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Topological Relations on Witten–Kontsevich and Hodge Potentials
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 equation for generating series for certain intersection indices on the moduli spaces. The genus homogeneous components of the Witten–Kontsevich potential, as well as of the more general Hodge potential, which include, in addition to psi-classes, intersection indices for lambda-classes, are special cases of these generating series, and the well-known partial differential equations for them are instances of our general construction.
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., / Cornell University. Series math "arxiv.org". 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. ...
Added: March 3, 2015
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., 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
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
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
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
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
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
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
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
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
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
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
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
Fonarev A., Kuznetsov A., Journal of London Mathematical Society 2018 Vol. 97 No. 2 P. 24-46
We prove that the derived category D(C) of a generic curve of genus greater than one embeds into the derived category D(M) of the moduli space M of rank two stable bundles on C with fixed determinant of odd degree. ...
Added: November 7, 2017
Tikhomirov A. S., Matemática Contemporânea 2020 Vol. 47 P. 301-316
In this article, we will give a review of recent results on the geography and geometry of the Gieseker-Maruyama moduli scheme M = M (c1 , c2 ) of rank 2 semi-stable coherent sheaves with first Chern class c1 = 0 or −1, second Chern class c2 , and third Chern class 0 on the ...
Added: December 22, 2020
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
Maxim Kazarian, Journal of the Institute of Mathematics of Jussieu 2021 P. 1-6
We derive a quadratic recursion relation for the linear Hodge integrals of the form $⟨τ_2^nλ_k⟩$. These numbers are used in a formula for Masur-Veech volumes of moduli spaces of quadratic differentials discovered by Chen, Möller and Sauvaget. Therefore, our recursion provides an efficient way of computing these volumes. ...
Added: April 15, 2021
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
Kazaryan M., Lando S., Prasolov V., Switzerland : Springer, 2018
This book offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Over the last few decades, this notion has become central not only in algebraic geometry, but in mathematical physics, including string theory, as well.
The book begins by studying individual smooth algebraic curves, including the most beautiful ones, ...
Added: November 19, 2018