Direct uniform estimates of Bott–Chern transgression forms at infinity of Calabi–Yau moduli spaces

Lu and Douglas proved a Gauss–Bonnet–Chern theorem on the moduli space of polarized Calabi–Yau manifolds — for every invariant polynomial, the Chern–Weil integral of the singular Hodge connection equals the corresponding topological integer — and deduced finiteness and rationality of Chern–Weil integrals with respect to the Weil–Petersson and Hodge metrics, the analytic input behind finiteness statements for type IIB flux-vacuum counting. The standard route to such a comparison, writing the difference of the Chern–Weil forms of two connections in terms of Bott–Chern transgression forms, was explicitly beyond reach for them near the normal-crossings boundary divisor (the infinity of the moduli space), an obstruction they attributed to the lack of intrinsic background metrics; they instead built a delicately matched smooth connection. The expansion of the Weil–Petersson volume form with log-integrable residual factor is already established (Lu–Sun, Lemma 6.4). The surviving open problem is to obtain intrinsic, uniform, boundary-asymptotic control of the Bott–Chern transgression forms comparing the singular Hodge connection with smooth connections, in the generality of the multivariable SL(2)-orbit theorem, sufficient for a direct proof of the Chern–Weil comparison.

Background

A Calabi–Yau manifold is a compact Kähler manifold with trivial canonical bundle and vanishing intermediate Hodge numbers $h^{i,0}$; in string theory it is the internal space of the most studied superstring compactifications. Fixing an ample line bundle (a polarization) and passing to a finite cover with level structure to remove automorphisms, the deformations of such a manifold are parametrized by a quasi-projective moduli space $\mathcal{M}$, a complex manifold that is typically noncompact. After semistable reduction one may compactify it to a smooth $\overline{\mathcal{M}}$ whose boundary $Y=\overline{\mathcal{M}}\setminus\mathcal{M}$ is a divisor with normal crossings, i.e. locally the union of coordinate hyperplanes $\{s_1\cdots s_a=0\}$ in a polydisc with coordinates $(s_1,\dots,s_a,w_1,\dots,w_b)$; approaching $Y$ is referred to as going to infinity of the moduli space.

The geometry of $\mathcal{M}$ is governed by Hodge theory. The primitive cohomology of the varying fibers forms flat vector bundles over $\mathcal{M}$ (Hodge bundles) under the Gauss–Manin connection, carrying a polarized variation of Hodge structure: a flat bundle with a holomorphic Hodge filtration whose flat sections satisfy Griffiths transversality. The period map sends $\mathcal{M}$ into a classifying domain of polarized Hodge structures, and pulling back the natural $L^2$ inner product on the Hodge bundles gives the Hodge metric on $\mathcal{M}$; the Weil–Petersson metric is the induced $L^2$ metric on first-order deformations of the holomorphic volume form of the fibers. For a Calabi–Yau threefold moduli space of complex dimension $m$ the two are related by $\omega_H=(m+3)\omega_{WP}+\mathrm{Ric}(\omega_{WP})$.

Both metrics degenerate at $Y$, and their boundary asymptotics are governed by the multivariable SL(2)-orbit theorem of Cattani, Kaplan and Schmid: near $Y$, after a singular change of frame (a transition matrix $e_j$ that blows up at the boundary), the connection and curvature matrices of the Hodge bundle become bounded on each of finitely many cones covering a neighborhood of $Y$, with respect to the coframe $ds_i/(s_i\log|s_i|)$, $dw_j$ of the Poincaré metric on the punctured polydisc $(\Delta^*)^a\times\Delta^b$ — a property Lu and Douglas call Poincaré boundedness.

The source paper addresses Chern–Weil theory on such a noncompact space. If $f$ is an invariant polynomial of matrices and $R$, $R_0$ are the curvatures of two connections on the same bundle, the Chern–Weil forms $f(R)$ and $f(R_0)$ differ by an exact form; on a compact manifold the classical Gauss–Bonnet–Chern theorem makes $\int f(\frac{\sqrt{-1}}{2\pi}R)$ a topological integer. On $\mathcal{M}$ the natural Hodge connection $\Gamma$ is singular at $Y$, and the standard comparison route writes $f(R)-f(R_0)$ in terms of Bott–Chern transgression forms: secondary forms, built from the two connections and their curvatures, whose exterior derivative (and locally $\partial\bar\partial$, up to conventions) equals the difference of the two Chern–Weil forms. Lu and Douglas proved that nevertheless $\int_{\mathcal{M}} f(\frac{\sqrt{-1}}{2\pi}R)\in\mathbb{Z}$ for the Hodge connection after a finite cover, and deduced finiteness and rationality of all Chern–Weil integrals for the Weil–Petersson and Hodge metrics on Calabi–Yau moduli. This matters in physics: by the Ashok–Douglas analysis, counting flux compactifications of type IIB string theory reduces to integrals of exactly these forms, and finiteness of the counting indices follows. But the proof did not go through the Bott–Chern route: the authors state that estimating the Bott–Chern form at infinity should be interesting by itself but that they were not able to get the appropriate results, attributing the difficulty to the lack of intrinsic background metrics. Their workaround expands $f(R)-f(R_0)$ by linearity (the polarization of $f$) and constructs a special smooth connection $\Gamma_0$ that stays Poincaré bounded under the singular frame changes (their Lemma 4.8, which they call the technical heart of the paper).

