On genus one mirror symmetry in higher dimensions and the BCOV conjectures

Eriksson, Dennis; Montplet, Gerard Freixas i; Mourougane, Christophe

doi:10.1017/fmp.2022.13

Cited by 6 publications

(2 citation statements)

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“…Eriksson, Freixas i Montplet and Mourougane [EFM21] constructed the BCOV invariant for Calabi–Yau manifolds of arbitrary dimension and established the asymptotics of the BCOV invariant for one-parameter normal crossings degenerations. In another paper [EFM22], they confirmed the (B-side) genus-one mirror symmetry conjecture of Bershadsky, Cecotti, Ooguri and Vafa [BCOV93, BCOV94] for Calabi–Yau hypersurfaces of arbitrary dimension, which is compatible with the results of Zinger [Zin08, Zin09] on the A-side.…”

Section: Introductionsupporting

confidence: 74%

BCOV invariant and blow-up

Zhang¹

2023

Compositio Math.

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Bershadsky, Cecotti, Ooguri and Vafa constructed a real-valued invariant for Calabi–Yau manifolds, which is now called the BCOV invariant. In this paper, we extend the BCOV invariant to such pairs $(X,D)$ , where $X$ is a compact Kähler manifold and $D$ is a pluricanonical divisor on $X$ with simple normal crossing support. We also study the behavior of the extended BCOV invariant under blow-ups. The results in this paper lead to a joint work with Fu proving that birational Calabi–Yau manifolds have the same BCOV invariant.

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Section: Introductionsupporting

confidence: 74%

BCOV invariant and blow-up

Zhang¹

2023

Compositio Math.

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“…The corresponding holomorphic torsion invariant of Calabi-Yau threefolds, called the BCOV invariant, was introduced by Fang, Lu and the second author [19], who verified some predictions in [3]. Very recently, the BCOV invariant is extended to Calabi-Yau manifolds of arbitrary dimension by Eriksson, Freixas i Montplet and Mourougane [17], who have established the mirror symmetry at genus one for the Dwork family in arbitrary dimension [18]. The notion of the BCOV invariant is further extended to a certain class of pairs by Y. Zhang [49], who, together with L. Fu, has established the birational invariance of the BCOV invariants [50], [20].…”

Section: Introductionmentioning

confidence: 98%

Analytic torsion for log-Enriques surfaces and Borcherds product

Dai

Yoshikawa

2022

Forum of Mathematics, Sigma

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We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type $\frac {1}{4}(1,1)$ . The moduli space of such log-Enriques surfaces with k singular points is a modular variety of orthogonal type associated with a unimodular lattice of signature $(2,10-k)$ . We prove that the invariant, viewed as a function of the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension $2$ . As a consequence, the BCOV invariant in this case is not a birational invariant, unlike the Calabi-Yau case.

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Motivic integration and birational invariance of BCOV invariants

Zhang²

2023

Sel. Math. New Ser.

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In this paper, we discuss the birational invariance of the class of balanced hyperbolic manifolds. IntroductionIn his celebrated paper [Gro91], M. Gromov introduces an important notion called the Kähler hyperbolicity. It is pinched between the real hyperbolicity and the Kobayashi hyperbolicity [Kob98], and helps to settle the Kähler case of the Chern conjecture [Gro91]. After that, it leads to fruitful applications and improvements such as [CX01, CY18, Kol95, Eys97, Hit00, McM00] and so on.However, since the class of Kähler manifolds in general is not invariant under the birational transform, it would be desirable to have a birational variant of the Kähler hyperbolicity developed. It is an open problem posed by J. Kollár in [Kol95]. Kollár suggests to require Gromov's condition for a degenerated Kähler form, and [BDET22, BCDT24] introduce the weakly Kähler hyperbolicity by asking the cohomology class to be nef and big rather than Kähler. Weakly Kähler hyperbolic manifolds possess many key features as Kähler hyperbolic manifolds, and are invariant under the birational transform. Whereas in this paper, we are trying to investigate a more general situation, namely the balanced hyperbolicity.More precisely, let X be a compact complex manifold of dimension n. A Hermitian metric ω on X is called balanced if dω n−1 = 0. X is called a balanced manifold if it possesses a balanced metric. Obviously a Kähler metric must be balanced, but there do exist non-Kähler balanced metrics. Hence a balanced form is also regarded as a degenerated Kähler form in some aspects. A celebrated theorem in [AB95] asserts that the class of compact balanced manifolds is invariant under the birational transform, which directly inspires this paper. Now, let π : X → X be the universal cover, and fix a Riemannian metric g on X. Recall that a k-form α on X is called d-bounded, if there exists a (k − 1)-form β on X such that π * α = dβ and sup X β π * g < ∞. Note since X is compact, this notion is actually independent of the choice of g. We say that a Hermitian metric ω on X is balanced hyperbolic if ω is balanced and ω n−1 is d-bounded. It is notable that the balanced hyperbolicity was first introduced in [MP22, MP23], and we will continue this streamline to make a discussion in a wider range.Observe that if ω is balanced, [ω n−1 ] must be nef and big as an (n−1, n−1)-class. It allows us to talk about more degenerated cases. We say that a real smooth (1, 1)-form ω is semi-balanced hyperbolic if ω n−1 is d-closed, non-negative, strictly positive on a Zariski open set, and d-bounded. Fix an integer k | (n − 1), say kt = n − 1. We say that a real smooth (k, k)-form β is weakly

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On genus one mirror symmetry in higher dimensions and the BCOV conjectures

Cited by 6 publications

References 106 publications

BCOV invariant and blow-up

BCOV invariant and blow-up

Analytic torsion for log-Enriques surfaces and Borcherds product

Motivic integration and birational invariance of BCOV invariants

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