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## Примеры решёточно-поляризованных K3-поверхностей с автоморфным дискриминантом и лоренцевы алгебры Каца—Муди

TRANSACTIONS OF THE MOSCOW MATHEMATICAL SOCIETY. 2017. Т. 78. № 1. С. 89-100.

Gritsenko V., Никулин В. В.

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: автоморфные формызеркальная симметрияалгебры Каца-Мудиmirror symmetryK3 surfacesautomorphic formsKac-Moody algebraK3-поверхность

Publication based on the results of:

Galkin S., Rybakov S., / Cornell University. Series math "arxiv.org". 2019. No. 1910.14379.

For a family of K3 surfaces we implement a variation of a general construction of towers of algebraic curves over finite fields given in a previous paper. As a result we get a good tower over k=𝔽_{p^2}, that is optimal if p=3. ...

Added: November 6, 2019

Coates T., Corti A., Galkin S. et al., Geometry and Topology 2016 Vol. 20 No. 1 P. 103-256

The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3-dimensional Fano manifolds. In particular we show that 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by ...

Added: November 18, 2014

Coates T., Corti A., Galkin S. et al., / Cornell University. Series math "arxiv.org". 2012. No. 1212.1722.

We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas. ...

Added: September 14, 2013

Gusein-Zade S., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2020 Vol. 16 No. 051 P. 1-15

P. Berglund, T. Hübsch, and M. Henningson proposed a method to construct mirror symmetric Calabi–Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group
of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal ...

Added: October 27, 2020

Akhtar M., Coates T., Galkin S. et al., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2012 Vol. 8 No. 094 P. 1-707

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with ...

Added: September 14, 2013

Ebeling W., Gusein-Zade S., International Mathematics Research Notices 2021 Vol. 2021 No. 16 P. 12305-12329

A.Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito duality between Burnside rings to a case of non-abelian groups and prove a "non-abelian" generalization of the statement about the equivariant Saito duality ...

Added: August 26, 2021

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

Cruz Morales J. A., Galkin S., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2013 Vol. 9 No. 005 P. 1-13

In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1–52]. ...

Added: May 27, 2013

Galkin S., Belmans P., Mukhopadhyay S., / Cornell University. Series math "arxiv.org". 2020. No. 2009.05568.

We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. These graphs encode degenerations of curves to rational curves, and graph potentials encode degenerations of the moduli space of rank 2 bundles with fixed determinant. We show that the birational type of the graph potential only depends on the homotopy type of ...

Added: April 15, 2021

Galkin S., Iritani H., / Cornell University. Series math "arxiv.org". 2015. No. 1508.00719.

The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class G_F associated to Euler's Γ-function. We illustrate in ...

Added: August 5, 2015

Galkin S., / Cornell University. Series math "arxiv.org". 2018. No. 1809.02738.

We show that G-Fano threefolds are mirror-modular.
1. Mirror maps are inversed reversed Hauptmoduln for moonshine subgroups of SL2(ℝ).
2. Quantum periods, shifted by an integer constant (eigenvalue of quantum operator on primitive cohomology) are expansions of weight 2 modular forms (theta-functions) in terms of inversed Hauptmoduln.
3. Products of inversed Hauptmoduln with some fractional powers of shifted ...

Added: September 25, 2018

Coates T., Galkin S., Kasprzyk A. et al., / Cornell University. Series math "arxiv.org". 2014. No. 1406.4891.

We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles. ...

Added: June 20, 2014

Coates T., Galkin S., Kasprzyk A. et al., Experimental Mathematics 2020 Vol. 29 No. 2 P. 183-221

We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles. ...

Added: September 1, 2018

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

Galkin S., Golyshev V., Iritani H., Duke Mathematical Journal 2016 Vol. 165 No. 11 P. 2005-2077

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the ...

Added: November 18, 2014

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

Sawada T., Li Y., Pizlo Z., Symmetry 2011 Vol. 3 No. 2 P. 365-388

Added: September 23, 2014

Ivan Cheltsov, Karzhemanov I., Advances in Mathematics 2010 Vol. 223 P. 594-618

For any smooth quartic threefold in P4 we classify pencils on it whose general element is an irreducible surface birational to a surface of Kodaira dimension zero. ...

Added: December 6, 2013

Gritsenko V., Nikulin V. V., Transactions of the Moscow Mathematical Society 2017 Vol. 78 P. 75-83

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: January 29, 2018

Galkin S., Shinder E., / Cornell University. Series math "arxiv.org". 2014. No. 1405.5154.

We find a relation between a cubic hypersurface Y and its Fano variety of lines F(Y) in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational ...

Added: May 21, 2014

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

Ilten N. O., Lewis J., Victor Przyjalkowski, Journal of Algebra 2013 Vol. 374 P. 104-121

We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of ...

Added: July 2, 2013

Galkin S., / Cornell University. Series math "arxiv.org". 2014. No. 1404.7388.

Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates. As a consequence we obtain a positive answer ...

Added: May 4, 2014

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