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## Dubrovin's conjecture for IG(2,6)

Cornell University
,
2014.
No. 1405.3857.

Galkin S., Mellit A., Smirnov M.

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.

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

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., Russian Mathematical Surveys 2006 Vol. 61 No. 1 P. 171-173

Added: September 14, 2013

Kuznetsov A., Polishchuk A., / Cornell University. Series math "arxiv.org". 2011. No. 1110.5607.

We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups. ...

Added: October 4, 2013

Przyjalkowski V., Shramov K., Успехи математических наук 2014 Т. 69 № 6(420) С. 181-182

В Московском математическом обществе.
Сообщения Московского математического общества ...

Added: February 26, 2015

Galkin S., Iritani H., , in : Primitive Forms and Related Subjects — Kavli IPMU 2014. : Tokyo : Mathematical Society of Japan, 2019. P. 55-115.

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 associated to Euler's Gamma-function.
We illustrate in the case of ...

Added: September 1, 2018

Gorbounov V., Корфф К., Строппель К., Успехи математических наук 2020 Т. 75 № 5(455) С. 3-58

We survey a recent development which connects quantum integrable models with Schubert calculus for quiver varieties: there is a purely geometric construction of solutions to the Yang-Baxter equation and their associated Yang-Baxter algebras which play a central role in quantum integrable systems and exactly solvable lattice models in statistical physics. We will provide a simple ...

Added: September 9, 2020

Netay I. V., Функциональный анализ и его приложения 2013 Т. 47 № 3 С. 54-74

We describe the syzygy spaces for the Segre embedding~$\bbP(U)\times\bbP(V)\subset\bbP(U\otimes V)$ in terms of representations of $\GL(U)\times \GL(V)$ and construct the minimal resolutions of the sheaves~$\mathscr{O}_{\bbP(U)\times\bbP(V)}(a,b)$ in~$D(\bbP(U\otimes V))$ for~$a\geqslant-\dim(U)$ and~$b\geqslant-\dim(V)$. Also we prove some property of multiplication on syzygy spaces of the Segre embedding. ...

Added: June 21, 2013

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., 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

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

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

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

Kuznetsov A., Polishchuk A., Journal of the European Mathematical Society 2016 Vol. 18 No. 3 P. 507-574

We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups. ...

Added: December 22, 2013

Penkov I., Tikhomirov A. S., Pure and Applied Mathematics Quarterly 2014 Vol. 10 No. 2 P. 289-323

We consider ind-varieties obtained as direct limits of chains of embeddings $X_1\stackrel{\phi_1}{\hookrightarrow}\dots\stackrel{\phi_{m-1}}{\hookrightarrow} X_m\stackrel{\phi_m}{\hookrightarrow}X_{m+1}\stackrel{\phi_{m+1}}{\hookrightarrow}\dots$, where each $X_m$ is a grassmannian or an isotropic grassmannian (possibly mixing grassmannians and isotropic grassmannians), and the embeddings $\phi_m$ are linear in the sense that they induce isomorphisms of Picard groups. We prove that any such ind-variety is isomorphic to one ...

Added: October 9, 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

Stanislav Fedotov, Transactions of the American Mathematical Society 2013 Vol. 365 No. 8 P. 4153-4179

In this work we study a realization of moduli spaces of framed quiver representations as Grassmannians of submodules devised by Marcus Reineke. Obtained is a generalization of this construction for finite dimensional associative algebras and for quivers with oriented cycles. As an application we get an explicit realization of fibers for the moduli space bundle ...

Added: November 5, 2015

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

Gusein-Zade S., 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 ...

Added: October 27, 2020

A. Kuznetsov, Mathematische Zeitschrift 2014 Vol. 276 No. 3 P. 655-672

Given a generic family $Q$ of 2-dimensional quadrics over a smooth 3-dimensional base $Y$ we consider the relative Fano scheme $M$ of lines of it. The scheme $M$ has a structure of a generically conic bundle $M \to X$ over a double covering $X \to Y$ ramified in the degeneration locus of $Q \to Y$. ...

Added: December 22, 2013

Kuznetsov A., Известия РАН. Серия математическая 2019 Т. 83 № 3 С. 127-132

Мы показываем, что ограниченная производная категория когерентных пучков на общей поверхности Энриквеса может быть реализована как полуортогональная компонента в производной категории гладкого многообразия Фано с диагональным ромбом Ходжа. ...

Added: June 4, 2019

Kuznetsov A., Perry A., Compositio Mathematica 2018 Vol. 154 No. 7 P. 1362-1406

We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, ...

Added: September 13, 2018

Kuznetsov A., / Cornell University. Series math "arxiv.org". 2010. No. 1011.4146.

Given a generic family $Q$ of 2-dimensional quadrics over a smooth 3-dimensional base $Y$ we consider the relative Fano scheme $M$ of lines of it. The scheme $M$ has a structure of a generically conic bundle $M \to X$ over a double covering $X \to Y$ ramified in the degeneration locus of $Q \to Y$. ...

Added: October 4, 2013

Bondal A. I., Известия РАН. Серия математическая 2013 Т. 77 № 4 С. 5-30

В работе вводится понятие согласованных пар и согласованных цепей t-структур. Доказывается, что две согласованных цепи t-структур порождают дистрибутивную решетку. Развитая техника применяется к случаю пары цепей, полученных из стандартной t-структуры на производной категории когерентных пучков и двойственной к ней применением функтора сдвига. В результате получается семейство t-структур, сердцевины которых известны как превратные когерентные пучки. ...

Added: February 6, 2013