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Ken-Ichi Yoshikawa

Publications and source records attributed to Ken-Ichi Yoshikawa.

17 recordsLinked to original sources
arXiv2026

Degeneration of Riemann surfaces and small eigenvalues of the Laplacian

For a one-parameter degeneration of compact Riemann surfaces endowed with the Kähler metric induced from the Kähler metric on the total space of the family, we determine the exact magnitude of the small eigenvalues of the Laplacian as a function on the parameter space, under the assumption that the singular fiber is reduced. The novelty in our approach is that we compute the asymptotic behavior of certain difference of (logarithm of) analytic torsions in the degeneration in two ways. On the one hand, via heat kernel estimates, it is shown that the leading asymptotic is determined by the product of the small eigenvalues. On the other hand, using Quillen metrics, the leading asymptotic is connected with the period integrals, which we explicitly evaluate.

math.DG↗
arXiv2026

On the behavior of analytic torsion for twisted canonical bundles under degenerations

Consider a degeneration of projective algebraic manifolds equipped with a compact group action over a curve. Suppose that the total space carries a Nakano semi-positive vector bundle, which is equivariant with respect to this action. We consider the relative canonical bundle twisted by this bundle. Under this setting, we prove that the logarithm of the equivariant analytic torsion of the regular fibers for this coefficient admits an asymptotic expansion near the discriminant locus. The leading term is given by a logarithmic singularity, while the subdominant term is given by a loglog-type singularity. In the non-equivariant case, we provide a formula for the coefficient of the leading term in terms of an integral of characteristic classes associated with the semi-stable reduction of the family. To establish these results, we prove the existence of an asymptotic expansion for both the equivariant Quillen metrics and the $L^{2}$-metrics in the above setting. We also calculate the leading term of the fiber integral of the Bott-Chern classes associated with the degeneration.

math.AG↗
arXiv2021

j-invariant and Borcherds Phi-function

We give a formula that relates the difference of the j-invariants with the Borcherds Phi-function, an automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor.

math.AG↗
arXiv2020

Analytic torsion for log-Enriques surfaces and Borcherds product

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 %of dimension $10-k$ associated with a unimodular lattice of signature $(2,10-k)$. We prove that the invariant, viewed as a function on 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.

math.DG↗
arXiv2018

K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space IV: the structure of invariant

A holomorphic torsion invariant of K3 surfaces with involution was introduced by the second-named author. In this paper, we completely determine its structure as an automorphic function on the moduli space of such K3 surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds' conjecture.

math.AG↗
arXiv2016

Analytic torsion for Borcea-Voisin threefolds

In their study of genus-one string amplitude, Bershadsky-Cecotti-Ooguri-Vafa discovered a remarkable identification between holomorphic Ray-Singer torsion and instanton numbers for Calabi-Yau threefolds. The holomorphic torsion invariant for Calabi-Yau threefolds corresponding to the genus-one string amplitude is called BCOV invariant. In this paper, we establish an identification between the BCOV invariants of Borcea-Voisin threefolds and another holomorphic torsion invariants for K3 surfaces with involution. We also introduce BCOV invariants for abelian Calabi-Yau orbifolds. Between Borcea-Voisin orbifold and its crepant resolution, we compare their BCOV invariants.

math.AG↗
arXiv2014

Degenerations of Calabi-Yau threefolds and BCOV invariants

In their papers published in 1993 and 1994, by expressing certain physical quantity 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 a paper published in 2008, a holomorphic torsion invariant for Calabi-Yau threefolds corresponding to the physical quantity was constructed and is called BCOV invariant. 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.

math.AG↗
arXiv2013

Resultants and the Borcherds Phi-function

The Borcherds Phi-function is the automorphic form on the moduli space of Enriques surfaces characterizing the discriminant locus. In this paper, we give an algebro-geometric construction of the Borcherds Phi-function.

math.AG↗
arXiv2010

K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space II: a structure theorem for r(M)>10

We study the structure of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion. It was known before that the invariant is expressed as the Petersson norm of an automorphic form on the moduli space. When the rank of the invariant sublattice of the K3-lattice with respect to the involution is strictly bigger than 10, we prove that this automorphic form is expressed as the tensor product of an explicit Borcherds lift and Igusa's Siegel modular form.

math.AG↗
arXiv2010

Singularities and analytic torsion

We prove the logarithmic divergence of equivariant analytic torsion for one-parameter degenerations of projective algebraic manifolds, when the coefficient vector bundle is given by a Nakano semi-positive vector bundle twisted by the relative canonical bundle.

math.AG↗
arXiv2010

On the boundary behavior of the curvature of L2-metrics

For one-parameter degenerations of compact Kähler manifolds, we determine the asymptotic behavior of the first Chern form of the direct image of a Nakano semi-positive vector bundle twisted by the relative canonical bundle, when the direct image is equipped with the L2-metric.

math.AG↗
arXiv2006

Analytic torsion for Calabi-Yau threefolds

After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau threefolds, which we call the BCOV invariant and which we obtain using analytic torsion. We give an explicit formula for the BCOV invariant as a function on the compactified moduli space, when it is isomorphic to a projective line. As a corollary, we prove the formula for the BCOV invariant of quintic mirror threefolds conjectured by Bershadsky-Cecotti-Ooguri-Vafa.

math.DG↗
arXiv2006

Real $K3$ surfaces without real points, equivariant determinant of the Laplacian, and the Borcherds Phi-function

We consider an equivariant analogue of a conjecture of Borcherds. Let $Y$ be a real $K3$ surface without real points. Let $g$ be a Ricci-flat Kaehler metric on $Y$ invariant under the complex conjugation. We shall prove that the equivariant determinant of the Laplacian of $(Y,g)$ with respect to the complex conjugation is expressed as the norm of the Borcherds Phi-function at the "period point". Here the period is not the one in algebraic geometry.

math.DG↗
arXiv1999

K3 Surfaces with Involution and Analytic Torsion

This is the abstruct of the revised paper. We study the equivariant analytic torsion for K3 surfaces with an anti-symplectic involution with the invariant lattice M (such a surface is called a 2-elementary K3 surface of type M in this paper), and show that it (together with the analytic torsion of the fixed curves) can be identified with the automorphic form on the moduli space characterizing the discriminant locus. Three lattices A_1, II_{1,1}(2), II_{1,9}(2) are of particular interest, because they consist of the building blocks of 2-elementary lattices. An explicit formula is given for them. In particular, if M is twice the Enriques lattice, the automorphic form coincides with Borcherds's Phi-function which confirms an observation by Jorgenson-Todorov and Harvey-Moore. Some other examples are shown to be related to Borcherds's product and generalized Kac-Moody algebras.

math.AG↗