arXiv subjects
Publications and source records attributed to Nikon Kurnosov.
Let $S$ be a minimal compact complex surface of class VII with $b_2(S)=3$. We prove that if $S$ carries two distinct singular holomorphic foliations, then $S$ contains a global spherical shell. Moreover, it is an Inoue-Hirzebruch surface. The proof is based on Teleman's theorem on the existence of a cycle of rational curves and results of Dloussky and is inspired by work of Brunella.
We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works of D. Guan and the first author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kähler manifold $W_F$ which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction, and prove that the automorphism group of $Q$ satisfies the Jordan property.
In 1995, Dan Guan constructed examples of non-Kahler, simply-connected holomorphically symplectic manifolds. An alternative construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. We investigate topology and deformation theory of Bogomolov-Guan manifolds and show that it is similar to that of hyperkahler manifolds. We prove the local Torelli theorem, showing that holomorphically symplectic deformations of BG-manifolds are unobstructed, and the corresponding period map is locally a diffeomorphism. Using the local Torelli theorem, we prove the Fujiki formula for a BG-manifold $M$, showing that there exists a symmetric form q on the second cohomology such that for any $w\in H^2(M)$ one has $\int_M w^{2n}=q(w,w)^n$. This form is a non-Kahler version of the Beauville-Bogomolov-Fujiki form known in hyperkahler geometry.
We give a new proof of a theorem of Bogomolov, that the only $VII_0$ surfaces with $b_2 = 0$ are those constructed by Hopf and Inoue. The proof follows the strategy of the original one, but it is of purely group-theoretic nature.
Let M be a simple hyperkahler manifold. Kuga-Satake construction gives an embedding of H^2(M,C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H^*(M,C) \to H^{*+l}(T,C) for some l, which is compatible with the Hodge structures and the Poincare pairing. Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz sl(2)-triples on M.
We study groups of bimeromorphic and biholomorphic automorphisms of projective hyperkähler manifolds. Using an action of these groups on some non-positively curved space, we deduce many of their properties, including finite presentation, strong form of Tits' alternative and some structural results about groups consisting of transformations with infinite order.
In this paper we study the Lagrangian fibrations for projective irreducible symplectic fourfolds and exclude the case of non-smooth base. Our method could be extended to the higher-dimensional cases.
Let $M$ be a compact irreducible hyperkahler manifold, from Bogomolov inequality [V1] we obtain forbidden values of the second Betti number $b_2$ in arbitrary dimension. UPD: Unfortunately, decomposition of dual to BBF-form is not right in the main theorem. Instead of this work, take a look on recent preprints of Sawon and me on boundedness of $b_2$ for hyperkähler manifolds.
We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori.
We prove that $b_2$ is bounded for hyperkähler manifolds with vanishing odd-Betti numbers. The explicit upper boundary is conjectured. Following the method described by Sawon we prove that $b_2$ is bounded in dimension eight and ten in the case of vanishing odd-Betti numbers by 24 and 25 respectively.