We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson–Drinfeld aﬃne Grassmannians. The answer is given in terms of global Demazure modules over the current Lie algebra.
The convolution ring of loop rotation equivariant K-homology of the affine Grassmannian of GL(n) was identified with a quantum unipotent cell of the loop group of SL(2) by Cautis and Williams. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell.
A cubic polynomial with a marked fixed point 0 is called an IS-capture polynomial if it has a Siegel disk D around 0 and if D contains an eventual image of a critical point. We show that any IS-capture polynomial is on the boundary of a unique bounded hyperbolic component of the polynomial parameter space determined by the rational lamination of the map and relate IS-capture polynomials to the cubic principal hyperbolic domain and its closure.
We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup to the category of GL(N − 1, C[[t]])-equivariant perverse sheaves on the affine Grassmannian of GLN . We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.
We study exceptional collections of line bundles on surfaces. We prove that any full cyclic strong exceptional collection of line bundles on a rational surface is an augmentation in the sense of Lutz Hille and Markus Perling. We find simple geometric criteria of exceptionality (strong exceptionality, cyclic strong exceptionality) for collections of line bundles on weak del Pezzo surfaces. As a result, we classify smooth projective surfaces admitting a full cyclic strong exceptional collection of line bundles. Also, we provide an example of a weak del Pezzo surface of degree 2 and a full strong exceptional collection of line bundles on it which does not come from augmentations, thus answering a question by Hille and Perling.
We classify triangulated categories that are equivalent to finitely generated thick subcategories $T\subset D^b(cohC)$ for smooth projective curves C over an algebraically closed field.
In this note we discuss three notions of dimension for triangulated categories: Rouquier dimension, diagonal dimension and Serre dimension. We prove some basic properties of these dimensions, compare them and discuss open problems.
We apply Weyl n-algebras to prove formality theorems for higher Hochschild cohomology. We present two approaches: via propagators and via the factorization complex. It is shown that the second approach is equivalent to the first one taken with a new family of propagators we introduce.
We develop the formal analogue of the Morse theory for a pair of commuting gradient-like vector fields. The resulting algebraic formalism turns out to be very similar to the algebra of the infrared of Gaiotto, Moore and Witten (see Gaiotto et al., and Kapranov et al.): from a manifold M with the pair of gradient-like commuting vector fields, subject to some general position conditions we construct an L∞-algebra and Maurer–Cartan element in it. We also provide Morse-theoretic examples for the algebra of the infrared data.
A projective manifold M is algebraically hyperbolic if there exists a positive constant A such that the degree of any curve of genus g on M is bounded from above by A(g−1). A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here, we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.
We study automorphism groups and birational automorphism groups of compact complex surfaces. We show that the automorphism group of such a surface X is always Jordan, and the birational automorphism group is Jordan unless X is birational to a product of an elliptic and a rational curve.
We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan.
We prove that a finite group acting by birational automorphisms of a nontrivial Severi-Brauer surface over a field of characteristic zero contains a normal abelian subgroup of index at most 3. Also, we find an explicit bound for the orders of such finite groups in the case when the base field contains all roots of 1.