We prove that for a Q-Gorenstein degeneration $X$ of del Pezzo surfaces, the number of non-Du Val singularities is at most $\rho(X)+2$. Degenerations with $\rho(X)+2$ and $\rho(X)+1$ non-Du Val points are investigated.

In this paper, I construct noncompact analogs of the Chern classes of equivariant vector bundles over complex reductive groups. For the tangent bundle, these Chern classes yield an adjunction formula for the Euler characteristic of complete intersections in reductive groups. When a complete intersection is a curve, this formula gives an explicit answer for the Euler characteristic and the genus of the curve. An extension of these results to arbitrary spherical homogeneous spaces is outlined. This is the first step towards extension to the reductive case of the explicit answer given by D.Bernstein, Khovanskii and Kouchnirenko for the Euler characteristic of all complete intersections in the complex torus (C^*)^n.

The main result of this note, Theorem 1.3, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant and ergodic under the action of the infinite unitary group and that admits well-defined projections onto the quotient space of “corners" of finite size, must be finite. A similar result, Theorem 1.1, is also established for unitarily invariant measures on the space of all infinite complex matrices. These results imply that the infinite Hua-Pickrell measures of Borodin and Olshanski have finite ergodic components.

The proof is based on the approach of Olshanski and Vershik. First, it is shown that if the sequence of orbital measures assigned to almost every point is weakly precompact, then our ergodic measure must indeed be finite. The second step, which completes the proof, shows that if a unitarily-invariant measure admits well-defined projections onto the quotient space of finite corners, then for almost every point the corresponding sequence of orbital measures is indeed weakly precompact.

The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians IG(2,2n). We show that these rings are regular. In particular, by “generic smoothness”, we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for IG(2,2n). Further, by a general result of Hertling, the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type A_{n−1}. By the homological mirror symmetry conjecture, these results suggest the existence of a very special full exceptional collection in the derived category of coherent sheaves on IG(2,2n). Such a collection is constructed in the appendix by Alexander Kuznetsov.

We study the billiard on a square billiard table with a one-sided vertical mirror.

We associate trajectories of these billiards with double rotations and study orbit behavior and questions of complexit

For any noncompact semisimple real Lie group G, we construct a group of affine transformations of its Lie algebra g whose linear part is Zariskidense in Ad G and which is free, nonabelian and acts properly discontinuously on g.

We show that an everywhere regular foliation F on a quasi-projective manifold, such that all of its leaves are compact with semi-ample canonical bundle, has isotrivial family of leaves when the orbifold base of this family is special. The specialness condition means that for any p > 0, the p-th exterior power of the logarithmic extension of its conormal bundle does not contain any rank-one subsheaf of maximal possible Kodaira dimension p. This condition is satisfied, for example, in the very particular case when the Kodaira dimension of the determinant of the logarithmic extension of the conormal bundle vanishes. Motivating examples are given by the “algebraically coisotropic” submanifolds of irreducible hyperkähler projective manifolds.