# Publications

We study a coproduct in type A quantum open Toda lattice in terms of a coproduct in the shifted Yangian of sl2. At the classical level this corresponds to the multiplication of scattering matrices of euclidean SU(2) monopoles. We also study coproducts for shifted Yangians for any simply-laced Lie algebra.

We propose an r-variable version of Kostka-Shoji polynomials K_{λμ} for r-multipartitions λ, μ. Our version has positive integral coefficients and encodes the graded multiplicities in the space of global sections of a line bundle over Lusztig’s iterated convolution diagram for the cyclic quiver Ã_{r−1}.

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety. We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kahler form taking values in a local system L, called the conformal weight bundle. The L-valued cohomology of M is called Morse-Novikov cohomology. It was conjectured that (just as it happens for Kahler manifolds) the Morse-Novikov complex satisfies the dd^c-lemma. If true, it would have far-reaching consequences for the geometry of LCK manifolds. Counterexamples to the Morse-Novikov dd^c-lemma on Vaisman manifolds were found by R. Goto. We prove that dd^c-lemma is true with coefficients in a sufficiently general power L^a of L on any LCK manifold with potential (this includes Vaisman manifolds). We also prove vanishing of Dolbeault and Bott-Chern cohomology with coefficients in L^a. The same arguments are used to prove degeneration of the Dolbeault-Frohlicher spectral sequence with coefficients in any power of L.

We study Calabi-Yau threefolds fibered by abelian surfaces, in particular, their arithmetic properties, e.g., Neron models and Zariski density.

A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which follows from Kobayashi hyperbolicity. We prove that hyperkähler manifolds are not algebraically hyperbolic when the Picard rank is at least 3, or if the Picard rank is 2 and the SYZ conjecture on existence of Lagrangian fibrations is true. We also prove that if the automorphism group of a hyperkähler manifold is infinite then it is algebraically nonhyperbolic.

The aim of this short note is to give a simple proof of the non-rationality of the double cover of the three-dimensional projective space branched over a sufficiently general quartic.

We study two rational Fano threefolds with an action of the icosahedral group 𝔄5. The first one is the famous Burkhardt quartic threefold, and the second one is the double cover of the projective space branched in the Barth sextic surface. We prove that both of them are 𝔄5-Fano varieties that are 𝔄5-birationally superrigid. This gives two new embeddings of the group 𝔄5 into the space Cremona group.

We prove that the characteristic foliation F on a nonsingular divisor D in an irreducible projective hyperk¨ahler manifold X cannot be algebraic, unless the leaves of F are rational curves or X is a surface. More generally, we show that if X is an arbitrary projective manifold carrying a holomorphic symplectic 2-form, and D and F are as above, then F can be algebraic with non-rational leaves only when, up to a finite ´etale cover, X is the product of a symplectic projective manifold Y with a symplectic surface and D is the pull-back of a curve on this surface. When D is of general type, the fact that F cannot be algebraic unless X is a surface was proved by Hwang and Viehweg. The main new ingredient for our results is the observation that the canonical class of the (orbifold) base of the family of leaves is zero. This implies, in particular, the isotriviality of the family of leaves of F. We show this, more generally, for regular algebraic foliations by curves defined by the vanishing of a holomorphic (d − 1)-form on a complex projective manifold of dimension d.

In this article, we calculate the ring of unstable (possibly nonadditive) operations from algebraic Morava K-theory K(n)^∗ to Chow groups with ℤ_(p) -coefficients. More precisely, we prove that it is a formal power series ring on generators c_i:K(n)^∗→CH^i⊗ℤ_(p) , which satisfy a Cartan-type formula.

W. Thurston constructed a combinatorial model of the Mandelbrot set M2M2such that there is a continuous and monotone projection of M2M2to this model. We propose the following related model for the space MD3MD3of critically marked cubic polynomials with connected Julia set and all cycles repelling. If (P,c1,c2)∈MD3(P,c1,c2)∈MD3, then every point *z* in the Julia set of the polynomial *P * defines a unique maximal finite set AzAzof angles on the circle corresponding to the rays, whose impressions form a continuum containing *z *. Let G(z)G(z)denote the convex hull of AzAz. The convex sets G(z)G(z)partition the closed unit disk. For (P,c1,c2)∈MD3(P,c1,c2)∈MD3let <img height="16" border="0" style="vertical-align:bottom" width="14" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si6.gif">c1⁎be the *co-critical point of *c1c1. We tag the marked dendritic polynomial (P,c1,c2)(P,c1,c2)with the set <img height="18" border="0" style="vertical-align:bottom" width="159" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si14.gif">G(c1⁎)×G(P(c2))⊂D‾×D‾. Tags are pairwise disjoint; denote by <img height="18" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si10.gif">MD3combtheir collection, equipped with the quotient topology. We show that tagging defines a continuous map from MD3MD3to <img height="18" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si10.gif">MD3combso that <img height="18" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si10.gif">MD3combserves as a model for MD3MD3.

We study a coproduct in type *A* quantum open Toda lattice in terms of a coproduct in the shifted Yangian of sl2. At the classical level this corresponds to the multiplication of scattering matrices of euclidean SU(2) monopoles. We also study coproducts for shifted Yangians for any simply-laced Lie algebra.

In this paper we prove the indicated conjecture in the last case of known infinite series of theta-blocks of weight 2.

Let M be an irreducible holomorphic symplectic (hyperkähler) manifold. If b 2 (M ) > 5, we construct a deformation M 0 of M which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real (1, 1)-classes is hyperbolic. If b 2 (M ) > 14, similarly, we construct a deformation which admits a parabolic automorphism (and many other automorphisms as well).

On del Pezzo surfaces, we study effective ample ℝ -divisors such that the complements of their supports are isomorphic to 𝔸1 -bundles over smooth affine curves. All considered varieties are assumed to be algebraic and defined over an algebraically closed field of characteristic 0 throughout this article.

We study the connection between the affine degenerate Grassmannians in type A, quiver Grassmannians for one vertex loop quivers and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type GL(n) and identify it with semi-infinite orbit closure of type A_{2n-1}. We show that principal quiver Grassmannians for the one vertex loop quiver provide finite-dimensional approximations of the degenerate affine Grassmannian. Finally, we give an explicit description of the degenerate affine Grassmannian of type A_1^{(1)}, propose a conjectural description in the symplectic case and discuss the generalization to the case of the affine degenerate flag varieties.