On the singularity of Quillen metrics

Yoshikawa, Kenichi

doi:10.1007/s00208-006-0027-5

Cited by 16 publications

(26 citation statements)

References 19 publications

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“…• The metric on a Deligne pairing of line bundles over a smooth variety has some regularity as the variety degenerates [12]. • The Quillen metric for hermitian vector bundles over a smooth variety can be controlled as the variety degenerates [3,15].…”

Section: Deligne Pairing and Quillen Metricmentioning

confidence: 99%

Deligne pairing and Quillen metric

Biswas

Schumacher

2014

Int. J. Math.

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Let X → S be a smooth projective surjective morphism of relative dimension n, where X and S are integral schemes over C. Let L → X be a relatively very ample line bundle. For every sufficiently large positive integer m, there is a canonical isomorphism of the Deligne pairing L, . . . , L → S with the determinant line bundle Det((L − O X ) ⊗(n+1) ⊗ L ⊗m ) (see [D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the knudsen-Mumford expansion, J. Differential Geom. 78 (2008) 475-496]). If we fix a hermitian structure on L and a relative Kähler form on X, then each of the line bundles Det((L − O X ) ⊗(n+1) ⊗ L ⊗m ) and L, . . . , L carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between L, . . . , L → S and Det((L−O X ) ⊗(n+1) ⊗L ⊗m ) is compatible with these hermitian structures. This holds also for the isomorphism in [Deligne pairing and determinant bundle, Electron. Res. Announc. Math. Sci. 18 (2011) 91-96] between a Deligne paring and a certain determinant line bundle.

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Section: Deligne Pairing and Quillen Metricmentioning

confidence: 99%

Deligne pairing and Quillen metric

Biswas

Schumacher

2014

Int. J. Math.

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“…where the second equality follows from (3.14) and the third equality follows from [18,Cor. 4.6] and the equalities of divisors div(σ * Ξ) = σ * K (X ,X0) , Σ f = div(d f ) on X .…”

Section: 7mentioning

confidence: 99%

“…In this direction, in [7], the following results were obtained as an application of the theory of Quillen metrics [5], [3], [18]: log τ BCOV always has logarithmic singularity for arbitrary algebraic one-parameter degenerations of Calabi-Yau threefolds and the logarithmic singularity of log τ BCOV is determined for smoothings of Calabi-Yau varieties with at most one ordinary double point under an additional assumption of the dimension of moduli space. These results, together with the formula for dd c log τ BCOV and the known boundary behaviors of Weil-Petersson and its Ricci forms, are sufficient to determine τ BCOV for quintic mirror threefolds [7].…”

Section: Introductionmentioning

confidence: 99%

Degenerations of Calabi–Yau threefolds and BCOV invariants

Yoshikawa

2015

Int. J. Math.

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anomalies in topological field theories, Nucl. Phys. B 405 (1993) 279-304; M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311-427], by expressing the physical quantity F 1 in two distinct ways, Bershadsky-Cecotti-Ooguri-Vafa discovered a remarkable equivalence between Ray-Singer analytic torsion and elliptic instanton numbers for Calabi-Yau threefolds. After their discovery, in [H. Fang, Z. Lu and K.-I. Yoshikawa, Analytic torsion for Calabi-Yau threefolds, J. Differential Geom. 80 (2008) 175-250], a holomorphic torsion invariant for Calabi-Yau threefolds corresponding to F 1 , called BCOV invariant, was constructed. In this article, we study the asymptotic behavior of BCOV invariants for algebraic one-parameter degenerations of Calabi-Yau threefolds. We prove the rationality of the coefficient of logarithmic divergence and give its geometric expression by using a semi-stable reduction of the given family.

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“…A Weil–Petersson form for the space of compact submanifolds of a Kähler manifold was constructed in [10], and further results given in [2]. We will use Yoshikawa's theorem [33] on the singularities of Quillen metrics in an essential way.…”

Section: Introductionmentioning

confidence: 99%

“…The proof requires Yoshikawa's theorem about the singularities of Quillen metrics [33] for families over curves with smooth total space, which in our case allows logarithmic poles for

- \log h_{ξ}^{Q}

but no simple poles along the divisor B so that the current

T_{v}

from the previous theorem is not present.…”

Section: Introductionmentioning

confidence: 99%

The Weil–Petersson current on Douady spaces

Axelsson

Schumacher

2021

Mathematische Nachrichten

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The Douady space of compact subvarieties of a Kähler manifold is equipped with the Weil–Petersson current, which is everywhere positive with local continuous potentials, and of class C∞ when restricted to the locus of smooth fibers. There a Quillen metric is known to exist, whose Chern form is equal to the Weil–Petersson form. In the algebraic case, we show that the Quillen metric can be extended to the determinant line bundle as a singular hermitian metric. On the other hand the determinant line bundle can be extended in such a way that the Quillen metric yields a singular hermitian metric whose Chern form is equal to the Weil–Petersson current. We show a general theorem comparing holomorphic line bundles equipped with singular hermitian metrics which are isomorphic over the complement of a snc divisor B. They differ by a line bundle arising from the divisor and a flat line bundle. The Chern forms differ by a current of integration with support in B and a further current related to its normal bundle. The latter current is equal to zero in the case of Douady spaces due to a theorem of Yoshikawa on Quillen metrics for singular families over curves.

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On the singularity of Quillen metrics

Cited by 16 publications

References 19 publications

Deligne pairing and Quillen metric

Deligne pairing and Quillen metric

Degenerations of Calabi–Yau threefolds and BCOV invariants

The Weil–Petersson current on Douady spaces

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