The transcendental Hodge lattice of a projective manifold M is the smallest Hodge substructure in pth cohomology which contains all holomorphic p-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkähler manifold. As an application, we obtain a theorem about dimension of a compact torus T admitting a holomorphic symplectic embedding to a hyperkähler manifold M. If M is generic in a d-dimensional family of deformations, then dimT≥2^[(d+1)/2].
Let M be a closed symplectic manifold of volume V. We say that M admits an unobstructed symplectic packing by balls if any collection of symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in terms of the Kahler cones of their blow-ups. When M is a Kahler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple) these Kahler cones can be described explicitly using the Demailly and Paun structure theorem. We prove that any Campana simple Kahler manifold, as well as any manifold which is a limit of Campana simple manifolds in a smooth deformation, admits an unobstructed symplectic packing by balls. This is used to show that all even-dimensional tori equipped with Kahler symplectic forms and all hyperkahler manifolds of maximal holonomy admit unobstructed symplectic packings by balls. This generalizes a previous result by Latschev-McDuff-Schlenk. We also consider symplectic packings by other shapes and show, using Ratner's orbit closure theorem, that any even-dimensional torus equipped with a Kahler form whose cohomology class is not proportional to a rational one admits a full symplectic packing by any number of equal polydisks (and, in particular, by any number of equal cubes).
For every commutative ring A, one has a functorial commutative ring W(A) of p-typical Witt vectors of A, an iterated extension of A by itself. If A is not commutative, it has been known since the pioneering work of L. Hesselholt that W(A) is only an abelian group, not a ring, and it is an iterated extension of the Hochschild homology group HH_0(A) by itself. It is natural to expect that this construction generalizes to higher degrees and arbitrary coefficients, so that one can define “Hochschild–Witt homology” WHH∗(A,M) for any bimodule M over an associative algebra A over a field k. Moreover, if one want the resulting theory to be a trace theory, then it suffices to define it for A=k. This is what we do in this paper, for a perfect field k of positive characteristic p. Namely, we construct a sequence of polynomial functors W_m, m≥1 from k-vector spaces to abelian groups, related by restriction maps, we prove their basic properties such as the existence of Frobenius and Verschiebung maps, and we show that W_m are trace functors. The construction is very simple, and it only depends on elementary properties of finite cyclic groups.
Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of sln[s±1,t]) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of cohomology. This basis is an affine analogue of the Gelfand-Tsetlin basis. The affine analogue of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space Mn,d of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is naturally embedded into the cohomology ring of certain affine Laumon space. It is the image of the center Z of the Yangian of gln naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on Mn,d is the image of a noncommutative power sum in Z.