Vector Bundles Near Negative Curves: Moduli and Local Euler Characteristic

Abstract

We study moduli of vector bundles on a two-dimensional neighbourhood Z _k of an irreducible curve ℓ ≅ ℙ¹ with ℓ² = −k and give an explicit construction of their moduli stacks. For the case of instanton bundles, we stratify the stacks and construct moduli spaces. We give sharp bounds for the local holomorphic Euler characteristic of bundles on Z _k and prove existence of families of bundles with prescribed numerical invariants. Our numerical calculations are performed using a Macaulay 2 algorithm, which is available for download at http://www.maths.ed.ac.uk/~s0571100/Instanton/.

Key Words:

• Holomorphic vector bundles
• Local holomorphic Euler characteristic
• Local moduli

2000 Mathematics Subject Classification:

• 14J60

ACKNOWLEDGMENTS

We would like to thank the editor for having done such a wonderful job, and we thank the referee for several helpful suggestions.

Notes

¹A rank-2 bundle whose extension class is a polynomial not divisible by u is in fact not of splitting type j, but of a lower splitting type. See Remark 2.13.

²But the bundle with p(z, u) = z ⁻¹ u + z ⁴ u ² has charge 7. Also, it seems always possible to create a new bundle with charge +k via elementary transformations, and since charges 2 and 3 exist by Corollary 2.18, we expect these are the only charge gaps on Z ₃.

Communicated by I. Swanson.