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Valery Gritsenko

Publications and source records attributed to Valery Gritsenko.

At least 19 recordsLinked to original sources
arXiv2025

Modular differential equations of minimal orders of the elliptic genus of Calabi--Yau varieties

We study modular differential equations (MDEs) of high orders for weak Jacobi forms and find necessary conditions for weak Jacobi forms to satisfy MDEs of order 3 with respect to the heat operator. We investigate all possible MDEs for the elliptic genus of six-dimensional manifolds with a trivial first Chern class. We prove that the minimal possible order of the MDE for the elliptic genus of a strict six-dimensional Calabi--Yau variety is four, and find MDEs of order 7 for hyperkähler varieties of dimension 6. The latter MDEs correspond to the generic case. The non-generic weak Jacobi forms of weight 0 and index 3 form a divisor that contains two cubic plane curves in the coefficient space.

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arXiv2023

Orbits of Jacobi forms and Theta relations

Jacobi theta functions with rational characteristics can be viewed as vector-valued Jacobi forms. Theta relations usually correspond to different constructions of certain Jacobi forms. From this observation, we extract a new approach, which is called orbits of Jacobi forms, to produce identities on Jacobi theta functions. Our approach not only provides simple proofs of many known theta relations but also produces a large number of new identities, which can be considered as generalizations of Riemann's theta relations.

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arXiv2022

Elliptic genus and modular differential equations

We study modular differential equations for the basic weak Jacobi forms in one abelian variable with applications to the elliptic genus of Calabi--Yau varieties. We show that the elliptic genus of any $CY_3$ satisfies a differential equation of degree one with respect to the heat operator. For a $K3$ surface or any $CY_5$ the degree of the differential equation is $3$. We prove that for a general $CY_4$ its elliptic genus satisfies a modular differential equation of degree $5$. We give examples of differential equations of degree two with respect to the heat operator similar to the Kaneko--Zagier equation for modular forms in one variable. We find modular differential equations of Kaneko--Zagier type of degree $2$ or $3$ for the second, third and fourth powers of the Jacobi theta-series.

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arXiv2020

Graded rings of integral Jacobi forms

We determine the structure of the bigraded ring of weak Jacobi forms with integral Fourier coefficients. This ring is the target ring of a map generalising the Witten and elliptic genera and a partition function of $(0,2)$-model in string theory. We also determine the structure of the graded ring of all weakly holomorphic Jacobi forms of weight zero and integral index with integral Fourier coefficients. These forms are the data for Borcherds products for the Siegel paramodular groups.

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arXiv2019

The D_8-tower of weak Jacobi forms and applications

We construct a tower of arithmetic generators of the bigraded polynomial ring J_{*,*}^{w, O}(D_n) of weak Jacobi modular forms invariant with respect to the full orthogonal group O(D_n) of the root lattice D_n for 2\le n\le 8. This tower corresponds to the tower of strongly reflective modular forms on the orthogonal groups of signature (2,n) which determine the Lorentzian Kac-Moody algebras related to the BCOV (Bershadsky-Cecotti-Ooguri-Vafa)-analytic torsions. We prove that the main three generators of index one of the graded ring satisfy a special system of modular differential equations. We found also a general modular differential equation of the generator of weight 0 and index 1 which generates the automorphic discriminant of the moduli space of Enriques surfaces.

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arXiv2019

Theta block conjecture for paramodular forms of weight 2

In this paper we construct an infinite family of paramodular forms of weight $2$ which are simultaneously Borcherds products and additive Jacobi lifts. This proves an important part of the theta-block conjecture of Gritsenko--Poor--Yuen (2013) related to the only known infinite series of theta-blocks of weight $2$ and $q$-order $1$. We also consider some applications of this result.

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arXiv2019

Theta Blocks

We define theta blocks as products of Jacobi theta functions divided by powers of the Dedekind eta-function and show that they give a powerful new method to construct Jacobi forms and Siegel modular forms, with applications also in lattice theory and algebraic geometry. One of the central questions is when a theta block defines a Jacobi form. It turns out that this seemingly simple question is connected to various deep problems in different fields ranging from Fourier analysis over infinite-dimensional Lie algebras to the theory of moduli spaces in algebraic geometry. We give several answers to this question.

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arXiv2019

Antisymmetric paramodular forms of weight 3

The problem on the construction of antisymmetric paramodular forms of canonical weight 3 was open since 1998. Any cusp form of this type determines a canonical differential form on any smooth compactification of the moduli space of Kummer surfaces associated to $(1,t)$-polarised abelian surfaces. In this paper, we construct the first infinite family of antisymmetric paramodular forms of weight 3 as Borcherds products whose first Fourier-Jacobi coefficient is a theta block.

