Algebraic Geometry

We show that the moduli space M of marked cubic surfaces is biholomorphic to the quotient by a discrete group generated by complex reflections of the complex four-ball minus the reflection hyperplanes of the group. Thus M carries a complex hyperbolic structure: an (incomplete) metric of constant holomorphic sectional curvature.

Each choice of a Kähler class on a compact complex manifold defines an action of the Lie algebra $\slt$ on its total complex cohomology. If a nonempty set of such Kähler classes is given, then we prove that the corresponding $\slt$-copies generate a semisimple Lie algebra. We investigate the formal properties of the resulting representation and we work things out explicitly in the case of complex tori, hyperkähler manifolds and flag varieties. We pay special attention to the cases where this leads to a Jordan algebra structure or a graded Frobenius algebra.

In order to use the technique of dimensional reduction, it is usually necessary for there to be a symmetry coming from a group action. In this paper we consider a situation in which there is no such symmetry, but in which a type of dimensional reduction is nevertheless possible. We obtain a relation between the Coupled Vortex equations on a closed Kahler manifold, X , and the Hermitian-Einstein equations on certain P 1 -bundles over X . Our results thus generalize the dimensional reduction results of Garcia-Prada, which apply when the Hermitian-Einstein equations are on X× P 1 .

The rational cohomology of the moduli space of rank two, odd degree stable bundles over a curve (of genus g > 1) has been studied intensely in recent years and in particular the invariant subring generated by Newstead's generators alpha, beta, gamma. Several authors have independently found a minimal complete set of relations for this subring. Their methods are very different from the methods originally employed by Kirwan to prove Mumford's conjecture -- that relations derived from the vanishing Chern classes of a particular rank 2g-1 bundle are a complete set of relations for the entire cohomology ring. This note contains two theorems which readily follow from Kirwan's original calculations. We rederive the above result showing that the first three invariant Mumford relations generate the relation ideal of the invariant subring. Secondly we prove a stronger version of Mumford's conjecture and show that the relations coming from the first vanishing Chern class generate the relation ideal of the entire cohomology ring as a Q[alpha,beta]-module. (Only a few typos have been amended in this revised version).

Mumford defined a natural isomorphism between the intermediate jacobian of a conic-bundle over P 2 and the Prym variety of a naturally defined étale double cover of the discrminant curve of the conic-bundle. Clemens and Griffiths used this isomorphism to give a proof of the irrationality of a smooth cubic threefold and Beauville later generalized the isomorphism to intermediate jacobians of odd-dimensional quadric-bundles over P 2 . We further generalize the isomorphism to the primitive cohomology of a smooth cubic hypersurface in P n .

The union of two quintic elliptic scrolls in P^4 intersecting transversally along an elliptic normal quintic curve is a singular surface Z which behaves numerically like a bielliptic surface. In the appendix to the paper [W. Decker et al.: Syzygies of abelian and bielliptic surfaces in P^4, alg-geom/9606013] where the equations of this singular surface were computed, we proved that Z defines a smooth point in the appropriate Hilbert scheme and that Z cannot be smoothed in P^4. Here we consider the analogous situation in higher dimensional projective spaces P^{n-1}, where, to our surprise, the answer depends on the dimension n-1. If n is odd the union of two scrolls cannot be smoothed, whereas it can be smoothed if n is even. We construct an explicit smoothing.

We review what is known about the Hodge conjecture for abelian varieties, with some emphasis on how Mumford-Tate groups have been applied to this problem.

Let Q be a finite quiver without oriented cycles. Denote by U --> M the fine moduli space of stable thin sincere representations of Q with respect to the canonical stability notion. We prove Ext^i(U,U) = 0 for all i >0 and compute the endomorphism algebra of the universal bundle U. Moreover, we obtain a necessary and sufficient condition for when this algebra is isomorphic to the path algebra kQ of the quiver Q. If so, then the bounded derived category of finitely generated right kQ-modules is embedded into that of coherent sheaves on M.

We prove a base point free theorem for nef and log big divisors on log canonical surfaces.

There are several ways to generalize characteristic classes for singular algebraic varieties. The simplest ones to describe are Chern-Mather classes obtained by Nash blow up. They serve as an ingredient to construct Chern-MacPherson-Schwartz classes. Unfortunately, they all are defined in homology. There are examples showing, that they do not lie in the image of Poincaré morphism. On the other hand they are represented by an algebraic cycles. Barthel, Brasselet, Fiesler, Kaup and Gabber have shown that, any algebraic cycle can be lifted to intersection homology. Nevertheless, a lift is not unique. The Chern-Mather classes are represented by polar varieties. We show how to define a canonical lift of Chern-Mather classes to intersection homology. Instead of the polar variety alone, we consider it as a term in the whole sequence of inclusions of polar varieties. The inclusions are of codimension one. In this case the lifts are unique.