On the geometry of moduli spaces

We construct a Kähler metric on the moduli spaces of compact complex manifolds with c₁,<0 and of polarized compact Kähler manifolds with c₁=0, which is a generalization of the Petersson-Well metric. It is induced by the variation of the Kähler-Einstein metrics on the fibers that exist according to the Calabi-Yau theorem. We compute the above metric on the moduli spaces of polarized tori and symplectic manifolds. It turns out to be the Maaß metric on the Siegel upper half space and the Bergmann metric on a symmetric space of type III resp. In particular it is Kähler-Einstein with negative curvature.Dedicated to Karl SteinHeisenberg-Stipendiat der Deutschen Forschungsgemeinschaft     

Path-connectedness of the moduli spaces of metrics with positive isotropic curvature on four-manifolds

In the paper we prove the path-connectedness of the moduli spaces of metrics with positive isotropic curvature on certain compact four-dimensional manifolds.     

On the symplectic structure of instanton moduli spaces

We study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of SU(2) instantons on both commutative and non-commutative R⁴. We identify global Darboux coordinates and quadratic Hamiltonians on classical phase spaces for which these quiver varieties are natural completions. We also show that the group of non-commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non-commutative instantons. This paper should be viewed as a step towards extending known results for Calogero–Moser spaces to the instanton moduli spaces.     

On the Euler characteristic of Kronecker moduli spaces

Combining the MPS degeneration formula for the Poincaré polynomial of moduli spaces of stable quiver representations and localization theory, it turns that the determination of the Euler characteristic of these moduli spaces reduces to a combinatorial problem of counting certain trees. We use this fact in order to obtain an upper bound for the Euler characteristic in the case of the Kronecker quiver. We also derive a formula for the Euler characteristic of some of the moduli spaces appearing in the MPS degeneration formula.     

On the geometry of moduli spaces of symplectic structures

 An approach to a natural geometry of moduli spaces of symplectic structures is presented. Received: 21 March 2002 / Revised version: 28 August 2002 This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG).     

On the rationality of moduli spaces of pointed curves

Motivated by several recent results on the geometry of the modulispaces of stable curves of genus g with n marked points, we determine the birational structureof these spaces for small values of g and n by exploiting suitableplane models of a general curve. More precisely, _g,n is shownto be rational for g = 2 and 1 n 12, g = 3 and 1 n 14, g= 4 and 1 n 15, and g = 5 and 1 n 12     

On the moduli spaces of multipolygonal linkages in the plane

One can form a polygonal linkage by identifying initial and terminal points of two free linkages. Likewise, one can form a multipolygonal linkage by identifying initial and terminal points of three free linkages. The geometric and topological properties of moduli spaces of multipolygonal linkages in the plane are studied. These spaces are compact algebraic varieties. Some conditions under which these spaces are smooth manifolds, cross products or disjoint unions of moduli spaces of polygonal linkages, or connected, are determined. Dimensions in smooth manifold cases and some Euler characteristics are computed. A classification of generic multiquadrilateral linkages is also made.     

On moduli spaces for stable bundles on quadrics

We investigate the relation between stable representations of quivers and stable sheaves. A construction of thin smooth compact moduli spaces for stable sheaves on quadrics based on this relation is presented. Translated fromMatematicheskie Zametki, Vol 62, No. 6, pp. 843–864, December, 1997 Translated by S. K. Lando     

On relations among moduli of continuity in Lorentz spaces

Estimates are obtained among moduli of continuity of functions in several variables that belong to various Lorentz spaces. The functions considered are periodic with period 1 in each variable. More exactly the following theorem is proved: If 0<,<(t) and (t) are so-called -functions such that _;>₁>1, and

Singular symplectic moduli spaces

Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of symmetric products of a K3 or abelian surface and the class of moduli spaces found by O’Grady. Consequently, since singular moduli space that do not belong to these exceptional cases have singularities in codimension ≥4 they do no admit projective symplectic resolutions. A la mémoire de Joseph Le Potier Mathematics Subject Classification (1991) 14J60, 14D20, 14J28, 32J27