Arnaud Beauville

Conformal Blocks and Generalized Theta Functions

LetSUXr be the moduli space of rankr vector bundles with trivial determinant on a Riemann surfaceX. This space carries a natural line bundle, the determinant line bundleL. We describe a canonical isomorphism of the space of global sections ofLk with the space of conformal blocks defined in terms of representations of the Lie algebraslr(C((z))). It follows in particular that the dimension ofH0(SUXr,Lk) is given by the Verlinde formula.

Counting rational curves on

The aim of these notes is to explain the remarkable formula found by Yau and Zaslow [Y-Z] to express the number of rational curves on a K3 surface. Projective K3 surfaces fall into countably many families (Fg)g≥1 ; a surface in Fg admits a gdimensional linear system of curves of genus g . A naive count of constants suggests that such a system will contain a positive number, say n(g) , of rational (highly singular) curves. The formula is∑

{\text{K3}}

We show that the Chow group of 0-cycles on a K3 surface contains a class of degree 1 with remarkable properties: any product of divisors is proportional to this class, and so is the second Chern class c2.

surfaces

Let G be a complex semi-simple group, and X a compact Riemann surface. The moduli space of principal G-bundles on X, and in particular the holomorphic line bundles on this space and their global sections, play an important role in the recent applications of Conformal Field Theory to algebraic geometry. In this paper we determine the Picard group of this moduli space when G is of classical or G2 coarse moduli space and the moduli stack).

ON THE CHOW RING OF A K3 SURFACE

We prove in this note the following result: Theorem .− A smooth complex projective hypersurface of dimension ≥ 2 and degree ≥ 3 admits no endomorphism of degree > 1 . Since the case of quadrics is treated in [PS], this settles the question of endomorphisms of hypersurfaces. We prove the theorem in Section 1, using a simple but efficient trick devised by Amerik, Rovinsky and Van de Ven [ARV]. In Section 2 we collect some general results on endomorphisms of projective manifolds; we classify in particular the Del Pezzo surfaces which admit an endomorphism of degree > 1 .

The Picard group of the moduli of

This note, written in 1994, answers a question of Dolgachev by constructing a Calabi–Yau threefold whose fundamental group is the quaternion group H = H8. The construction is reminiscent of Reid’s unpublished construction of a surface with pg = 0, K2 = 2 and π1 = H; I explain below the link between the two problems.

G

The usual structures of symplectic geometry (symplectic, contact, Poisson) make sense for complex manifolds; they turn out to be quite interesting on projective, or compact Kahler, manifolds. In these notes we review some of the recent results on the subject, with emphasis on the open problems and conjectures.

-bundles on a curve

For a smooth projective variety X, let CH(X) be the Chow ring (with rational coefficients) of algebraic cycles modulo rational equivalence. The conjectures of Bloch and Beilinson predict the existence of a functorial ring filtration of CH(X). We want to investigate for which varieties this filtration splits, that is, comes from a graduation on CH(X) -- this occurs for K3 surfaces and, conjecturally, for abelian varieties. We observe that, though the Bloch-Beilinson filtration is only conjectural, the fact that it splits has some simple consequences which can be tested in concrete examples. Namely, for a regular variety X, it implies that the sub-Q-algebra of CH(X) spanned by divisor classes injects into the cohomology of X . We give examples of Calabi-Yau threefolds which do not satisfy this property. On the other hand we conjecture that the property does indeed hold for (holomorphic) symplectic manifolds, and we give some (weak) evidence in favour of this conjecture.

Endomorphisms of hypersurfaces and other manifolds

We prove that a double covering of P^3 branched along a very general sextic surface is not stably rational.

New Trends in Algebraic Geometry: A Calabi–Yau threefold with non-Abelian fundamental group

Let J be the Jacobian of a smooth curve C of genus g, and let A(J) be the ring of algebraic cycles modulo algebraic equivalence on J , tensored with Q. We study in this paper the smallest Q-vector subspace R of A(J) which contains C and is stable under the natural operations of A(J): intersection and Pontryagin products, pull back and push down under multiplication by integers. We prove that this “tautological subring” is generated (over Q) by the classes of the subvarieties W1 = C,W2 = C + C, . . . ,Wg−1. If C admits a morphism of degree d onto P, we prove that the last d− 1 classes suffice.