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.
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' conject…
We study the boundary behavior of the invariant of $K3^{[2]}$-type manifolds with antisymplectic involution, which we obtained using equivariant analytic torsion. We show the algebraicity of the singularity of the invariant by using the asymptotic of equivariant Quillen metrics and equivariant $L^2$-metrics. We prove that, in some cases, the invariant coincides with Yoshikawa's invariant for 2-elementary K3 surfaces. Hence, in these cases, our invariant is expressed as the Petersson norm of a Borcherds product and a Siegel modular form.
For the Grothendieck group of a split simple linear algebraic group, the twisted gamma-filtration provides a useful tool for constructing torsion elements in gamma-rings of twisted flag varieties. In this paper, we construct a non-trivial torsion element in the gamma-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline. We use this torsion element to study algebras with orthogonal involutions.
We prove the automorphic property of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion, in the case where the dimension of the moduli space is less than or equal to 2.
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.
In this paper we classify the elliptic fibrations on K3 surfaces which are the double cover of a blow up of $\mathbb{P}^2$ branched along rational curves and we give equations for many of these elliptic fibrations. Thus we obtain a classification of the van Geemen--Sarti involutions (which are symplectic involutions induced by a translation by a 2-torsion section on an elliptic fibration) on such a surface. Each van Geemen--Sarti involution induces a 2-isogeny between two K3 surfaces, which is described in this paper.
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-V…
The moduli space $M$ of semi-stable rank 2 bundles with trivial determinant over a complex curve carries involutions naturally associated to 2-torsion points on the Jacobian of the curve. For every lift of a 2-torsion point to a 4-torsion point, we define a lift of the involution to the determinant line bundle $Ł$. We obtain an explicit presentation of the group generated by these lifts in terms of the order 4 Weil pairing. This is related to the triple intersections of the components of the fixed point sets in $M$, which we also determine completely using the order 4 Weil pairing. The lifted …
Minimal algebraic surfaces of general type with the smallest possible invariants have geometric genus zero and K^2=1 and are usually called "numerical Godeaux surfaces". Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical Godeaux surfaces with an involution, i.e. with an automorphism of order 2. We prove that they are birationally equivalent either to double covers of Enriques surfaces, or to double planes of two different types: the branch curve either has degree 10 and suitable singularities, originally suggested…
We give a simple axiomatic description of the degree 0 part of the polylogarithm on abelian schemes and show that its realisation in analytic Deligne cohomology can be described in terms of the Bismut-Köhler higher analytic torsion form of the Poincaré bundle.
In this paper we generalized the variational formulas for the determinants of the Laplacians on functions of CY metrics to forms of type (0,q) on CY manifolds. We also computed the Ray Singer Analytic torsion on CY manifolds we proved that it is bounded by a constant. In case of even dimensional CY manifolds the Ray Singer Analytic torsion is zero. The interesting case is the odd dimensional one.
We prove that Godeaux--Reid surfaces with torsion group Z/3 have topological fundamental group Z/3. For this purpose, we describe degenerations to stable KSBA surfaces with one 1/4(1,1) singularity, whose minimal resolution are elliptic fibrations with two multiplicity 3 fibres and one I_4 singular fibre. We study special such degenerations which have an involution, describing the corresponding Campedelli double plane construction. We also find some stable rational degenerations, some of which have more singularities, and one of which has a single 1/9(1,2) singularity, the minimal possible ind…
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.
We extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for arbitrary projective morphisms.
Using different forms of the arithmetic Riemann-Roch theorem and the computations of Bott-Chern secondary classes, we compute the analytic torsion and the height of Hirzebruch surfaces.
The purpose of this paper is first to give an asymptotic formula for the holomorphic analytic torsion forms of a fibration associated with increasing powers of a given line bundle. Secondly, we generalize this formula, thanks to the theory of Toeplitz operators, in the case where the powers of the line bundle is replaced by the direct image of powers of a line bundle on a bigger manifold. In both cases we have to make fiberwise positivity assumption on the line bundle. This results are the family versions of the results of Bimsut and Vasserot on the asymptotic of the holomorphic torsion.
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.
Given a choice of metric on the Riemann surface, the regularized determinant of Laplacian (analytic torsion) is defined via the complex power of elliptic operators: $$ \det(Δ)=\exp(-ζ'(0)) $$ In this paper we gave an asymptotic effective estimate of analytic torsion under Arakelov metric. In particular, after taking the logarithm it is asymptotically upper bounded by $g$ for $g>1$. The construction of a cohomology theory for arithmetic surfaces in Arakelov theory has long been an open problem. In particular, it is not known if $h^{1}(X,L)\ge 0$. We view this as an indirect piece of evidence th…
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, th…
Spectrum of singularities, analytic torsion and Calabi--Yau degenerations
Chalmers University of Technology and University of Gothenburg（查尔姆斯理工大学和哥德堡大学）；CNRS – Centre de Mathématiques Laurent Schwartz - École Polytechnique - Institut Polytechnique de Paris（法国国家科学研究中心-洛朗·施瓦茨数学中心-巴黎综合理工学院-巴黎理工学院）
