Instantons and curves on class VII surfaces

Andrei Teleman¹

¹ LATP, UMR 6632; CMI, Université de Provence<br/>Centre de Mathématiques et Informatique, Université de Provence<br/>39 Rue F. Joliot-Curie<br/>13453 Marseille Cedex 13<br/>France

Annals of mathematics, Tome 172 (2010) no. 3, pp. 1749-1804.

Résumé

We develop a general strategy, based on gauge theoretical methods, to prove existence of curves on class VII surfaces. We prove that, for $b_2=2$, every minimal class VII surface has a cycle of rational curves hence, by a result of Nakamura, is a global deformation of a one parameter family of blown up primary Hopf surfaces. The case $b_2=1$ was solved in a previous article. The fundamental object intervening in our strategy is the moduli space ${\mathcal M}^{\rm pst}(0,{\mathcal K})$ of polystable bundles ${\mathcal E}$ with $c_2({\mathcal E})=0$, $\det({\mathcal E})={\mathcal K}$. For large $b_2$ the geometry of this moduli space becomes very complicated. The case $b_2=2$ treated here in detail requires new ideas and difficult techniques of both complex geometric and gauge theoretical nature. We explain the substantial obstacles which must be overcome in order to extend our methods to the case $b_2\geq 3$.

Détail

DOI : 10.4007/annals.2010.172.1749