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 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}.

Математическая физика и математика

В этой заметке мы вычисляем когомологическое препятствие на существование некоторых пучков вершинных алгебр на гладких многообразиях. Эти пучки были введены и изучены в предыдущей работе Маликова, Вайнтроба и одного из авторов. Надеемся, что наш результат в какой-то мере разъясняет конструкции вышеуказанной работы. В этой заметке мы вычисляем когомологическое препятствие на существование некоторых пучков вершинных алгебр на гладких многообразиях. Эти пучки были введены и изучены в предыдущей работе Маликова, Вайнтроба и одного из авторов. Надеемся, что наш результат в какой-то мере пояснит конструкции вышеупомянутой работы.

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.

We consider the lower central filtration of the free associative algebra An with n generators as a Lie algebra. We consider the associated graded Lie algebra. It is shown that this Lie algebra has a huge center which belongs to the cyclic words, and on the quotient Lie algebra by the center there acts the Lie algebraWn of polynomial vector fields on Cn. We compute the space [An,An]/[An, [An,An]] and show that it is isomorphic to the space 2 closed(Cn) ⊕ 4 closed(Cn) ⊕ 6closed(Cn)