Left and right centers in quasi-Poisson geometry of moduli spaces

We introduce left central and right central functions and left and right leaves in quasi-Poisson geometry, generalizing central (or Casimir) functions and symplectic leaves from Poisson geometry. They lead to a new type of (quasi-)Poisson reduction, which is both simpler and more general than known quasi-Hamiltonian reductions. We study these notions in detail for moduli spaces of flat connections on surfaces, where the quasi-Poisson structure is given by an intersection pairing on homology.     

Any component of moduli of polarized hyperkähler manifolds is dense in its deformation space

Let M be a compact hyperkähler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v   in H²(M)$H^2(M)$ defines a divisor _Dv$D_v$ in W consisting of all algebraic manifolds polarized by v. We prove that every connected component of this divisor is dense in W.     

Siegel coordinates and moduli spaces for morphisms of Abelian varieties

We describe the moduli spaces of morphisms between polarized complex abelian varieties. The discrete invariants, derived from a Poincaré decomposition of morphisms, are the types of polarizations and of lattice homomorphisms occurring in the decomposition. For a given type of morphisms the moduli variety is irreducible, and is obtained from a product of Siegel spaces modulo the action of a discrete group. Mathematics Subject Classification (2000) 14K20     

Betti Numbers of parabolic U(2,1)-Higgs bundles moduli spaces

Let X be a compact Riemann surface together with a finite set of marked points. We use Morse theoretic techniques to compute the Betti numbers of the parabolic U(2,1)-Higgs bundles moduli spaces over X. We give examples for one marked point showing that the Poincaré polynomials depend on the system of weights of the parabolic bundle.      

Geometry of tropical moduli spaces and linkage of graphs

We prove the following “linkage” theorem: two p-regular graphs of the same genus can be obtained from one another by a finite alternating sequence of one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the linkage theorem to prove that various moduli spaces of tropical curves are connected through codimension one.     

Chow groups of the moduli spaces of weighted pointed stable curves of genus zero

The moduli space of weighted pointed stable curves of genus zero is stratified according to the degeneration types of such curves. We show that the homology groups of are generated by the strata of and give all additive relations between them. We also observe that the Chow groups and the homology groups are isomorphic. This generalizes Kontsevich-Manin's and Losev-Manin's theorems to arbitrary weight data A.     

On the structure of certain natural cones over moduli spaces of genus-one holomorphic maps

We show that certain naturally arising cones over the main component of a moduli space of J₀-holomorphic maps into Pn have a well-defined Euler class. We also prove that this is the case if the standard complex structure J₀ on Pn is replaced by a nearby almost complex structure J. The genus-zero analogue of the cone considered in this paper is a vector bundle. The genus-zero Gromov-Witten invariant of a projective complete intersection can be viewed as the Euler class of such a vector bundle. As shown in a separate paper, this is also the case for the “genus-one part” of the genus-one GW-invariant. The remaining part is a multiple of the genus-zero GW-invariant.     

The moduli of substructures of a compact complex space

We construct a space of fine moduli for the substructures of an arbitrary compact complex space . A substructure of is given by a subalgebra of the structure sheaf with the additional feature that is also a complex space; and are called equivalent if and only if and are isomorphic as subalgebras of .

Since substructures are quotients, it is only natural to start with the fine moduli space of all complex-analytic quotients of . In order to obtain a representable moduli functor of substructures, we are forced to concentrate on families of quotients which satisfy some flatness condition for relative differential modules of higher order. Considering the corresponding flatification of , we realize that its open subset consisting of all substructures turns out to be a complex space which has the required universal property.

     

Hilbert schemes of surfaces are dense in moduli

We prove that the locus of Hilbert schemes of n points on a projective K 3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds is proven as well. Along the way we prove an integral constraint on the monodromy group of generalized Kummer manifolds.     

Quantum cohomology of moduli spaces of genus zero stable curves

We investigate the (small) quantum cohomology ring of the moduli spaces of stable n-pointed curves of genus 0. In particular, we determine an explicit presentation in the case n = 5 and we outline a computational approach to the case n = 6. This research was partially supported by MIUR and GNSAGA of INdAM (Italy).     

Computation of the cohomology rings of Kac-Moody groups,their flag manifolds and classifying spaces

In this paper we introduce the history and present situation of the computation of the cohomology rings of Kac-Moody groups, their flag manifolds and classifying spaces, and give some problems and conjectures that deserve further study.     

The Mumford relations and the moduli of rank three stable bundles

The cohomology ring of the moduli space M(n,d) of semistable bundles of coprime rank n and degree d over a Riemann surface M of genus g 2 has again proven a rich source of interest in recent years. The rank two, odd degree case is now largely understood. In 1991 Kirwan [8] proved two long standing conjectures due to Mumford and to Newstead and Ramanan. Mumford conjectured that a certain set of relations form a complete set; the Newstead-Ramanan conjecture involved the vanishing of the Pontryagin ring. The Newstead–Ramanan conjecture was independently proven by Thaddeus [15] as a corollary to determining the intersection pairings. As yet though, little work has been done on the cohomology ring in higher rank cases. A simple numerical calculation shows that the Mumford relations themselves are not generally complete when n>2. However by generalising the methods of [8] and by introducing new relations, in a sense dual to the original relations conjectured by Mumford, we prove results corresponding to the Mumford and Newstead-Ramanan conjectures in the rank three case. Namely we show (Sect. 4) that the Mumford relations and these dual Mumford relations form a complete set for the rational cohomology ring of M(3,d) and show (Sect. 5) that the Pontryagin ring vanishes in degree 12g-8 and above.     

The universal theta divisor over the moduli space of curves

By computing the class of the universal antiramification locus of the Gauss map, we obtain a complete birational classification by Kodaira dimension of the universal theta divisor over the moduli space of curves.     

Picard groups of the moduli spaces of semistable sheaves I

We compute the Picard group of the moduli spaceU′ of semistable vector bundles of rankn and degreed on an irreducible nodal curveY and show thatU′ is locally factorial. We determine the canonical line bundles ofU′ andU _L ′, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata in the compactification ofU′.     

Witten's formulas for intersection pairings on moduli spaces of flat G-bundles

In a 1992 paper (J. Geom. Phys. 9 (1992) 303), Witten gave a formula for the intersection pairings of the moduli space of flat G-bundles over an oriented surface, possibly with markings. In this paper, we give a general proof of Witten's formula, for arbitrary compact, simple groups, and any markings for which the moduli space has at most orbifold singularities.