A desingularization of arbitrary quiver Grassmannians for representations of Dynkin quivers is constructed in terms of quiver Grassmannians for an algebra derived equivalent to the Auslander algebra of the quiver.

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 spectral sequences may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on the structure of the graph cohomology as a whole.

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 are spires of two helices.

A hypercomplex manifold M is a manifold equipped with three complex structures I,J,K satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called the Obata connection. A quaternionic Hermitian metric is a Riemannian metric which is invariant with respect to unitary quaternions. Such a metric is called hyperkähler with torsion (HKT for short) if it is locally obtained as the Hessian of a function averaged with quaternions. An HKT metric is a natural analogue of a Kähler metric on a complex manifold. We push this analogy further, proving a quaternionic analogue of the result of Buchdahl and of Lamari that a compact complex surface M admits a Kähler structure if and only if b1(M) is even. We show that a hypercomplex manifold M with the Obata holonomy contained in SL(2,H) admits an HKT structure if and only if H1(O(M,I)) is even-dimensional.

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit t→∞ and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to coincide in the untwisted setting with the extremal degrees of the so-called PBW-filtration in the corresponding finite-dimensional irreducible representations of the simple Lie algebras for any root systems. This is a particular case of a general conjecture in terms of the level-one Demazure modules. We prove this coincidence for all Lie algebras of classical type and for G2, and also establish the relations of our extremal degrees to minimal q-degrees of the extremal terms of the Kostant q-partition function; they coincide with the latter only for some root systems. © 2015 Elsevier Inc.

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact manifold with $b_2 \geq 7$ admits only finitely many deformation types of holomorphic Lagrangian fibrations. We also prove that all known hyperkahler manifolds are never Kobayashi hyperbolic.

Let G be an almost simple simply connected group over complex numbers. For a positive element of the coroot lattice of G, we consider the open zastava space of based maps from the projective line to the flag variety of G, having the above prescribed degree. This space is known to be isomorphic to the space of framed eucledian G-monopoles with maximal symmetry breaking at infinity of the above prescribed topological charge. In the previous work of Finkelberg-Kuznetsov-Markarian-Mirkovic, a system of etale rational coordinates on the open zastava was introduced. In this note we compute various known structures on the open zastava in terms of the above coordinates. As a byproduct we give a natural interpretation of the Gaiotto-Witten superpotential and relate it to the theory of Whittaker D-modules developed by D.Gaitsgory.

Let G be an almost simple simply connected group over C. For a positive element α of the coroot lattice of G let View the MathML source denote the space of maps from P1 to the flag variety B of G sending ∞∈P1 to a fixed point in B of degree α. This space is known to be isomorphic to the space of framed G -monopoles on R3 with maximal symmetry breaking at infinity of charge α. In [6] a system of (étale, rational) coordinates on Z source is introduced. In this note we compute various known structures on Z source in terms of the above coordinates. As a byproduct we give a natural interpretation of the Gaiotto–Witten superpotential studied in [8] and relate it to the theory of Whittaker D-modules discussed in [9].

We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric q-Whittaker functions. In particular, we show that the series part of the nonsymmetric q-Whittaker function is a generating function for the graded characters of generalized global Weyl modules.

For any smooth quartic threefold in *P*4 we classify pencils on it whose general element is an irreducible surface birational to a surface of Kodaira dimension zero.

We solve the spectral synthesis problem for exponential systems on an interval. Namely, we prove that any complete and minimal system of exponentials in L^2(-a,a) is hereditarily complete up to a one-dimensional defect. However, this one-dimensional defect is possible and, thus, there exist nonhereditarily complete exponential systems. Analogous results are obtained for systems of reproducing kernels in de Branges spaces. For a wide class of de Branges spaces we construct nonhereditarily complete systems of reproducing kernels, thus answering a question posed by N. Nikolski.

In a previous paper, we have defined polynomial Witt vectors functor from vector spaces over a perfect field k of positive characteristic p to abelian groups. In this paper, we use polynomial Witt vectors to construct a functorial Hochschild- Witt complex WCH_∗(A) for any associative unital k-algebra A, with homology groups WHH∗(A). We prove that the group WHH_0(A) coincides with the group of non-commutative Witt vectors defined by Hesselholt, while if A is commutative, finitely generated, and smooth, the groups WHH_i(A) are naturally identified with the terms WΩ^i_A of the de Rham- Witt complex of the spectrum of A.

An affine algebraic variety *X* of dimension ≥2 is called *flexible* if the subgroup SAut(X)⊂Aut(X) generated by the one-parameter unipotent subgroups acts *m*-transitively on reg(X) for any m≥1. In the previous paper we proved that any nondegenerate toric affine variety *X* is flexible. In the present paper we show that one can find a subgroup of SAut(X) generated by a finite number of one-parameter unipotent subgroups which has the same transitivity property, provided the toric variety *X* is smooth in codimension two. For X=A^n with n≥2, three such subgroups suffice.

Let M be a simple hyperkähler manifold. Kuga-Satake construction gives an embedding of H^2(M, C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H^•(M, C) → H^{•+l}(T, C) for some l, which is compatible with the Hodge structures and the Poincaré pairing. Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz sl(2)-triples on M.

We consider the conjectures of Katzarkov, Kontsevich, and Pantev about Landau--Ginzburg Hodge numbers associated to tamely compactifiable Landau--Ginzburg models. We test these conjectures in case of dimension two, verifying some and giving a counterexample to the other.

We obtain certain Mellin-Barnes integrals which present Whittaker wave functions related to classical real split forms of simple complex Lie groups