Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten-Dijkgraaf-Verlinde-Verlinde and Dubrovin equations). We show that each N-dimensional Frobenius manifold can locally be represented by a special flat N-dimensional submanifold with flat normal bundle in a 2N-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.     

Holomorphic deformations of the Ricci-flat $$\partial \overline \partial $$-manifolds

We present a construction of globally convergent power series of integrable Beltrami differentials on the Ricci-flat \(\partial \overline \partial \)-manifolds and also a construction of global canonical family of holomorphic (n, 0)-forms on the deformation spaces of the Ricci-flat \(\partial \overline \partial \)-manifolds.     

On the deformations of the incompressible Euler equations

We consider systems of deformed system of equations, which are obtained by some transformations from the system of incompressible Euler equations. These have similar properties to the original Euler equations including the scaling invariance. For one form of deformed system we prove that finite time blow-up actually occurs for ‘generic’ initial data, while for the other form of the deformed system we prove the global in time regularity for smooth initial data. Moreover, using the explicit functional relations between the solutions of those deformed systems and that of the original Euler system, we derive the condition of finite time blow-up of the Euler system in terms of solutions of one of its deformed systems. As another application of those relations we deduce a lower estimate of the possible blow-up time of the 3D Euler equations. This research was supported partially by the KOSEF Grant no. R01-2005-000-10077-0     

The second cohomology with symmplectic coefficients of the moduli space of smooth projcetive curves

Each finite dimensional irreducible rational representation V of the symplectic group Sp_2g(Q) determines a generically defined local system V over the moduli space M_g of genus g smooth projective curves. We study H² (M_g; V) and the mixed Hodge structure on it. Specifically, we prove that if g 6, then the natural map IH²(M~_g; V) H²(M_g; V) is an isomorphism where M~_{_g} is tfhe Satake compactification of M_g. Using the work of Saito we conclude that the mixed Hodge structure on H²(M_g; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H²(M_g; V) for 3 g < 6. Results of this article can be applied in the study of relations in the Torelli group T_g.     

Rank one Eisenstein cohomology of local systems on the moduli space of abelian varieties

We give a formula for the Eisenstein cohomology of local systems on the partial compactification of the moduli of principally polarized abelian varieties given by rank 1 degenerations.     

On the measure with maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials

The Teichmüller flow g _t on the moduli space of Abelian differentials with zeros of given orders on a Riemann surface of a given genus is considered. This flow is known to preserve a finite absolutely continuous measure and is ergodic on every connected component ? of the moduli space. The main result of the paper is that µ/µ(?) is the unique measure with maximal entropy for the restriction of g _t to ?. The proof is based on the symbolic representation of g _t .     

The birational type of the moduli space of even spin curves

We determine the Kodaira dimension of the moduli space Sg of even spin curves for all g. Precisely, we show that Sg is of general type for g>8 and has negative Kodaira dimension for g<8.     

On twisting of comodule algebras

We study twistings of H-comodule algebras for a bialgebra H, recently introduced by M. Beactie, C-Y. Chen, aud J. J. Zhang. We describe a close connection between twistings and opposite smash products. We also prove the draracterization of crossed products R_σ, H as twisted algebrar of R?H (proved by Beattie et d. for Hopf algebras) for more geberal bialgebras, including all coconunutative bialgebras.     

Integrable cocycles and global deformations of Lie algebra of type G2 in characteristic 2

All classes of integrable cocycles in H²(L,L) are obtained for Lie algebra of type G₂ over an algebraically closed field of characteristic 2. It is proved that there exist only two orbits of classes of integrable cocycles with respect to automorphism group. The global deformation is shown to exist for any nontrivial class of integrable cocycles. These deformations are isomorphic to one of the two algebras of Cartan type, one of which being S(3:1,ω) while the other H(4:1,ω).     

On the intersection theory of the moduli space of rank two bundles

We give an algebro-geometric derivation of the known intersection theory on the moduli space of stable rank 2 bundles of odd degree over a smooth curve of genus g. We lift the computation from the moduli space to a Quot scheme, where we obtain the intersections via equivariant localization with respect to a natural torus action.     

关于非主Hopf曲面的模空间

构造了Ｓ＾１上Ｓ＾３｜Ｈ－丛的所有复结构的模空间,其中丛的转换函数ｕ：ｓ＾３／Ｈ→Ｓ＾３／Ｈ是Ｓ＾３／Ｈ的一个对合,ＨＵ（２）,Ｈ为有限群且在Ｓ＾３／Ｈ上的作用是自由的。真不连续的。     

Cohomology of moduli spaces of Del Pezzo surfaces

We compute the rational Betti cohomology groups of the coarse moduli spaces of geometrically marked Del Pezzo surfaces of degree 3 and 4 as representations of the Weyl groups of the corresponding root systems. The proof uses a blend of methods from point counting over finite fields and techniques from arrangement complements.     

On a desingularization of the moduli space of noncommutative tori

It is shown that the moduli space of the noncommutative tori admits a natural desingularization by the group . Namely, we prove that the moduli space of pairs is homeomorphic to a punctured two-dimensional sphere. The proof is based on a correspondence (a covariant functor) between the complex and noncommutative tori.

     

Isotrivially fibred isles in the moduli space of surfaces of general type

We complement Catanese's results on isotrivially fibred surfaces by completely describing the components containing an isotrivial surface with monodromy group . We also give an example for deformation equivalent isotrivial surfaces with different monodromy group.

     

On the smoothness of the moduli space of mathematical instanton bundles

In this paper we prove that the moduli spaces MI _2n+1(k) of mathematical instanton bundles on ²ⁿ⁺¹ with quantum number k are singular for n 2 and k 3 ,giving a positive answer to a conjecture made by Ancona and Ottaviani in 1993.     

Hochschild cohomologies and deformations of the pointwise superproduct

We consider the associative superalgebra of smooth compactly supported functions on ℝ ⁿ taking values in a Grassmann algebra. We find the lower Hochschild cohomologies and also the deformations of this superalgebra under certain additional conditions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 323–346, March, 2009.     

Relations and deformations of odd Hamiltonian superalgebras

The following (co)homology groups of the Lie superalgebraH(0,n) are calculated: theH ² (H, H) cohomology with coefficients in the adjoint module and theH ₂ (LH) homology of the nitpotent subalgebra. It is shown that dimH ² (H,H) = 1. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 391–401, March, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-01263 and by the International Science Foundation under grant No. RO 4300.