On line bundles over real algebraic curves

Let X be a geometrically irreducible smooth projective curve defined over the real numbers. Let nX be the number of connected components of the locus of real points of X. Let x₁,…,x? be real points from ? distinct components, with ?<nX. We prove that the divisor x₁+?+x? is rigid. We also give a very simple proof of the Harnack's inequality.     

The moduli space of stable vector bundles over a real algebraic curve

We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension of \mathbbZ/2{\mathbb{Z}/2} by the fundamental group. By comparison with the space of real or quaternionic connections, some of the basic topological invariants of these spaces are calculated.     

On real moduli spaces of holomorphic bundles over M-curves

Let F be a genus g curve and σ:F→F a real structure with the maximal possible number of fixed circles. We study the real moduli space N^′=Fix(σ#) where σ#:N→N is the induced real structure on the moduli space N of stable holomorphic bundles of rank 2 over F with fixed non-trivial determinant. In particular, we calculate H^?(N^′,Z) in the case of g=2, generalizing Thaddeus' approach to computing H^?(N,Z).     

On rationality of moduli spaces of vector bundles on real Hirzebruch surfaces

Let X be a real form of a Hirzebruch surface. Let M _H (r,c ₁, c ₂) be the moduli space of vector bundles on X. Under some numerical conditions on r, c ₁ and c ₂, we identify those M _H (r,c ₁,c ₂) that are rational.     

On real algebraic vector bundles

Supported by the Netherlands Organisation for Scientific Research (NWO)     

Coherent algebraic sheaves in real algebraic geometry

Summary In the following paper we give examples of vector bundlesF→X defined on a regular, compact, connected affine varietyX such thatF has no algebraic structure.

---

Riassunto Nel presente lavoro si danno esempi di fibrati vettorialiF→X, definiti su varietà affini, regolari, compatte, connesseX che non ammettono struttura algebrica.

     

Vector bundles over real algebraic varieties

The object of this paper is to study continuous vector bundles, over real algebraic varieties, admitting an algebraic structure. For large classes of real varieties, we obtain explicit information concerning the Grothendieck group of algebraic vector bundles. We show that in many cases this group is small compared to the corresponding group of continuous vector bundles. These results are used elsewhere to study the geometry of real algebraic varieties.Dedicated to Professor Alexander Grothendieck on the occasion of his 60th birthdaySupported by the NSF Grant DMS-8602672.     

Algebraic loop groups and moduli spaces of bundles

We study algebraic loop groups and affine Grassmannians in positive characteristic. The main results are normality of Schubert-varieties, the construction of line-bundles on the affine Grassmannian, and the proof that they induce line-bundles on the moduli-stack of torsors. Received March 15, 2001 / final version received April 11, 2002?Published online November 19, 2002     

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     

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