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.
We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension n as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed n+1. Based on this bound we classify all smooth Fano complete intersections of dimensions 4 and 5, and compute their invariants.
Given a symmetric monoidal (∞,2)-category ℰ we promote the trace construction to a functor. We then apply this formalism to the case when ℰ is the (∞,2)-category of k-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah-Bott formula (also known as the Holomorphic Lefschetz fixed point formula).
We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all 𝔖6-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman–Edge pencil. As an application, we check that 𝔖6-invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class groups as 𝔖6-representations.
Consider the space M = O(p, q)/O(p) × O(q) of positive p-dimensional subspaces in a pseudo-Euclidean space V of signature (p, q), where p > 0, q > 1 and (p,q)≠(1,2), with integral structure: V=Vℤ⊗ℤ. Let Γ be an arithmetic subgroup in G=O(Vℤ), and R⊂Vℤ a Γ-invariant set of vectors with negative square. Denote by R⊥ the set of all positive p-planes W ⊂ V such that the orthogonal complement W⊥ contains some r ∈ R. We prove that either R⊥ is dense in M or Γ acts on R with finitely many orbits. This is used to prove that the squares of primitive classes giving the rational boundary of the Kähler cone (i.e., the classes of “negative” minimal rational curves) on a hyperkähler manifold Xare bounded by a number which depends only on the deformation class of X. We also state and prove the density of orbits in a more general situation when M is the space of maximal compact subgroups in a simple real Lie group.
We show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A given by generators; (2) as an algebra given by generators and relations; (3) as an algebra of differentialreflection operators preserving some spaces of functions; (4) as equivariant Borel-Moore homology of a certain variety. Also, we define a new q-deformation of this algebra, which we call cyclotomic DAHA. Namely, we give a q-deformation of each of the above four descriptions of the partially spherical rational Cherednik algebra, replacing differential operators with difference operators, degenerate DAHA with DAHA, and homology with K-theory, and show that they give the same algebra. In addition, we show that spherical cyclotomic DAHA are quantizations of certain multiplicative quiver and bow varieties, which may be interpreted as K-theoretic Coulomb branches of a framed quiver gauge theory. Finally, we apply cyclotomic DAHA to prove new flatness results for various kinds of spaces of q-deformed quasiinvariants.
For a variety 𝑋, a big ℚ-divisor 𝐿 and a closed connected subgroup 𝐺⊂Aut(𝑋,𝐿) we define a 𝐺-invariant version of the 𝛿-threshold. We prove that for a Fano variety (𝑋,−𝐾_𝑋) and a connected subgroup 𝐺⊂Aut(𝑋) this invariant characterizes 𝐺-equivariant uniform 𝐾-stability. We also use this invariant to investigate 𝐺-equivariant 𝐾-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of 𝐺 being a finite group.
We use the methods introduced by Cheltsov–Rubinstein–Zhang (Sel Math (N.S.) 25(2):25–34, 2019) to estimate δ-invariants of the seven singular del Pezzo surfaces with quotient singularities studied by Cheltsov–Park–Shramov (J Geom Anal 20(4):787–816, 2010) that have α-invariants less than 2/3. As a result, we verify that each of these surfaces admits an orbifold Kähler–Einstein metric.
Let G be a complex Lie group acting on a compact complex Hermitian manifold M by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott–Chern cohomology is trivial. We also apply this result to compute the Dolbeault cohomology of Vaisman manifolds.
Let G be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semi-infinite orbits in the affine Grassmannian Gr G . We prove Simon Schieder’s conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semi-infinite orbits with U (n ∨ ) (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra g ∨ ). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac–Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.