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## Upper Bounds for Mutations of Potentials

arxiv.org.
math.
Cornell University
,
2013.
No. 1301.4541.

Cruz Morales J. A., Galkin S.

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].

Braverman A., Michael Finkelberg, Nakajima H., Instanton moduli spaces and W-algebras / Cornell University. Series math "arxiv.org". 2014.

We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincar\'e pairing) in terms of representation theory of some vertex operator algebras ("W-algebras"). ...

Added: January 30, 2015

Brav C. I., Thomas H., Mathematische Annalen 2011 Vol. 351 No. 4 P. 1005-1017

We establish faithfulness of braid group actions generated by twists along an ADE configuration of 22-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity. ...

Added: September 29, 2014

Braverman A., Michael Finkelberg, Twisted zastava and q-Whittaker functions / Cornell University. Series math "arxiv.org". 2014.

In this note, we extend the results of arxiv:1111.2266 and arxiv:1203.1583 to the non simply laced case. To this end we introduce and study the twisted zastava spaces. ...

Added: February 5, 2015

Michael Finkelberg, Schechtman V., Microlocal approach to Lusztig's symmetries / Cornell University. Series math "arxiv.org". 2014.

We reformulate the De Concini -- Toledano Laredo conjecture about the monodromy of the Casimir connection in terms of a relation between the Lusztig's symmetries of quantum group modules and the monodromy in the vanishing cycles of factorizable sheaves. ...

Added: January 30, 2015

Michael Finkelberg, Leonid Rybnikov, Quantization of Drinfeld Zastava in type C / Cornell University. Series math "arxiv.org". 2013.

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 ...

Added: December 27, 2013

Bezrukavnikov R., Finkelberg M. V., Wreath Macdonald polynomials and categorical McKay correspondence (with Appendices by Ivan Losev, Vadim Vologodsky) / Cornell University. Series math "arxiv.org". 2012. No. 1208.3696.

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_n\ltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert ...

Added: February 6, 2013

Braverman A., Dobrovolska G., Michael Finkelberg, Gaiotto-Witten superpotential and Whittaker D-modules on monopoles / Cornell University. Series math "arxiv.org". 2014.

Let G be an almost simple simply connected group over complex numbers. For a positive element α of the coroot lattice of G let Z^α denote the space of based maps from the projective line to the flag variety of G of degree α. This space is known to be isomorphic to the space of ...

Added: February 3, 2015

Alexander I. Efimov, Derived categories of Grassmannians over integers and modular representation theory / Cornell University. Series math "arxiv.org". 2014.

In this paper we study the derived categories of coherent sheaves on Grassmannians Gr(k,n), defined over the ring of integers. We prove that the category D^b(Gr(k,n)) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of GL_k. This in particular implies existence of a full exceptional collection, which is ...

Added: February 2, 2015

Braverman A., Rybnikov L. G., Feigin B. L. et al., Communications in Mathematical Physics 2011 Vol. 308 No. 2 P. 457-478

Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P^2. More precisely, it predicts ...

Added: May 12, 2012

Brav C. I., International Mathematics Research Notices 2009 No. 8 P. 1355-1387

Kirillov has described a McKay correspondence for finite subgroups of that associates to each “height” function an affine Dynkin quiver together with a derived equivalence between equivariant sheaves on and representations of this quiver. The equivalences for different height functions are then related by reflection functors for quiver representations. The main goal of this article ...

Added: September 29, 2014

Fedor Bogomolov, Yuri Prokhorov, On stable conjugacy of finite subgroups of the plane Cremona group, I / Cornell University. Series math "arxiv.org". 2013.

We discuss the problem of stable conjugacy of finite subgroups of Cremona
groups. We show that the group $H^1(G,Pic(X))$ is a stable birational invariant
and compute this group in some cases. ...

Added: November 21, 2014

Fedor Bogomolov, Böhning C., Stable cohomology of alternating groups / Cornell University. Series math "arxiv.org". 2012.

In this article we determine the stable cohomology groups H^i_s (A_n, Z/p) of the alternating groups A_n for all integers n and i, and all primes p. ...

Added: December 4, 2013

Positselski L., Efimov A., Coherent analogues of matrix factorizations and relative singularity categories / Cornell University. Series math "arxiv.org". 2013. No. arXiv:1102.0261.

We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations ...

Added: December 22, 2013

Bershtein M., Gavrylenko P., Marshakov A., Twist-field representations of W-algebras, exact conformal blocks and character identities / . 2017. No. 1705.00957.

