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## Reflective modular forms in algebraic geometry

Cornell University
,
2010.
No. 3753.

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 K\"ahler 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.

Buryak A., Letters in Mathematical Physics 2015 Vol. 105 No. 10 P. 1427–1448

In a recent paper R. Pandharipande, J. Solomon and R. Tessler initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. The authors conjectured KdV and Virasoro type equations that completely determine all intersection numbers. In this paper we study these equations in detail. In particular, we prove that ...

Added: September 29, 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

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

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

Buryak A., Guere J., Journal de Mathématiques Pures et Appliquées 2016 Vol. 106 No. 5 P. 837–865

The double ramification hierarchy is a new integrable hierarchy of hamiltonian PDEs introduced recently by the first author. It is associated to an arbitrary given cohomological field theory. In this paper we study the double ramification hierarchy associated to the cohomological field theory formed by Witten's r-spin classes. Using the formula for the product of ...

Added: September 28, 2020

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

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

Buryak A., Moscow Mathematical Journal 2016 Vol. 16 No. 1 P. 27–44

Recently R. Pandharipande, J. Solomon and R. Tessler 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 is a specific solution of a system of PDEs, that they called the open KdV equations. In this paper we show ...

Added: September 28, 2020

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

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

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

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

Gritsenko V., Russian Mathematical Surveys 2018 Vol. 73 No. 5 P. 797–864

The reflective modular forms of orthogonal type are fundamental automorphic objects generalizing the classical Dedekind eta-function. This article describes two methods for constructing such modular forms in terms of Jacobi forms: automorphic products and Jacobi lifting. In particular, it is proved that the first non-zero Fourier–Jacobi coefficient of the Borcherds modular form Φ12 (the generating ...

Added: October 29, 2019

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

Alexandrov A., Buryak A., Tessler R., Journal of High Energy Physics 2017 Vol. 2017 No. 123 P. 123

A study of the intersection theory on the moduli space of Riemann surfaces with boundary was recently initiated in a work of R. Pandharipande, J. P. Solomon and the third author, where they introduced open intersection numbers in genus 0. Their construction was later generalized to all genera by J. P. Solomon and the third ...

Added: September 27, 2020

Gritsenko V., Rendiconti del Seminario Matematico Università e Politecnico di Torino 2010 Vol. 68 No. 3 P. 298–299

We show how to analyse the cusp forms of small weights for the moduli spaces of polarised K3 surfaces with these degrees 2d. ...

Added: March 3, 2015

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

Gritsenko V., Hulek K., Journal of Algebraic Geometry 2014 Vol. 23 No. 4 P. 711–725

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 ...

Added: February 26, 2015

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

Buryak A., Clader E., Tessler R., Journal of Geometry and Physics 2019 Vol. 137 P. 132–153

We study a generalization of genus-zero r-spin theory in which exactly one insertion has a negative-one twist, which we refer to as the "closed extended" theory, and which is closely related to the open r-spin theory of Riemann surfaces with boundary. We prove that the generating function of genus-zero closed extended intersection numbers coincides with the ...

Added: September 27, 2020

Aleksei Golota, Mathematische Nachrichten 2023 Vol. 296 No. 11 P. 5012–5029

The aim of this paper is to classify codimension 1 foliations ℱ with canonical
singularities and 𝜈(𝐾 ℱ ) < 3 on threefolds of general type. I prove a classification
result for foliations satisfying these conditions and having nontrivial algebraic
part. We also describe purely transcendental foliations ℱ with the canonical class
𝐾 ℱ being not big on manifolds ...

Added: September 4, 2023

Popov V. L., Zarhin Y., / Cornell University. Series math "arxiv.org". 2018. No. 1808.01136.

We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $\mathcal L(K)$ generated by ${\rm Aut} (K)$ and multipli\-ca\-tions by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are ...

Added: August 8, 2018