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arXiv2017

Powers of Jacobi triple product, Cohen's numbers and the Ramanujan $Δ$-function

We show that the eighth power of the Jacobi triple product is a Jacobi--Eisenstein series of weight $4$ and index $4$ and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of theta-constants of arbitrary order and the Fourier coefficients of the Ramanujan Delta-function $Δ(τ)=η^{24}(τ)$, $η^{12}(τ)$ and $η^{8}(τ)$ in terms of Cohen's numbers $H(3,N)$ and $H(5,N)$. We give new formulas for the number of representations of integers as sums of eight higher figurate numbers. We also calculate the sixteenth and the twenty-fourth powers of the Jacobi theta-series using the basic Jacobi forms.

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arXiv2016

Antisymmetric Paramodular Forms of Weights 2 and 3

We define an algebraic set in $23$~dimensional projective space whose $\mathbb Q$-rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on these lines give holomorphic Borcherds products and thus construct examples of Siegel modular forms on degree two paramodular groups. Weight $3$ examples provide antisymmetric canonical differential forms on Siegel modular threefolds. Weight $2$ is the minimal weight and these examples, via the Paramodular Conjecture, give evidence for the modularity of some rank one abelian surfaces defined over $\mathbb Q$.

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arXiv2016

Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections

We describe a new large class of Lorentzian Kac--Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices S with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac--Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac--Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2-reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings.

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arXiv2015

Moduli of polarized Enriques surfaces

In this paper we consider moduli spaces of polarized and numerically polarized Enriques surfaces. The moduli spaces of numerically polarized Enriques surfaces can be described as open subsets of orthogonal modular varieties of dimension 10. One of the consequences of our description is that there are only finitely many isomorphism classes of moduli spaces of polarized and numerically polarized Enriques surfaces. We use modular forms to prove for a number of small degrees that the Kodaira dimension of the moduli space of numerically polarized Enriques surfaces is negative. Finally we prove that there are infinitely many polarizatons for which the moduli space of numerically polarized Enriques surfaces is birational to the moduli space of unpolarized Enriques surfaces with a level 2 structure.

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arXiv2013

Borcherds Products Everywhere

We prove the Borcherds Products Everywhere Theorem, Theorem 6.6, that constructs holomorphic Borcherds Products from certain Jacobi forms that are theta blocks without theta denominator. The proof uses generalized valuations from formal series to partially ordered abelian semigroups of closed convex sets. We present nine infinite families of paramodular Borcherds Products that are simultaneously Gritsenko lifts. This is the first appearance of infinite families with this property in the literature.

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arXiv2012

24 faces of the Borcherds modular form Phi_{12}

The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of Phi_{12} in any of 23 Niemeier cusps is equal to the Weyl-Kac denominator function of the affine Lie algebra of the root system of the corresponding Niemeier lattice. This is an automorphic answer (in the case of the fake monster Lie algebra) on the old question of I. Frenkel and A. Feingold (1983) about possible relations between hyperbolic Kac-Moody algebras, Siegel modular forms and affine Lie algebras.

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arXiv2012

Uniruledness of orthogonal modular varieties

A strongly reflective modular form with respect to an orthogonal group of signature (2,n) determines a Lorentzian Kac--Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than n then the corresponding modular variety is uniruled. We also construct new reflective modular forms and thus provide new examples of uniruled moduli spaces of lattice polarised K3 surfaces. Finally we prove that the moduli space of Kummer surfaces associated to (1,21)-polarised abelian surfaces is uniruled.

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arXiv2011

Modular forms of orthogonal type and Jacobi theta-series

In this paper we consider Jacobi forms of half-integral index for any positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A_1=<2>). We give a lot of examples of Jacobi forms of singular and critical weights for root systems using Jacobi theta-series. We describe the Jacobi lifting for Jacobi forms of half-integral indices. In some case it gives additive lifting construction of new reflective modular forms.

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arXiv2010

Reflective modular forms in algebraic geometry

We prove that the existence of a strongly reflective modular form of a large weight implies that the Kodaira dimension of the corresponding modular variety is negative or, in some special case, it is equal to zero. Using the Jacobi lifting we construct three towers of strongly reflective modular forms with the simplest possible divisor. In particular we obtain a Jacobi lifting construction of the Borcherds-Enriques modular form Phi_4 and Jacobi liftings of automorphic discriminants of the Kähler moduli of Del Pezzo surfaces constructed recently by Yoshikawa. We obtain also three modular varieties of dimension 4, 6 and 7 of Kodaira dimension 0.

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