The remaining boundary ingredient sometimes confused with this gap is settled: Lu and Sun proved that the Weil–Petersson volume form near $Y$ factors as a monomial $\prod_j|s_j|^{p_j}$ (with $p_j$ the degeneration orders along the boundary components) times a residual density $g$ such that $\log g$ is integrable for the Poincaré metric — Lemma 6.4 of their 2006 paper, proved within their general framework of Weil–Petersson varieties and cited by Lu–Douglas in the proof of their $L^1$ curvature-integrability theorem (Theorem 6.4). Building on their earlier proof of finiteness of the Weil–Petersson and Hodge volumes, they used it to establish rationality of the Weil–Petersson volume and of the first Chern class of the Weil–Petersson metric; Todorov independently proved rationality of the Weil–Petersson volume. What the literature still lacks is transgression-level control: a direct, intrinsic, uniform estimate of the Bott–Chern forms comparing the singular Hodge connection with smooth connections near infinity — the component the source authors explicitly left open.

Problem Statement

Let $\mathcal{M}$ be the moduli space of polarized Calabi–Yau manifolds, with a smooth compactification $\overline{\mathcal{M}}$ whose boundary $Y=\overline{\mathcal{M}}\setminus\mathcal{M}$ is a divisor of normal crossings. Let $H\to\mathcal{M}$ be the Hodge bundle with the Hodge-metric (Griffiths) connection $\Gamma$ of its variation of Hodge structure — singular along $Y$ — with curvature $R$; let $\Gamma_0$ be a smooth connection on an extension of $H$ across $Y$, with curvature $R_0$; and let $f$ be an invariant polynomial. The Chern–Weil difference $f(R)-f(R_0)$ is exact on $\mathcal{M}$ and is represented, in the standard way, by Bott–Chern transgression forms $BC_f(\Gamma,\Gamma_0)$ constructed from the two connections.

Lu and Douglas, having proved the Gauss–Bonnet–Chern theorem on this moduli space by an indirect route, state that the estimation of the Bott–Chern form at infinity of the moduli space should be interesting by itself but that they were not able to get the appropriate results. Their remark is made in the more general setting of their Theorem 4.1 — Hodge bundles over quasi-projective subvarieties of the moduli space of a polarized Kähler manifold — of which the Calabi–Yau moduli space above is the specialization underlying their Weil–Petersson conclusions; a uniform estimate established in that general setting also resolves the problem as stated here. The problem is to close this gap: obtain a direct, intrinsic, uniform estimate of the Bott–Chern transgression forms $BC_f(\Gamma,\Gamma_0)$ near the boundary divisor $Y$, valid in the generality of the multivariable SL(2)-orbit asymptotics — over all boundary strata and all cones of the Cattani–Kaplan–Schmid decomposition, with constants independent of the cutoff parameter used to exhaust $\mathcal{M}$. The estimate should be intrinsic (valid for any smooth comparison connection $\Gamma_0$, not only for a specially matched choice) and strong enough that the boundary contributions in the Chern–Weil comparison are directly seen to vanish, so that

$$\int_{\mathcal{M}} f\Big(\frac{\sqrt{-1}}{2\pi}R\Big)=\int_{\mathcal{M}} f\Big(\frac{\sqrt{-1}}{2\pi}R_0\Big)\in\mathbb{Z}$$

and the $L^1$ integrability of the Hodge and Weil–Petersson curvature on Calabi–Yau moduli follow directly, without the auxiliary construction of a specially matched smooth connection.

References

1. Zhiqin Lu and Michael R. Douglas, "Gauss-Bonnet-Chern theorem on moduli space", Mathematische Annalen 357 (2013), no. 2, 469–511. DOI: 10.1007/s00208-013-0907-4, https://doi.org/10.1007/s00208-013-0907-4; arXiv:0902.3839, https://arxiv.org/abs/0902.3839.
2. Zhiqin Lu and Xiaofeng Sun, "On the Weil-Petersson volume and the first Chern class of the moduli space of Calabi-Yau manifolds", Communications in Mathematical Physics 261 (2006), no. 2, 297–322. DOI: 10.1007/s00220-005-1441-3, https://doi.org/10.1007/s00220-005-1441-3; arXiv:math/0510021, https://arxiv.org/abs/math/0510021 (Lemma 6.4: Weil–Petersson volume expansion and log-integrability).
3. Zhiqin Lu and Xiaofeng Sun, "Weil-Petersson geometry on moduli space of polarized Calabi-Yau manifolds", Journal of the Institute of Mathematics of Jussieu 3 (2004), no. 2, 185–229. DOI: 10.1017/S1474748004000076, https://doi.org/10.1017/S1474748004000076; arXiv:math/0510020, https://arxiv.org/abs/math/0510020 (finiteness of the Weil–Petersson and Hodge volumes).
4. Eduardo Cattani, Aroldo Kaplan and Wilfried Schmid, "Degeneration of Hodge structures", Annals of Mathematics (2) 123 (1986), no. 3, 457–535. DOI: 10.2307/1971333, https://doi.org/10.2307/1971333 (multivariable SL(2)-orbit theorem).
5. Andrey Todorov, "Weil-Petersson volumes of the moduli spaces of CY manifolds", Communications in Analysis and Geometry 15 (2007), no. 2, 381–405. DOI: 10.4310/cag.2007.v15.n2.a8, https://doi.org/10.4310/cag.2007.v15.n2.a8; arXiv:hep-th/0408033, https://arxiv.org/abs/hep-th/0408033 (independent proof of rationality of the Weil-Petersson volume).
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