Monads and bundles on rational surfaces (Q1880817)

The description of vector bundles as the cohomology of a monad, i.e. a complex of vector bundles \(A\to B\to C\) with the first map injective and the second map surjective, was first used by \textit{G. Horrocks} [Proc. Lond. Math. Soc. (3) 14, 689--713 (1964; Zbl 0126.16801)] to study bundles on \(\mathbb{P}^n\) and is established especially for the classification of bundles on surfaces since the results of \textit{W. Barth} [Invent. Math. 42, 63--91 (1977; Zbl 0386.14005)] for \(\mathbb{P}^2\). The author studies holomorphic vector bundles on the \(n\)-fold blowup \(\pi:\widetilde{\mathbb P}^2\to {\mathbb P}^2\). The main result is a description of bundles \(E\) such that \(\pi_*E\) is normalized and semi-stable as the cohomology of a monad \({\mathcal A}\rightarrow W\rightarrow{\mathcal B}\), where \(W\) is trivial and \(\mathcal A\) and \(\mathcal B\) are explicitly given extensions of direct sums of line bundles. Unfortunately, this monad turns out to jump for families of bundles and is quite complicated in the case of multiple blowups. However, for the blowup of \({\mathbb P}^2\) in \(n\) distinct points, the author derives a monad of a simpler form \[ H^1(E^*(-L))^*\otimes {\mathcal O}(-L)\oplus \bigoplus_{i=1}^n H^1(E(-L))\otimes {\mathcal O}(-L-E_i)\to W \to \bigoplus_{i=0}^n H^1(E(-L-E_i))\otimes {\mathcal O}(L+E_i) \] (here, \(E_i\) are the exceptional divisors and \({\mathcal O}(L)\simeq \pi^*{\mathcal O}_{{\mathbb{P}}^2}(1)\)). The author gives several applications of this description. First, he classifies bundles on \(\widetilde{\mathbb P}^2\) with Chern classes \(c_1=0\), \(c_2=2\). This example illustrates nicely the conjectural concept that ``the moduli space of a blowup is a blowup of the moduli space''. Then, bundles trivialized on a line apart from the exceptional locus are studied (this moduli space is related to based instantons, see \textit{S. K. Donaldson} [Commun. Math. Phys. 93, 453--460 (1984; Zbl 0581.14008)] and, for newer results, \textit{N. P. Buchdahl} [J. Differ. Geom. 24, 19--52 (1986; Zbl 0586.32034); J. Differ. Geom. 37, 669-687 (1993; Zbl 0793.53025)]; \textit{H. Kurke} and \textit{A. Matuschke} [in: Algebraic geometry: Hirzebruch 70. Contemp. Math. 241, 239--271 (1999; Zbl 0951.14028)] and \textit{J.P Santos} [Blowups of surfaces and moduli of holomorphic vector bundles, preprint, \texttt{http://arxiv.org/abs/math.AG/0212176}]). The monad description is explicitly computed and used to disproof a conjecture of \textit{J. Bryan} and \textit{M. Sanders} [Proc. Am. Math. Soc. 125, No.12, 3763--3768 (1997; Zbl 0885.58010)] about the homotopy type of the moduli space of rank-stable instantons (similar results were obtained by \textit{J.P Santos} [Rank stable instantons over positive definite four manifolds, preprint, \texttt{http://arxiv.org/abs/math.AG/0301158}]). Finally, bundles on the blowup of \(\mathbb{P}^2\) in one point which are trivial on a neigbourhood of a non-exceptional line are classified.