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## Orbifold Milnor lattice and orbifold intersection form

Manuscripta Mathematica. 2018. Vol. 155. No. 3-4. P. 335-353.

For a germ of a quasihomogeneous function with an isolated critical point at the origin invariant with respect to an appropriate action of a finite abelian group, H. Fan, T. Jarvis, and Y. Ruan defined the so-called quantum cohomology group. It is considered as the main object of the quantum singularity theory (FJRW-theory). We define orbifold versions of the monodromy operator on the quantum (co)homology group, of the Milnor lattice, of the Seifert form and of the intersection form. We also describe some symmetry properties of invariants of invertible polynomials refining the known ones.

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

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

Ebeling W., Gusein-Zade S., Pure and Applied Mathematics Quarterly 2020 Vol. 16 No. 4 P. 1099-1113

In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P.Berglund, T.Hubsch and M.Henningson considered a pair (f,G) consisting of an invertible polynomial f and a finite abelian group G of its diagonal symmetries and associated to this pair a dual pair (f~, G~). A.Takahashi suggested a generalization of this construction to pairs (f, ...

Added: February 3, 2021

Gusein-Zade S., Journal of Algebra and its Applications 2018 Vol. 17 No. 10 P. 1-13

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here, a so-called enhanced Burnside ring Bˆ(G) of a finite group G is defined. An ...

Added: October 27, 2020

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

Gusein-Zade S., Mathematische Nachrichten 2018 Vol. 291 No. 17-18 P. 2543-2556

Let a finite abelian group G act (linearly) on the space R^n and thus on its complexification C^n. Let W be the real part of the quotient C^n/G (in general W \neq R^n/G). We give an algebraic formula for the radial index of a 1-form \omega on the real quotient W. It is shown that ...

Added: October 27, 2020

Klimenko A. V., Bufetov A. I., Труды Математического института им. В.А. Стеклова РАН 2012 Т. 277 С. 33-48

Устанавливается сходимость почти всюду средних по Чезаро сферических средних произвольной функции из класса L^p, p>1, для действий марковских полугрупп, и в частности конечно порожденных гиперболических групп. ...

Added: February 13, 2013

Gusein-Zade S., Функциональный анализ и его приложения 2018 Т. 52 № 4 С. 72-85

The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with certain finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. We discuss a universal additive topological invariant of V-manifolds, the universal Euler characteristic. It takes values in the ring ...

Added: October 27, 2020

Alvarez S., Filimonov D., Kleptsyn V. et al., Journal of Topology 2019 Vol. 12 No. 4 P. 1315-1367

This article is inspired by two milestones in the study of non-minimal group actions on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves, and Ghys' freeness result in real-analytic regularity. Our first result concerns groups of real-analytic diffeomorphisms with infinitely many ends: if the action is non expanding, then the group ...

Added: July 13, 2019

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., Golyshev V., Iritani H., / Cornell University. Series math "arxiv.org". 2014. No. 1404.6407.

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: May 4, 2014

Galkin S., / Cornell University. Series math "arxiv.org". 2018. No. 1809.02737.

Given a singular variety I discuss the relations between quantum cohomology of its resolution and smoothing. In particular, I explain how toric degenerations helps with computing Gromov--Witten invariants, and the role of this story in Fanosearch programme. The challenge is to formulate enumerative symplectic geometry of complex 3-folds in a way suitable for extracting invariants ...

Added: September 25, 2018

Galkin S., Mellit A., Smirnov M., International Mathematics Research Notices 2015 Vol. 2015 No. 18 P. 8847-8859

We show that the big quantum cohomology of the symplectic isotropic Grassmanian IG(2,6) is generically semisimple, whereas its small quantum cohomology is known to be non-semisimple. This gives yet another case where Dubrovin's conjecture holds and stresses the need to consider the big quantum cohomology in its formulation. ...

Added: October 20, 2014

Kleptsyn V., Alvarez S., Malicet D. et al., / Cornell University. Series math "arxiv.org". 2015.

Added: June 22, 2016

Gusein-Zade S., Алгебра и анализ 2021 Т. 33 № 3 С. 73-84

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related with corresponding analogues of indices of singular points. Earlier, there was defined ...

Added: May 2, 2021

Filimonov D., Клепцын В. А., Nonlinearity 2014 Vol. 27 No. 6 P. 1205-1223

We study possible one-end finitely presented subgroups of <img />, acting without finite orbits. Our main result, theorem 1, establishes that any such action possesses the so-called property (<img />), that allows one to make distortion-controlled expansion and is thus sufficient to conclude that the action is Lebesgue-ergodic. We also propose a path towards full ...

Added: October 23, 2014

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

Galkin S., Mellit A., Smirnov M., / Cornell University. Series math "arxiv.org". 2014. No. 1405.3857.

We show that the big quantum cohomology of the symplectic isotropic Grassmanian IG(2,6) is generically semisimple, whereas its small quantum cohomology is known to be non-semisimple. This gives yet another case where Dubrovin's conjecture holds and stresses the need to consider the big quantum cohomology in its formulation. ...

Added: May 16, 2014

Gusein-Zade S., Раух А. Я., Функциональный анализ и его приложения 2021 Т. 55 № 1 С. 56-64

V.I.Arnold classified simple (i.e. having no moduli for the classification) singularities (function germs) and also simple boundary singularities: function germs invariant with respect to the action
σ(x1;y1,…,yn)=(−x1;y1,…,yn) of the group Z2. In particular, it was shown that a function germ (a germ of a boundary singularity) is simple if and only if the intersection form (respectively, ...

Added: February 3, 2021

Gusein-Zade S., Математические заметки 2020 Т. 107 № 6 С. 855-864

V.I.Arnold has classified simple (i.e., having no moduli for the classification) singularities (function germs), and also simple boundary singularities: function germs invariant with respect to the action σ (x1; y1, …, yn) = (−x1; y1, …, yn) of the group ℤ2. In particular, it was shown that a function germ (a boundary singularity germ) is ...

Added: October 27, 2020

Gusein-Zade S., Функциональный анализ и его приложения 2018 Т. 52 № 2 С. 78-81

Let G be a finite Abelian group acting (linearly) on space ℝn and, therefore, on its complexification ℂn, and let W be the real part of the quotient ℂn/G (in the general case, W ≠ ℝn/G). The index of an analytic 1-form on the space W is expressed in terms of the signature of the ...

Added: October 27, 2020

Galkin S., Golyshev V., Russian Mathematical Surveys 2006 Vol. 61 No. 1 P. 171-173

Added: September 14, 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

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