BCOV invariants of Calabi–Yau manifolds and degenerations of Hodge structures

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

doi:10.1215/00127094-2020-0045

Cited by 8 publications

(37 citation statements)

References 52 publications

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“…3.10] are explicitly stated for the whole Hodge bundles and describe the cokernels in terms of the semi-simple part of the monodromy acting on the limiting Hodge structure. For their minimal components, however, see Remark 2.7 (iii) in [EFiMM21] together with Proposition 3.13 and Lemma 4.2.…”

Section: Behaviour Of 𝜼 𝒌 At the Odp Pointsmentioning

confidence: 99%

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

Eriksson

Montplet

Mourougane

2022

Forum of Mathematics, Pi

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The mathematical physicists Bershadsky–Cecotti–Ooguri–Vafa (BCOV) proposed, in a seminal article from 1994, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck–Riemann–Roch theorem, we offer a mathematical description of the BCOV conjecture at genus one. As an application of the arithmetic Riemann–Roch theorem of Gillet–Soulé and our previous results on the BCOV invariant, we establish this conjecture for Calabi–Yau hypersurfaces in projective spaces. Our contribution takes place on the B-side, and together with the work of Zinger on the A-side, it provides the first complete examples of the mirror symmetry program in higher dimensions. The case of quintic threefolds was studied by Fang–Lu–Yoshikawa. Our approach also lends itself to arithmetic considerations of the BCOV invariant, and we study a Chowla–Selberg type theorem expressing it in terms of special $\Gamma $ -values for certain Calabi–Yau manifolds with complex multiplication.

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Section: Behaviour Of 𝜼 𝒌 At the Odp Pointsmentioning

confidence: 99%

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

Eriksson

Montplet

Mourougane

2022

Forum of Mathematics, Pi

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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: 99%

“…It is important to note that our torsion invariant is essentially the complex 2-dimensional analogue of the BCOV invariant (See [3], [19], [17], [20]). In higher dimensions, Bershadsky, Cecotti, Ooguri and Vafa [3] introduced a certain combination of holomorphic torsions, called the BCOV torsion, and predicted the mirror symmetry at genus one as an equivalence of the BCOV torsion and certain curve counting invariants at genus one.…”

Section: Introductionmentioning

confidence: 99%

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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“…BCOV torsion of quintic mirror threefolds. Eriksson, Freixas and Mourougane [15] extended these constructions to Calabi-Yau manifolds of arbitrary dimension.…”

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confidence: 99%

“…1 We will construct a real number τ (X, Y ) depending only on the complex structure of (X, Y ). For X Calabi-Yau, τ (X, ∅) is the logarithm of the BCOV invariant [15] of X.…”

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confidence: 99%

BCOV invariant for Calabi-Yau pairs

Zhang¹

2019

Preprint

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Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called BCOV invariant.We consider a pair (X, Y ), where X is a Kähler manifold and Y ⊆ X is a pluricanonical divisor. We will call such (X, Y ) a Calabi-Yau pair. In this paper, we extend the BCOV invariant to Calabi-Yau pairs. We show that the extended BCOV invariant for rigid del Pezzo surfaces is equivalent to Yoshikawa's equivariant BCOV invariant. We also study the behavior of the extended BCOV invariant under blowing up.We expect that these considerations may eventually lead to a positive answer to Yoshikawa's conjecture that the BCOV invariant for Calabi-Yau threefold is a birational invariant.

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BCOV invariants of Calabi–Yau manifolds and degenerations of Hodge structures

Cited by 8 publications

References 52 publications

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

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

Analytic torsion for log-Enriques surfaces and Borcherds product

BCOV invariant for Calabi-Yau pairs

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