $\check{X}$ exists and is not Hodge degenerate is Kobayashi non-hyperbolic. The proof identifies the genus-one topological string amplitude $F_{1}$ of $X$ (the generating function of its genus-one curve counts) with $\log T$ of the mirror, as predicted by the BCOV holomorphic anomaly framework (the formalism of Bershadsky–Cecotti–Ooguri–Vafa relating topological string amplitudes to Ray–Singer torsion): a non-vanishing non-constant contribution to $F_{1}$ forces genus 0 or genus 1 curves on $X$; this identification is rigorous so far only in cases such as quintic threefolds (smooth degree-five hypersurfaces in $\mathbb{P}^{4}$), where Zinger's computation of the genus-one Gromov–Witten invariants — invariants that virtually count stable maps from curves into $X$; the reduced invariants satisfy $N_{1,d}=N^{0}_{1,d}+\tfrac{1}{12}N_{0,d}$ — matches Fang–Lu–Yoshikawa's computation of the BCOV torsion of the mirror. All K3 surfaces and all complex tori are Hodge degenerate yet Kobayashi non-hyperbolic, so Hodge non-degeneracy is sufficient but not necessary for non-hyperbolicity in dimension two. The open questions are whether genuine (non-vacuous) Hodge degeneracy occurs at all among simply connected Calabi–Yau manifolds of dimension greater than two with maximal holonomy, and whether the condition is necessary for the mirror-symmetric criterion in higher dimensions.

Problem Statement

Let $X$ be a simply connected Calabi–Yau $n$-fold with maximal holonomy $\mathrm{SU}(n)$, $n>2$, whose deformation space $\mathrm{Def}(X)$ is positive-dimensional ($m:=h^{1}(X,T_{X})\ge 1$), and let $\pi\colon\mathcal{X}\to\mathrm{Def}(X)$ be its universal family. Say that $X$ is Hodge degenerate if the identity

$\sum_{i=1}^{n}\sum_{p=1}^{i}(-1)^{i}\,p\,c_{1}\bigl(R^{i-p}\pi_{*}\Omega^{p}_{\mathcal{X}/\mathrm{Def}}\bigr)=\frac{\chi(X)}{12}\,c_{1}\bigl(R^{0}\pi_{*}\Omega^{n}_{\mathcal{X}/\mathrm{Def}}\bigr)$

holds on $\mathrm{Def}(X)$, equivalently $\frac{\sqrt{-1}}{2\pi}\partial\bar\partial\log T=0$ for the BCOV torsion $T$ of the family, and for $n=3$ equivalently $\mathrm{Ric}(\omega_{WP})=-\bigl(m+3+\frac{\chi(X)}{12}\bigr)\omega_{WP}$.

(1) Existence: determine whether any such $X$ is Hodge degenerate. Either such an $X$ exists, with a verifiable computation establishing the identity, or no simply connected Calabi–Yau $n$-fold with maximal holonomy, $n>2$ and $m\ge 1$, satisfies it. The two requested outcomes serve one objective: deciding whether the Hodge-degeneracy locus in the Calabi–Yau moduli landscape is empty or not.

(2) Necessity (directly linked follow-up): if a Hodge-degenerate $X$ as in (1) exists and admits a mirror dual $\check{X}$, determine whether the Kamenova–Vafa criterion — a Calabi–Yau $n$-fold, $n>2$, whose mirror dual exists and is not Hodge degenerate is Kobayashi non-hyperbolic — is optimal: either exhibit a mirror pair $(Y,\check{Y})$ with $Y$ as in (1) and $\check{Y}$ Hodge degenerate together with a possibly singular rational or elliptic curve (or an entire curve) on $Y$ produced by any method, showing that Hodge non-degeneracy is sufficient but not necessary for the mirror-symmetric non-hyperbolicity criterion, or prove that Hodge degeneracy of the mirror genuinely obstructs the conclusion.

References

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