### ?

## Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac–Moody algebras

Transactions of the Moscow Mathematical Society. 2017. Vol. 78. P. 75-83.

Gritsenko V., Nikulin V. V.

Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant.

Keywords: moduli spaceautomorphic formsBorcherds productsпроизведения Борчердсамодули поляризованных К3 поверхностейавтоморфные дискриминантыK3-поверхность

Publication based on the results of:

Gritsenko V., Ванг Х., Успехи математических наук 2017 Т. 72 № 5 С. 191-192

In this paper we prove the conjecture above in the last case of known theta-blocks of weight 2. This gives a new intereting series of Borcherds products of weight 2. ...

Added: October 11, 2017

Gritsenko V., Никулин В. В., TRANSACTIONS OF THE MOSCOW MATHEMATICAL SOCIETY 2017 Т. 78 № 1 С. 89-100

Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant. ...

Added: October 11, 2017

Gritsenko V., Poor C., Yuen D. S., Journal of Number Theory 2015 Vol. 148 P. 164-195

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 ...

Added: February 26, 2015

Gritsenko V., Nikulin V., Proceedings of the London Mathematical Society 2018 Vol. 116 No. 3 P. 485-533

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 ...

Added: October 23, 2017

Bruzzo U., Markushevich D., Tikhomirov A. S., European Journal of Mathematics 2016 Vol. 2 P. 73-86

We study the moduli space $I_{n,r}$In,r of rank-2r symplectic instanton vector bundles on $\mathbb{P}^3$ℙ3 with $r\ge 2$r⩾2 and second Chern class $n\ge r+1, n-r\equiv 1(\mathrm{mod} 2)$n⩾r+1,n−r≡1(mod2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus $I_{n,r}^*$I∗n,r of tame symplectic instantons is irreducible and has the expected dimension equal to ...

Added: December 28, 2015

Gritsenko V., Poor C., Yuen D. S., International Mathematics Research Notices 2020 Vol. 2020 No. 20 P. 6926-6946

We define an algebraic set in 23-dimensional projective space whose 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 2 paramodular groups. Weight 3examples provide antisymmetric canonical differential forms on ...

Added: October 29, 2019

Buryak A., Netser Zernik A., Pandharipande R. et al., Advances in Mathematics 2022 Vol. 401 Article 108249

We define stationary descendent integrals on the moduli space of stable maps from disks to (CP^1,RP^1). We prove a localization formula for the stationary theory involving contributions from the fixed points and from all the corner-strata. We use the localization formula to prove a recursion relation and a closed formula for all genus 0 disk cover ...

Added: September 14, 2022

Alexandrov A., Basalaev A., Buryak A., International Mathematics Research Notices 2023 Vol. 2023 No. 17 P. 14840-14889

We present a construction of an open analogue of total descendant and total ancestor potentials via an “open version” of Givental’s action. Our construction gives a genus expansion for an arbitrary solution to the open WDVV equations satisfying a semisimplicity condition and admitting a unit. We show that the open total descendant potentials we define ...

Added: September 14, 2022

Gritsenko V., Wang H., Proceedings of the American Mathematical Society 2020 Vol. 148 P. 1863-1878

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. ...

Added: October 29, 2019

Sakharova N., / Cornell University. Series math "arxiv.org". 2018. No. 1802.03299.

In this paper we construct the modular Cauchy kernel $\Xi_N(z_1, z_2)$, i.e. the modular invariant function of two variables, $(z_1, z_2) \in \mathbb{H} \times \mathbb{H}$, with the first order pole on the curve $$D_N=\left\{(z_1, z_2) \in \mathbb{H} \times \mathbb{H}|~ z_2=\gamma z_1, ~\gamma \in \Gamma_0(N) \right\}.$$
The function $\Xi_N(z_1, z_2)$ is used in two cases and for ...

Added: February 23, 2018

Buryak A., Hernandez Iglesias F., Shadrin S., Epijournal de Geometrie Algebrique 2022 Vol. 6 Article 8595

We propose a conjectural formula for DR_g(a,−a)\lambda_g and check all its expected properties. Our formula refines the one point case of a similar conjecture made by the first named author in collaboration with Guéré and Rossi, and we prove that the two conjectures are in fact equivalent, though in a quite non-trivial way. ...

Added: September 14, 2022

Gritsenko V., Cléry F., Proceedings of the London Mathematical Society 2011 Vol. 102 No. 6 P. 1024-1052

We prove that there exist exactly eight Siegel modular forms with respect to the congruence subgroups of Hecke type of the paramodular groups of genus two vanishing precisely along the diagonal of the Siegel upper half-plane. This is a solution of a question formulated during the conference "Black holes, Black Rings and Modular Forms" (ENS, ...

Added: March 3, 2015

Gorsky E., Mathematical Research Letters 2009 Vol. 16 No. 4 P. 591-603

The generating function for Sn-equivariant Euler characteristics of moduli spaces of pointed hyperelliptic curves for any genus g ≥ 2 is calculated. This answer generalizes the known ones for genera 2 and 3 and the answers obtained by J. Bergstro ̈m for any genus and n ≤ 7 points. ...

