Douglas and Lu (2006), motivated by the counting of flux vacua over moduli spaces of polarized Calabi–Yau manifolds, posed two geometric questions that their paper left open: whether the volume of every projective special Kähler manifold (with respect to its natural metric) is finite, and whether Calabi–Yau moduli spaces are always incomplete for the Weil–Petersson metric — together with a third question, to find the relation between volume finiteness and metric incompleteness in this setting. The technical question deferred in the same paper (rationality of curvature integrals in arbitrary moduli dimension) was later resolved by the Gauss–Bonnet–Chern theorem for Calabi–Yau moduli of Lu and Douglas (2013). The two geometric questions and their proposed relation remain open: the Calabi–Yau case of volume finiteness is known (Lu–Sun 2004), finite Weil–Petersson distance along degenerations is characterized by Hodge-theoretic criteria (Wang 1997; Lee 2018, 2025), and complete projective special Kähler manifolds have been constructed (Cortés–Dyckmanns–Suhr 2017), but the general questions remain open.

Background

A Calabi–Yau manifold is a compact Kähler manifold XX with trivial canonical bundle and Hq(X,Ωp)=0H^q(X,\Omega^p)=0 for 0<p+q<dim⁡X0<p+q<\dim X; by the resolution of the Calabi conjecture, XX carries a unique Ricci-flat Kähler metric in each Kähler class. Fixing an ample polarization LL, the deformations of the pair (X,L)(X,L) form a quasi-projective moduli space M\mathcal{M} (an open subset of a projective variety), smooth of dimension hn−1,1(X)h^{n-1,1}(X), where n=dim⁡CXn=\dim_{\mathbb{C}}X, because Calabi–Yau deformations are unobstructed. The holomorphic family of nn-forms Ω\Omega over M\mathcal{M} defines the Weil–Petersson metric, the Kähler metric with local potential KWP=−log⁡(i∫XΩ∧Ωˉ).K_{WP}=-\log\Big(i\int_X\Omega\wedge\bar\Omega\Big). The same family is a variation of Hodge structure: the periods of Ω\Omega define the period map into a classifying space of Hodge structures, and pulling back its natural homogeneous metric gives the Hodge metric on M\mathcal{M}; the bundles of holomorphic pp-forms over the family are the Hodge bundles.

In type IIB string compactification on XX, the moduli space M\mathcal{M} is the field space of complex-structure fields, and the number of metastable flux vacua (string-theory solutions with quantized background flux) over M\mathcal{M} is controlled, in the flux-compactification analysis cited by Douglas and Lu, by integrals of curvature invariants of the Weil–Petersson metric over M\mathcal{M}; finiteness of such integrals implies finiteness of the flux-vacuum count. Douglas and Lu proved that every invariant polynomial of the Weil–Petersson curvature is integrable over M\mathcal{M} (their Theorem 4), and that integrals of invariant polynomials of the Hodge-bundle curvature are rational numbers when the complex dimension of M\mathcal{M} is at most 2 (their Theorem 5), deferring the arbitrary-dimension case to later work.

On the other side of the question sits special Kähler geometry. An affine special Kähler manifold, in the sense of Freed, is a Kähler manifold (M,g,J,ω)(M,g,J,\omega) equipped with a flat, torsion-free, symplectic connection ∇\nabla such that the complex structure JJ, viewed as a 1-form with values in TMTM, is parallel under the induced connection (d∇J=0d_\nabla J=0); the bases of algebraically completely integrable Hamiltonian systems, and the complex-structure moduli of Calabi–Yau manifolds with their Weil–Petersson-type metric, carry such structures. Equipping such a manifold with a compatible homothetic vector field (a vector field whose flow rescales the metric by a constant factor) gives a conical special Kähler manifold, whose quotient by the natural C∗\mathbb{C}^*-action is a projective special Kähler manifold — the class appearing in type IIB compactifications and in the duality between special Kähler geometry and Calabi–Yau moduli. Two boundary facts frame the open questions. First, Lu proved Freed's conjecture that every complete affine special Kähler manifold is flat, so nonflat affine special Kähler manifolds are always incomplete; second, Lu and Sun proved that the Weil–Petersson and Hodge volumes of Calabi–Yau moduli spaces are finite. Both proofs rest on a generalized maximum principle (an extension of the classical maximum principle to complete noncompact manifolds), and it is this common mechanism that motivated Douglas and Lu to expect a relationship between volume finiteness and metric incompleteness. Near singular degenerations of Calabi–Yau manifolds the Weil–Petersson metric was analyzed by Green, and Wang later characterized, via limiting mixed Hodge structures (the asymptotic formalism describing how the Hodge decomposition degenerates at a boundary point), which degenerations lie at finite Weil–Petersson distance; Lee extended this criterion to higher-dimensional bases. Note that a rigid Calabi–Yau manifold (one with no complex-structure moduli) has a zero-dimensional, hence trivially complete, moduli space, so the incompleteness question is substantive precisely for positive-dimensional moduli spaces.