We study twist-field representations of the W-algebras and generalize the construction of the corresponding vertex operators to D- and B-series. We demonstrate how the computation of characters of such representations leads to the nontrivial identities involving lattice theta-functions. We propose a construction of their exact conformal blocks, which for D-series express them in terms of ...

Added: May 4, 2017

A. Levin, Olshanetsky M., Zotov A., Classification of Isomonodromy Problems on Elliptic Curves / Cornell University. Series math "arxiv.org". 2013.

We consider the isomonodromy problems for flat $G$-bundles over punctured
elliptic curves $\Sigma_\tau$ with regular singularities of connections at
marked points. The bundles are classified by their characteristic classes.
These classes are elements of the second cohomology group
$H^2(\Sigma_\tau,{\mathcal Z}(G))$, where ${\mathcal Z}(G)$ is the center of
$G$. For any complex simple Lie group $G$ and arbitrary class we define ...

Added: December 27, 2013

Lev Soukhanov, On the phenomena of constant curvature in the diffusion-orthogonal polynomials / Cornell University. Series math "arxiv.org". 2014.

We consider the systems of diffusion-orthogonal polynomials, defined in the
work [1] of D. Bakry, S. Orevkov and M. Zani and (particularly) explain why
these systems with boundary of maximal possible degree should always come from
the group, generated by reflections. Our proof works for the dimensions $2$ (on
which this phenomena was discovered) and $3$, and fails in ...

Added: September 19, 2014

Ivan Cheltsov, Park J., Won J., Cylinders in singular del Pezzo surfaces / Cornell University. Series math "arxiv.org". 2013.

For each del Pezzo surface $S$ with du Val singularities, we determine
whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we
present an effective divisor $D$ that is $\mathbb{Q}$-linearly equivalent to
$-K_S$ and such that the open set $S\setminus\mathrm{Supp}(D)$ is a cylinder.
As a corollary, we classify all the del Pezzo surfaces with du ...

Added: December 27, 2013

Galkin S., Shinder E., Exceptional collections of line bundles on the Beauville surface / Cornell University. Series math "arxiv.org". 2012. No. 1210.3339.

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface S. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on S. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections ...

Added: September 14, 2013

Kharlamov V., Viktor Kulikov, On numerically pluricanonical cyclic coverings / Cornell University. Series math "arxiv.org". 2013.

In this article, we investigate some properties of cyclic coverings of complex surfaces of general type branched along smooth curves that are numerically equivalent to a multiple of the canonical class. The main results concern coverings of surfaces of general type with p_g=0 and Miyaoka--Yau surfaces; in particular, they provide new examples of multicomponent moduli ...

Added: December 27, 2013

Victor Kulikov, Shustin E., Duality of planar and spacial curves: new insight / Cornell University. Series math "arxiv.org". 2014.

We study the geometry of equiclassical strata of the discriminant in the space of plane curves of a given degree, which are families of curves of given degree, genus and class (degree of the dual curve). Our main observation is that the use of duality transformation leads to a series of new sufficient conditions for ...

Added: February 2, 2015

Fedor Bogomolov, De Oliveira B., Local structure of closed symmetric 2-differentials / Cornell University. Series math "arxiv.org". 2014.

In the authors's previous work on symmetric differentials and their
connection to the topological properties of the ambient manifold, a class of
symmetric differentials was introduced: closed symmetric differentials
([BoDeO11] and [BoDeO13]). In this article we give a description of the local
structure of closed symmetric 2-differentials on complex surfaces, with an
emphasis towards the local decompositions as products of ...

Added: November 21, 2014

Khoroshkin A., Willwacher T., Živković M., Differentials on graph complexes / Cornell University. Series math "arxiv.org". 2014. No. 1411.2369.

We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these series may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on ...

Added: December 9, 2014

Campana F., Demailly J., Misha Verbitsky, Compact Kahler 3-manifolds without non-trivial subvarieties / Cornell University. Series math "arxiv.org". 2013.

We prove that any compact K\"ahler 3-dimensional manifold which has no non-trivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of 'simple manifolds', central in the bimeromorphic classification of compact K\"ahler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of ...

Added: May 13, 2013

Lee K., Shabalin T., Exceptional collections on some fake quadrics / Cornell University. Series math "arxiv.org". 2014.

We construct exceptional collections of maximal length on four families of
surfaces of general type with $p_g=q=0$ which are isogenous to a product of
curves. From these constructions we obtain new examples of quasiphantom
categories as their orthogonal complements. ...

Added: October 17, 2014