Added: December 9, 2014

Tikhomirov A. S., Bruzzo U., Markushevich D., Central European Journal of Mathematics 2012 Vol. 10 No. 4 P. 1232-1245

Symplectic instanton vector bundles on the projective space $\mathbb{P}^3$ constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space $I_{n;r}$ of rank-$2r$ symplectic instanton vector bundles on $\mathbb{P}^3$ with $r\ge2$ and second Chern class $n\ge r, n\equiv r(\mod 2)$. We introduce the notion of tame symplectic instantons by excluding a kind ...

Added: October 21, 2014

Sakharova N., European Journal of Mathematics 2019 Vol. 5 No. 2 P. 528-539

The goal of this work is to present an alternative way of calculating the values of the Green's function associated with the Hirzebruch-Zagier divisor at the points of another Hirzebruch-Zagier divisor $T_m$. ...

Added: November 23, 2018

Moduli of symplectic instanton vector bundles of higher rank on projective space $\mathbb{P^3}$. II.

Tikhomirov A. S., Bruzzo U., Markushevich D., / Max Planck Institute for Mathematics. Series MPIM "MPIM". 2014. No. 2014-22.

Symplectic instanton vector bundles on the projective space $\mathbb{P^3}$ are a natural generalization of mathematical instantons of rank 2. We study the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vector bundles on $\mathbb{P^3}$ with $r\ge2$ and second Chern class $n\ge r+1,\ n-r \equiv 1(\mod2)$. We introduce the notion of tame symplectic instantons by excluding a ...

Added: October 19, 2014

Gritsenko V., Ванг Х., Математический сборник 2019 Т. 210 № 12 С. 43-66

Задача о построении антисимметричных парамодульных форм канонического веса 3 была поставлена в 1996 году. Любая параболическая форма этого типа определяет каноническую диф- ференциальную форму на любой гладкой компактификации про- странство модулей куммеровых поверхностей, отвечающих (1,t)- поляризованным абелевым поверхностям. В этой статье мы строим первое бесконечное семейство антисимметричных парамодулярных форм веса 3 как автоморфные произведения Борчердса, чьи пер- вые коэффициенты Фурье–Якоби ...

Added: October 29, 2019

Buryak A., Rossi P., Bulletin of the London Mathematical Society 2021 Vol. 53 No. 3 P. 843-854

In this paper we compute the intersection number of two double ramification (DR) cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic DR integrals are the main ingredients for the computation of the DR hierarchy associated to the ...

Added: February 1, 2021

Springer Publishing Company, 2020

This book offers an introduction to the research in several recently discovered and actively developing mathematical and mathematical physics areas. It focuses on: 1) Feynman integrals and modular functions, 2) hyperbolic and Lorentzian Kac-Moody algebras, related automorphic forms and applications to quantum gravity, 3) superconformal indices and elliptic hypergeometric integrals, related instanton partition functions, 4) ...

Added: September 9, 2020

Васильев Д. А., Siberian Mathematical Journal 2023 Vol. 64 No. 3 P. 525-541

We construct an infinite series of irreducible components of the moduli space of stable rank 3 sheaves on P3 with the zero first Chern class and establish the rationality of the components of this series. We also prove the rationality of the irreducible components of the moduli space of stable rank 2 sheaves on P3 belonging to an infinite subseries of the series ...

Added: May 29, 2023

Arsie A., Buryak A., Lorenzoni P. et al., Communications in Mathematical Physics 2023 Vol. 397 P. 141-197

In this paper, we generalize the Givental theory for Frobenius manifolds and cohomological field theories to flat F-manifolds and F-cohomological field theories. In particular, we define a notion of Givental cone for flat F-manifolds, and we provide a generalization of the Givental group as a matrix loop group acting on them. We show that this action is transitive on semisimple flat F-manifolds. We then extend this ...

Added: December 8, 2022

Брауэр О., Buryak A., Функциональный анализ и его приложения 2021 Т. 55 № 4 С. 22-39

In a recent paper, given an arbitrary homogeneous cohomological field theory (CohFT), Rossi, Shadrin, and the first author proposed a simple formula for a bracket on the space of local functionals, which conjecturally gives a second Hamiltonian structure for the double ramification hierarchy associated to the CohFT. In this paper we prove this conjecture in ...

Added: September 14, 2022

Gritsenko V., Wang H., Journal of Number Theory 2020 Vol. 214 P. 382-398

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 ...

Added: October 26, 2020

Buryak A., Gubarevich D., Mathematical Physics Analysis and Geometry 2023 Vol. 26 No. 3 Article 23

One of many manifestations of a deep relation between the topology of the moduli spaces of algebraic curves and the theory of integrable systems is a recent construction of Arsie, Lorenzoni, Rossi, and the first author associating an integrable system of evolutionary PDEs to an F-cohomological field theory (F-CohFT), which is a collection of cohomology ...

Added: November 20, 2023