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

Let G be a reductive group and let ·G be its Langlands dual. We give an interpretation of the dynamical Weyl group of ·G de¯ned in [5] in terms of the geometry of the a±ne Grassmannian Gr of G. In this interpretation the dynamical parameters of [5] correspond to equivariant parameters with respect to certain natural torus acting on Gr. We also present a conjectural generalization of our results to the case of a±ne Kac-Moody groups.

We classify threefolds with non-Jordan birational automorphism groups.

We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For GL_n and Gelfand{Zetlin polytopes, combinatorics of this algorithm coincides with that of the mitosis on pipe dreams introduced by Knutson and Miller. For Sp_4 and a Newton{Okounkov polytope of the symplectic flag variety, the algorithm yields a new combinatorial rule that extends to Sp_{2n}.

Mathematical Physics and Mathematics

In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by Malikov, Vaintrob and one of the authors. Hopefully our result clarifies to some extent the constructions of the above work.In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by Malikov, Vaintrob and one of the authors. Hopefully our result clarifies to some extent the constructions of the above work.

Let $(A,a)$ be an indecomposable principally polarized abelian threefold defined over a field $k \subset \CC$. Using a certain geometric Siegel modular form $\chi_{18}$ on the corresponding moduli space, we prove that $(A,a)$ is a Jacobian over $k$ if and only if $\chi_{18}(A, a)$ is a square over $k$. This answers a question of J.-P. Serre. Then, via a natural isomorphism between invariants of ternary quartics and Teichm\"uller modular forms of genus $3$, we obtain a simple proof of Klein formula, which asserts that $\chi_{18}(\Jac C, j)$ is equal to the square of the discriminant of $C$.

Of four types of Kaplansky algebras, type-2 and type-4 algebras have previously unobserved Z/2-gradings: nonlinear in roots. A method assigning a simple Lie superalgebra to every Z/2-graded simple Lie algebra in characteristic 2 is illustrated by seven new series. Type-2 algebras and one of the two type-4 algebras are demystified as nontrivial deforms (the results of deformations) of the alternate Hamiltonian algebras. The type-1 Kaplansky algebra is recognized as the derived of the nonalternate version of the Hamiltonian Lie algebra, the one that preserves a tensorial 2-form, not an exterior one.

Deforms corresponding to nontrivial cohomology classes can be isomorphic to the initial algebra, e.g., we confirm Grishkov's implicit claim and explicitly describe the Jurman algebra as such a "semitrivial" deform of the derived of the alternate Hamiltonian Lie algebra. This paper helps to sharpen the formulation of a conjecture describing all simple finite-dimensional Lie algebras over any algebraically closed field of nonzero characteristic and supports a conjecture of Dzhumadildaev and Kostrikin stating that all simple finite-dimensional modular Lie algebras are either of "standard" type or deforms thereof.

In characteristic 2, we give sufficient conditions for the known deformations to be semitrivial.

The Oeljeklaus-Toma (OT-) manifolds are complex manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces Sm. On each OT-manifold we construct a holomorphic line bundle with semipositive curvature form !0 and trivial Chern class. Using this form, we prove that the OT-manifolds admitting a locally conformally K¨ahler structure have no non-trivial complex subvarieties. The proof is based on the Strong Approximation theorem for number fields, which implies that any leaf of the null-foliation of !0 is Zariski dense.

Given a multigraded algebra A, it is a natural question whether or not for two homogeneous components A_u and A_v, the product A_nuA_nv is the whole component A_nu+nv for n big enough. We give combinatorial and geometric answers to this question.

The generating function for Sn-equivariant Euler characteristics of moduli spaces of pointed hyperelliptic curves for any genus g ≥ 2 is calculated. This answer generalizes the known ones for genera 2 and 3 and the answers obtained by J. Bergstro ̈m for any genus and n ≤ 7 points.