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

Proceedings of the London Mathematical Society. 2018. Vol. 116. No. 3. P. 485–533.

Gritsenko V., Nikulin V.

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.

Keywords: автоморфные формыпространства модулейmoduli spacesKac-Moody algebrasалгебры Каца-Мудиautomorphic forms

Publication based on the results of:

Gritsenko V., Hulek K., / Cornell University. Series math "arxiv.org". 2015. No. 02723.

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 birational equivalence classes of moduli spaces ...

Added: February 20, 2015

Gritsenko V., Hulek K., Sankaran G., Compositio Mathematica 2010 Vol. 146 No. 2 P. 404–434

We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of 2d polarised (split type) symplectic manifolds which are deformation equivalent to degree 2 Hilbert schemes of a K3 surface is of general type if d ...

Added: March 3, 2015

Gritsenko V., Hulek K., Sankaran G., , in : Handbook of Moduli. Vol. I. Vol. I.: Boston: International Press of Boston Inc, 2013. P. 469–525.

The name "K3 surfaces" was coined by A. Weil in 1957 when he formulated a research programme for these surfaces and theirmoduli. Since then, irreducible holomorphic symplectic manifolds have been introduced as a higher dimensional analogue of K3 surfaces. In this paper we present a review of this theory starting from the definition of K3 ...

Added: March 3, 2015

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

Verbitsky M., Duke Mathematical Journal 2013 Vol. 162 No. 15 (2013) P. 2929–2986

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hyperkähler manifold $M$, showing that it is commensurable to an arithmetic lattice in SO(3, b_2-3). A Teichmüller space of $M$ is a space of complex structures on $M$ up ...

Added: December 10, 2013

Natanzon S. M., Pratoussevitch A., Russian Mathematical Surveys 2016 Vol. 71 No. 2 P. 382–384

In this paper, we present all higher spinor structures on Klein surfaces. We present also topological invariants that describe the connected components of moduli of Klein surfaces with higher spinor structure. Each connected component is represented as a cell factorable by a discrete group . ...

Added: March 25, 2016

Gorinov A., / Cornell University. Series math "arxiv.org". 2014. No. 1402.5946.

We present a modification of the method of conical resolutions \cite{quintics,tom}. We apply our construction to compute the rational cohomology of the spaces of equations of nodal cubics in CP2, nodal quartics in CP2 and nodal cubics in CP3. In the last two cases we also compute the cohomology of the corresponding moduli spaces. ...

Added: February 26, 2014

Finkelberg M. V., Rybnikov L. G., Journal of the European Mathematical Society 2012

algebra $\hat{sl}_n$. We introduce an affine, reduced, irreducible, normal quiver variety $Z$ which maps to the Zastava space bijectively at the level of complex points. The natural Poisson structure on the Zastava space can be described on $Z$ in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) ...

Added: February 19, 2013

Poberezhny V. A., / ИТЭФ. Series "Препринты ИТЭФ". 2013. No. 52/13.

We give an w\explicit example of non-regular behaviour of fuchsian systems moduli space in the case of resonant singular points. Tha set of systems with same singularities, asymptotics and monodromy but still not globally equivalent is constructed. ...

Added: March 31, 2014

Costa A., Gusein-Zade S., Natanzon S. M., Indiana University Mathematics Journal 2011 Vol. 60 No. 3 P. 985–995

Klein foams are analogues of Riemann and Klein surfaces with one-dimensional singularities. We prove that the field of dianalytic functions on a Klein foam Ω coincides with the field of dianalytic functions on a Klein surface K Ω We construct the moduli space of Klein foams, and we prove that the set of classes of ...

Added: November 24, 2012

Cherednik I., Feigin B. L., Advances in Mathematics 2013 Vol. 248 No. 25 P. 1050–1088

Using the DAHA-Fourier transform of q-Hermite polynomials multiplied by level-one theta functions, we obtain expansions of products of any number of such theta functions in terms of the q-Hermite polynomials. An ample family of modular functions satisfying Rogers-Ramanujan type identities for arbitrary (reduced, twisted) affine root systems is obtained as an application. A relation to ...

Added: September 30, 2013

Buryak A., Clader E., Tessler R., International Mathematics Research Notices 2022 Vol. 2022 No. 14 P. 10458–10532

We lay the foundation for a version of r-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define the notion of r-spin disks, their moduli space, and the Witten bundle; we show that the moduli space is a compact smooth orientable orbifold with corners, and we prove that the Witten bundle is canonically ...

