Skew Invariant Theory of Symplectic Groups, Pluri-Hodge Groups and 3-Manifold Invariants

Abstract

IntroductionThis paper is concerned with four subjects. The second and third of these appear to be, at first sight, quite disparate. The fourth, however, shows how they are, somewhat surprisingly, related.The first has to do with skew invariant theory. Let V be a finite dimensional complex vector space with a skew symmetric non-degenerate form and let Sp(V ) be the corresponding symplectic group. For a complex vector space W we consider the action of Sp(V ) on the exterior algebra Λ(V ⊗ W ) (the action being trivial on W ). In Theorem 2.3 we give generators and relations for the algebra of Sp(V ) invariants for this action ("Fundamental theorem for skew invariant theory"). The relations are the so-called "P n " relations which appear in the study of certain 3-manifold invariants.Next we show that there is a natural action of Sp(g), where g = dim C W , on Λ(V ⊗ W ), by considering the spin representation of the orthogonal algebra associated to V ⊗ (W ⊕ W * ). The action of Sp(V ) × Sp(g) on Λ(V ⊗ W ) is multiplicity free and we determine the highest weights of the representations which occur, Theorem 2.4.The second subject is concerned with the study of Dolbeault cohomology groups with values in specific vector bundles. Let X be a compact closed complex manifold. Let Ω p X be the sheaf of holomorphic p-forms on X. The cohomology groups of interest areThese cohomology groups are not exotic. For example, the pluri-canonical sections, H 0 (X, K ⊗m X ), are well studied. In keeping with this terminology I call these cohomology groups, pluri-Hodge groups.Some of the motivation for studying pluri-Hodge groups, when X is holomorphic symplectic, comes from physics. In section 3.1 I give an alternative motivation. Namely one can show that knowing the dimensions of these groups (actually one needs somewhat less) one knows all the Chern numbers of X. Theorem 3.3 gives a precise formula for the Chern numbers in terms of the dimensions of the pluri-Hodge groups. Along the way a pluri-χ y genus is defined. 1In section 4 one specializes to holomorphic symplectic manifolds. In this setting physics suggests that there is an Sp(g) action on the direct sum of these pluri Hodge spaces. This is a straightforward consequence of the preparational material in section 1. The case g = 1 has been studied by Fujiki [5]. The multiplicities of the irreducible components of the total representations are determined in Theorem 4.4. Also in this section Theorem 4.3 shows that the graded trace of an Sp(g) element is essentially (up to normalization) the pluri χ y genus. This generalizes the situation for the usual χ y genus.For a hyper-Kähler manifold X there are good reasons (see below) to believe that the dimensions of the pluri-Hodge groups do not depend on the intrinsic P 1 of complex structures associated with X. In Theorem 5.1 the dimensions of the pluri-Hodge groups are determined in the case of K3 surfaces and this shows that indeed the dimensions do not depend on the specific K3 surface.Obviously, one unifying theme between the two subjec...

The zero section of the universal semiabelian variety and the double ramification cycle

“…Remark 3.2. While we give an elementary direct computational proof of the theorem below, Moonen explained to us that this result can be deduced from a special case of his work [Moo11], combined with the results of Thompson [Tho07], stated in terms of representation theory. Indeed, recall that given an ample divisor class L on a projective variety X, multiplication by L defines a degree 2 operator e on the cohomology ring H * (X, Q), which by the hard Lefschetz theorem extends to an sl 2 -action on H * (X, Q), known as the Lefschetz action.…”

Stable cohomology of the perfect cone toroidal compactification of 𝒜 g

“…The first author is grateful to Dmitry Zakharov for many stimulating discussions on topics related to the cohomology of families of abelian varieties. We are grateful to Ben Moonen for bringing results of Thompson [Tho07] to our attention, and indicating how they relate to the questions we address in this paper. We thank Jayce Getz who asked us in a talk whether a statement such as Theorem 1.2 could hold.…”

“…In this section we describe the stable cohomology of the n'th fiber product X ×n g of the universal family X g → A g , for a fixed n. It turns out (we thank Ben Moonen for pointing this out and explaining it to us) that the description of the subring in the cohomology of a very general ppav generated by divisors follows from the results of Thompson [Tho07] on invariant theory for the symplectic group (this construction is also a special case of a much more general deep construction of Looijenga and Lunts [LL97] in cohomology and of Moonen [Moo13] in the Chow ring). The results of Thompson [Tho07] are formulated in terms of representations of the symplectic group, which we think of as local systems on A g ; we give the reformulation in terms of cohomology classes.…”

“…Thompson's results allow us to describe completely the subalgebra of the rational cohomology of X ×n g generated by T i , P jk , also outside the stable range. Specifically, Theorem 3.4 of [Tho07] corresponds to the statement that the ideal of relations among the classes T i and P jk is generated by (g + 1)'st powers of divisors, i.e. by relations of the form ( m 2 i T i + m j m k P jk ) g+1 = 0, for arbitrary m 1 , .…”