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## Modified Elliptic Genus

P. 87-119.

Abstract This mini course is an additional part to my semester course on the theory of Jacobi modular forms given at the mathematical department of NRU HSE in Moscow (see Gritsenko Jacobi modular forms: 30 ans après; COURSERA (12 lectures and seminars), 2017–2019). This additional part contains some applications of Jacobi modular forms to the theory of elliptic genera and Witten genus. The subject of this course is related to my old talk given in Japan (see Gritsenko (Proc Symp “Automorphic forms and L-functions” 1103:71–85, 1999)).

Publication based on the results of:

Drozd Y., Gavran V., Central European Journal of Mathematics 2014 Vol. 12 No. 5 P. 675-687

We generalize the results of Kahn about a correspondence between Cohen-Macaulay modules and vector bundles to non-commutative surface singularities. As an application, we give examples of non-commutative surface singularities which are not Cohen-Macaulay finite, but are Cohen-Macaulay tame. ...

Added: May 14, 2018

Gritsenko V., Wang H., European Journal of Mathematics 2018 Vol. 4 No. 2 P. 561-584

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
$\Delta(\tau)=\eta^{24}(\tau)$, $\eta^{12}(\tau)$ and $\eta^{8}(\tau)$ in terms ...

Added: October 11, 2017

Adler D., Gritsenko V., Journal of Geometry and Physics 2020 Vol. 150 P. 103616

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) ...

Added: November 1, 2019

Tikhomirov A. S., Markushevich D., Trautmann G., Central European Journal of Mathematics 2012 Vol. 19 No. 4 P. 1331-1355

We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundled connections an in differential geometry. The new moduli spaces are algebraic spaces arising as quotients ...

Added: October 21, 2014

Valery Gritsenko, Wang H., Russian Mathematical Surveys 2017 Vol. 72 No. 5 P. 968-970

In this paper we prove the indicated conjecture in the last case of known infinite series of theta-blocks of weight 2. ...

Added: January 29, 2018

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

Solomadin G., Journal of the Mathematical Society of Japan 2020 Vol. 3 P. 765-776

In the present paper we construct two new explicit complex bordisms between any two projective bundles over $\mathbb{C} P^1$ of the same complex dimension, including the Milnor hypersurface $H_{1,n}$ and $\mathbb{C} P^1\times \mathbb{C} P^{n-1}$. These constructions reduce the bordism problem to the null-bordism of some projective bundle over $\mathbb{C} P^1$ with the non-standard stably complex ...

Added: September 20, 2021

Bogomolov F. A., Lukzen E., / Cornell University. Series arXiv "math". 2020.

We offer a new approach to proving the Chen-Donaldson-Sun theorem which we demonstrate with a series of examples. We discuss the existence of a construction of a special metric on stable vector bundles over the surfaces formed by a families of curves and its relation to the one-dimensional cycles in the moduli space of stable ...

Added: October 27, 2020

Adler D., / Cornell University. Series math "arxiv.org". 2020.

We prove the polynomiality of the bigraded ring J_{*,*}^{w, O}(F_4) of weak Jacobi forms for the root system F_4 which are invariant with respect to the corresponding Weyl group. This work is a continuation of the joint article with V.A. Gritsenko, where the structure of algebras of the weak Jacobi forms related to the root systems ...

Added: September 24, 2020

Gritsenko V., Cléry F., Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 2013 Vol. 83 No. 2 P. 187-217

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 ...

Added: February 26, 2015

Adler D., Gritsenko V., / Cornell University. Series math "arxiv.org". 2019.

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) ...

Added: November 5, 2019

Adler D., Gritsenko V., Journal of Geometry and Physics 2020

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) ...

Added: October 30, 2019

Adler D., Gritsenko Valery, Journal of Geometry and Physics 2023 Vol. 194 Article 104995

We study modular differential equations (MDEs) of the elliptic genus of four-dimensional complex varieties with trivial first Chern class. We construct modular differential equations of orders 3, 4, 5 and 6 with respect to the heat operator for every weak Jacobi form of weight 0 and index 2. We prove that the elliptic genus of a Calabi–Yau ...

Added: October 24, 2023

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

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

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

Adler D., Функциональный анализ и его приложения 2020 Т. 54 № 3 С. 8-25

We prove the polynomiality of the bigraded ring $J_{*,*}^{w, W}(F_4)$ of weak Jacobi forms for the root system $F_4$ which are invariant with respect to the corresponding Weyl group. This work is a continuation of the joint article with V.A. Gritsenko, where the structure of algebras of the weak Jacobi forms related to the root ...

Added: November 6, 2020