Added: October 29, 2021

Иванов А. Н., Математический сборник 2020 Т. 211 № 7 С. 72–92

We construct a new infinite series of irreducible components of the Gieseker-Maruyama moduli scheme M(k), k≥3, of semistable rank-2 sheaves on P3 with Chern classes c1=0, c2=k and c3=0, whose general points are sheaves with singularities of mixed dimension. These sheaves are constructed by elementary transformations of stable and properly μ-semistable reflexive sheaves along disjoint unions of collections of points and smooth irreducible curves which ...

Added: October 11, 2021

Maxim Kazarian, Norbury P., International Mathematics Research Notices 2024 Vol. 2024 No. 3 P. 1825–1867

We construct an infinite collection of universal—independent of (g,n)—polynomials in the Miller–Morita–Mumford classes κm ∈ H2m(Mg,n, Q), defined over the moduli space of genus g stable curves with n labeled points. We conjecture vanishing of these polynomials in a range depending on g and n. ...

Added: February 19, 2024

Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al., Selecta Mathematica, New Series 2011 Vol. 17 No. 3 P. 573–607

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal ...

Added: October 9, 2012

Gritsenko V., Nikulin V., / Cornell University. Series arXiv "math". 2016.

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: March 17, 2016

Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al., Selecta Mathematica, New Series 2011 Vol. 17 No. 2 P. 337–361

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand-Tsetlin subalgebra of U(gln), and formulate a conjectural answer for the small quantum cohomology rings in terms of ...

Added: October 9, 2012

Finkelberg M. V., Rybnikov L. G., Algebraic Geometry 2014 Vol. 1 No. 2 P. 166–180

Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic 0. The natural Poisson structure on ...

Added: October 25, 2013

Buryak A., Tessler R., Communications in Mathematical Physics 2017 Vol. 353 No. 3 P. 1299–1328

In a recent work, R. Pandharipande, J. P. Solomon and the second author have initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. They conjectured that the generating series of the intersection numbers satisfies the open KdV equations. In this paper we prove this conjecture. Our proof goes ...

Added: September 27, 2020

Kazaryan M., Lando S., Moscow Mathematical Journal 2012 Vol. 12 No. 2 P. 397–411

Let Mg;n denote the moduli space of genus g stable algebraic curves with n marked points. It carries the Mumford cohomology classes ki. A homology class in H*(Mg;n) is said to be k-zero if the integral of any monomial in the k-classes vanishes on it. We show that any k-zero class implies a partial differential ...

Added: May 24, 2012

Lavrov A., Tikhomirov A. S., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2017 Т. 476 № 6 С. 617–620

Мы описываем новую неприводимую компоненту схемы модулей Гизекера-Маруямы M(3) полустабильных когерентных пучков ранга 2 с классами Черна c_1=0, c_2=3, c_3=0 на P3, общая точка которой соответствует пучку с множеством особенностей, содержащим компоненты размерностей 0 и 1. Эти пучки получаются с помощью элементарных преобразований стабильных рефлексивных пучков ранга 2 с классами Черна c_1=0, c_2=2, c_3=2 вдоль ...

Added: June 19, 2017

Jardim M., Maican M., Tikhomirov A. S., Pacific Journal of Mathematics 2017 Vol. 291 No. 2 P. 399–424

We study the irreducible components of the moduli space of instanton sheaves on P^3, that is, µ-semistable rank 2 torsion-free sheaves E with c_1(E)= c_3(E)=0 satisfying h^1(E(−2))= h^2(E(−2))=0. In particular, we classify all instanton sheaves with c_2(E) ≤4, describing all the irreducible components of their moduli space. A key ingredient for our argument is the ...

Added: September 20, 2017

Boston: International Press of Boston Inc, 2013

The Handbook of Moduli, comprising three volumes, offers a multi-faceted survey of a rapidly developing subject aimed not just at specialists but at a broad community of producers of algebraic geometry, and even at some consumers from cognate areas. The thirty-five articles in the Handbook, written by fifty leading experts, cover nearly the entire range of the field. They ...

Added: February 27, 2015

Kochetkov Y., / Cornell University Library. 2013. No. 1301.6059.

We consider the space $\mathcal{M}_{2,1}$ --- the open moduli space of complex curves of genus 2 with one marked point. Using language of chord diagrams we describe the cell structure of $\mathcal{M}_{2,1}$ and cell adjacency. This allows one to construct matrices of boundary operators and compute Betty numbers of $\mathcal{M}_{2,1}$ over $\mathbb{Q}$. ...

Added: February 24, 2013