# Publications

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 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 M2M2 such that there is a continuous and monotone projection of M2M2 to this model. We propose the following related model for the space MD3MD3 of 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 AzAz of 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)∈MD3 let <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">MD3comb their collection, equipped with the quotient topology. We show that tagging defines a continuous map from MD3MD3 to <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">MD3comb so 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">MD3comb serves as a model for MD3MD3.

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.

Given a variety Y with a rectangular Lefschetz decomposition of its derived category, we consider a degree n cyclic cover X→Y ramified over a divisor Z⊂Y. We construct semiorthogonal decompositions of Db(X) and Db(Z) with distinguished components AX and AZ and prove the equivariant category of AX (with respect to an action of the nth roots of unity) admits a semiorthogonal decomposition into n−1n−1 copies of AZ. As examples, we consider quartic double solids, Gushel–Mukai varieties, and cyclic cubic hypersurfaces.

In this paper we study the derived categories of coherent sheaves on Grassmannians Gr(k,n), defined over the ring of integers. We prove that the category Db(Gr(k,n)) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of GLk. This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection [13], which was constructed over a field of characteristic zero. We also describe the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of GLn−k. The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors. We also construct a tilting vector bundle on Gr(k,n). We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over Z, and we identify the objects of Db(Gr(k,n)), which correspond to the standard and costandard modules in both structures. All the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the Grassmannian, by extension of scalars (base change). Similar results over fields of arbitrary characteristic were obtained independently in [7], by different methods.

We prove derived equivalence of Calabi–Yau threefolds constructed by Ito–Miura–Okawa– Ueda as an example of non-birational Calabi–Yau varieties whose difference in the Grothendieck ring of varieties is annihilated by the affine line.

A diffusion-orthogonal polynomial system is a bounded domain Ω in R d endowed with the measure μ and the second-order elliptic differential operator L , self adjoint w.r.t L 2 (Ω ,μ ) , preserving the space of polynomials of degree 6 n for any n . This notion was initially defined in [2], and 2 -dimensional models were classified. It turns out that the boundary of Ω is always an algebraic hypersurface of degree 6 2 d . It was pointed out in [2] that in dimension 2 , when the degree is maximal (so, equals 4 ), the symbol of L (denoted by g ij ) is a cometric of constant curvature. We present the self-contained classification-free proof of this property, and its multidimensional generalisation.

Let B be a simply-connected projective variety such that the first cohomology groups of all line bundles on B are zero. Let E be a vector bundle over B and X = P(E). It is easily seen that a power of any endomorphism of X takes fibers to fibers. We prove that if X admits an endomorphism which is of degree greater than one on the fibers, then E splits into a direct sum of line bundles.

We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis.

In the favourable case a basis of the module is parametrized by the lattice points of a normal polytope. The filtrations induce at degenerations of the corresponding ag variety to its abelianized version and to a toric variety, the special fibres of the degenerations being projectively normal and arithmetically Cohen-Macaulay. The polytope itself can be recovered as a Newton-Okounkov body. We conclude the paper by giving classes of examples for favourable modules.

We discuss a conjecture saying that derived equivalence of simply connected smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection X of three quadrics in P5 and the corresponding double cover Y→P2 branched over a sextic curve. We show that as soon as the natural Brauer class on Y vanishes, so that X and Y are derived equivalent, the difference [X]−[Y] is annihilated by the affine line class.