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**On the classification of stable rank-\(r\) vector bundles over the projective plane.**
*(English)*
Zbl 0446.14006

Vector bundles and differential equations, Proc., Nice 1979, Prog. Math. 7, 113-144 (1980).

Author’s introduction: “I give some results concerning the moduli of rank-\(r\) vector bundles. We shall restrict ourselves to \(c_i(\mathcal T) = 0\). The reason is that though most of the ideas should also work for \(c_i(\mathcal T) \ne 0\) the technical modifications which are necessary, are quite substantial, and so far I have not been able to bring them into a unified theory.

In the first chapter we shall characterize those vector bundles \(\mathcal T\)-with \(h^0(\mathcal T) = h^0(\mathcal T^*) = 0\) (which holds for stable bundles) by means of the map

\[ H^i(\mathcal T(-2)) \otimes \Gamma(\mathcal O(1)) \to H^i(\mathcal T(-1)). \]

This will lead us to equivalence classes of certain “Kronecker-modules” \(\alpha\colon \mathbb C^n\otimes \Gamma(\mathcal O(1)) \to \mathbb C^n\). One of the main points there will be to determine the rank of a bundle \(\mathcal T\) which belongs to a given \(\alpha\).

In the second chapter we shall make use of the above construction to prove that the moduli-scheme of stable vector bundles with given rank and given second Chern class will be irreducible.

The last chapter is concerned with stability. We first recall the definition of stability given by Mumford-Takemoto. Then we consider stability (in the sense of Mumford) of the Kronecker-modules under the operation of the group \(\mathrm{SL}(n,\mathbb C)\times \mathrm{SL}(n,\mathbb C)\). We prove that for all rank-2 bundles and that for rank-3 bundles with \(c_2(\mathcal T) \le 7\) the bundle \(\mathcal T\) is stable if and only if \(\alpha\) is stable. As an application we get the Grauert-Mülich theorem for rank-2 bundles.”

[For the entire collection see Zbl 0431.00023.]

In the first chapter we shall characterize those vector bundles \(\mathcal T\)-with \(h^0(\mathcal T) = h^0(\mathcal T^*) = 0\) (which holds for stable bundles) by means of the map

\[ H^i(\mathcal T(-2)) \otimes \Gamma(\mathcal O(1)) \to H^i(\mathcal T(-1)). \]

This will lead us to equivalence classes of certain “Kronecker-modules” \(\alpha\colon \mathbb C^n\otimes \Gamma(\mathcal O(1)) \to \mathbb C^n\). One of the main points there will be to determine the rank of a bundle \(\mathcal T\) which belongs to a given \(\alpha\).

In the second chapter we shall make use of the above construction to prove that the moduli-scheme of stable vector bundles with given rank and given second Chern class will be irreducible.

The last chapter is concerned with stability. We first recall the definition of stability given by Mumford-Takemoto. Then we consider stability (in the sense of Mumford) of the Kronecker-modules under the operation of the group \(\mathrm{SL}(n,\mathbb C)\times \mathrm{SL}(n,\mathbb C)\). We prove that for all rank-2 bundles and that for rank-3 bundles with \(c_2(\mathcal T) \le 7\) the bundle \(\mathcal T\) is stable if and only if \(\alpha\) is stable. As an application we get the Grauert-Mülich theorem for rank-2 bundles.”

[For the entire collection see Zbl 0431.00023.]

Reviewer: Steven L. Kleiman (Cambridge, MA)

### MSC:

14F06 | Sheaves in algebraic geometry |

32L05 | Holomorphic bundles and generalizations |

14D20 | Algebraic moduli problems, moduli of vector bundles |