Rank Two Bundles on the Blow-up ofC2

✍ Scribed by Elizabeth Gasparim

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 134 KB
• Volume: 199
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1997.7156

✦ Synopsis

In this paper we study holomorphic rank two vector bundles on the blow-up of C 2 with vanishing Chern class. The restriction of such a bundle over the excep-Ž . Ž . tional divisor splits as O O j [ O O yj for some integer j. We denote by M M the j moduli space of holomorphic bundles on the blow-up of C 2 whose restriction Ž . Ž . to the exceptional divisor is O O j [ O O yj . We prove that M M is generically a j complex projective space of dimension 2 j y 3. ᮊ 1998 Academic Press 1. INTRODUCTION Holomorphic vector bundles over complex surfaces have been extensively studied by several different methods. See, for example, the books of w x w x Kobayashi 9 , Okonek, Schneider, and Spindler 10 , and Donaldson and w x Kronhheimer 2 . A fundamental result on the classification of rational surfaces is: ''Every rational surface is obtained by blowing up points on 2 Ž w x. either P or on a rational ruled surface'' see Griffiths and Harris 6 . This result suggests that the understanding of vector bundles on rational surfaces depends on the analysis of the behavior of vector bundles under blow-ups.

Some works on holomorphic bundles on blow-ups are the papers by w x w x w x Friedman and Morgan 3, 4 , Brussee 1 , and Qin 11 . Roughly speaking we may see the ''difference'' between moduli spaces of bundles on a rational surface and moduli spaces of bundles on one of its minimal models by studying bundles on the blow-up of C 2 . In this work we concentrate on the study of bundles on blow-ups in the local sense, that is, 581