Problem Statement

Douglas and Lu pose, at the end of Section 2 of 'On the Geometry of Moduli Space of Polarized Calabi-Yau manifolds' (arXiv:math/0603414), the following questions, motivated by the duality between special Kähler geometry and Calabi–Yau moduli and by the common role of the generalized maximum principle in the known results:

(a) Is the volume of every projective special Kähler manifold finite, with respect to its natural projective special Kähler metric?

(b) Is the moduli space of polarized Calabi–Yau manifolds always incomplete with respect to the Weil–Petersson metric? (Zero-dimensional moduli spaces, i.e. rigid Calabi–Yau manifolds, are points and hence complete; the question is substantive for positive-dimensional moduli spaces.)

(c) Is there a precise relationship between volume finiteness and metric incompleteness for these classes of special Kähler and moduli metrics?

Each of (a) and (b) is posed at the stated generality, so that either a proof or a counterexample settles it; for (c), the two possible outcomes are a theorem establishing such a relationship and a refutation of its existence.

References

  1. Michael Douglas and Zhiqin Lu (2006). On the Geometry of Moduli Space of Polarized Calabi-Yau manifolds. arXiv:math/0603414, DOI 10.48550/arXiv.math/0603414 (RIMS Kokyuroku 1487, 2006). https://arxiv.org/abs/math/0603414
  2. Zhiqin Lu and Michael R. Douglas (2013). Gauss–Bonnet–Chern theorem on moduli space. Mathematische Annalen 357(2), 469–511, DOI 10.1007/s00208-013-0907-4 (arXiv:0902.3839). https://doi.org/10.1007/s00208-013-0907-4
  3. Zhiqin Lu and Xiaofeng Sun (2004). Weil–Petersson geometry on moduli space of polarized Calabi–Yau manifolds. Journal of the Institute of Mathematics of Jussieu 3(2), 185–229, DOI 10.1017/S1474748004000076. https://doi.org/10.1017/S1474748004000076
  4. Zhiqin Lu (1999). A note on special Kähler manifolds. Mathematische Annalen 313(4), 711–713, DOI 10.1007/s002080050278 (arXiv:math/0505577). https://doi.org/10.1007/s002080050278
  5. Daniel S. Freed (1999). Special Kähler manifolds. Communications in Mathematical Physics 203(1), 31–52, DOI 10.1007/s002200050604. https://doi.org/10.1007/s002200050604
  6. Paul S. Green (1991). Singular degenerations of Calabi-Yau manifolds and the Weil-Petersson metric. Proceedings of the American Mathematical Society 111(3), 599–605, DOI 10.1090/S0002-9939-1991-1042267-X. https://doi.org/10.1090/S0002-9939-1991-1042267-X
  7. Chin-Lung Wang (1997). On the incompleteness of the Weil-Petersson metric along degenerations of Calabi-Yau manifolds. Mathematical Research Letters 4(1), 157–171, DOI 10.4310/mrl.1997.v4.n1.a14. https://doi.org/10.4310/mrl.1997.v4.n1.a14
  8. Tsung-Ju Lee (2018). A Hodge theoretic criterion for finite Weil–Petersson degenerations over a higher dimensional base. Mathematical Research Letters 25(2), 617–647, DOI 10.4310/mrl.2018.v25.n2.a14 (arXiv:1604.06914). https://doi.org/10.4310/mrl.2018.v25.n2.a14
  9. Tsung-Ju Lee (2025). Finite distance problem on the moduli of non-Kähler Calabi–Yau ∂∂ˉ\partial\bar\partial-threefolds. Mathematische Annalen 392(2), 1541–1583, DOI 10.1007/s00208-025-03115-8 (arXiv:2404.19125). https://doi.org/10.1007/s00208-025-03115-8
  10. Zhiqin Lu and Eisuke Natsukawa (2005). On the Weil-Petersson Geometry of Calabi-Yau Moduli (announcement). arXiv:math/0509172, DOI 10.48550/arXiv.math/0509172. https://arxiv.org/abs/math/0509172
  11. Vicente Cortés, Malte Dyckmanns and Stefan Suhr (2017). Completeness of Projective Special Kähler and Quaternionic Kähler Manifolds. In: Special Metrics and Group Actions in Geometry, Springer INdAM Series, 81–106, DOI 10.1007/978-3-319-67519-0_4 (arXiv:1607.07232). https://doi.org/10.1007/978-3-319-67519-0_4