# Publications

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 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 *X*are 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 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 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.

This paper is a review of results on multiple flag varieties, i.e., varieties of the form *G/P*1*×· · ·×G/Pr*. We provide a classification of multiple flag varieties of complexity 0 and 1 and results on the combinatorics and geometry of *B*-orbits and their closures in double cominuscule flag varieties. We also discuss questions of finiteness for the number of *G*-orbits and existence of an open *G*-orbits on a multiple flag variety.

It was conjectured that multiplicity of a singularity is bi-Lipschitz invariant. We disprove this conjecture constructing examples of bi-Lipschitz equivalent complex algebraic singularities with different values of multiplicity.

We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, thereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated as an algebra. These results are deduced from a general sufficient criterion for smoothness.

We define symmetric Dellac configurations as the Dellac configurations that are symmetrical with respect to their centers. The even-length symmetric Dellac configurations coincide with the Fang-Fourier symplectic Dellac configurations. Symmetric Dellac configurations generate the Poincaré polynomials of (odd or even) symplectic or orthogonal versions of degenerate flag varieties. We give several combinatorial interpretations of the Randrianarivony-Zeng polynomial extension of median Euler numbers in terms of objects that we call extended Dellac configurations. We show that the extended Dellac configurations generate symmetric Dellac configurations. As a consequence, the cardinalities of odd and even symmetric Dellac configurations are respectively given by two sequences (1, 1, 3, 21, 267, ...) and (1, 2, 10, 98, 1594, ...), defined as specializations of polynomial extensions of median Euler numbers.

We construct a full exceptional collection of vector bundles in the bounded derived category of coherent sheaves on the Grassmannian IGr(3,8) of isotropic 3-dimensional subspaces in an 8-dimensional symplectic vector space.

The forms of the Segre cubic over non-algebraically closed fields, their automorphisms groups, and equivariant birational rigidity are studied. In particular, it is shown that all forms of the Segre cubic over any field have a point and are cubic hypersurfaces.

Let $F$ be an infinite division ring, $V$ be a left $F$-vector space, $r>0$ be an integer. We study the structure of the representation of the linear group $\mathrm{GL}_F(V)$ in the vector space of formal finite linear combinations of $r$-dimensional vector subspaces of $V$ with coefficients in a field.

This gives a series of natural examples of irreducible infinite-dimensional representations of projective groups. These representations are non-smooth if $F$ is locally compact and non-discrete.

For a family of K3 surfaces we implement a variation of a general construction of towers of algebraic curves over finite fields given in a previous paper. As a result we get a good tower over k=F_{p^2}, that is optimal if p=3.

The purpose of this article is to develop techniques for estimating basis log canonical thresholds on logarithmic surfaces. To that end, we develop new local intersection estimates that imply log canonicity. Our main motivation and application is to show the existence of Kähler–Einstein edge metrics on all but finitely many families of asymptotically log del Pezzo surfaces, partially confirming a conjecture of two of us. In an appendix we show that the basis log canonical threshold of Fujita–Odaka coincides with the greatest lower Ricci bound invariant of Tian.