Local Holomorphic Euler Characteristic and Instanton Decay

Abstract

We study the local holomorphic Euler characteristic χ x, F of sheaves near a surface singularity obtained from contracting a line ℓ inside a smooth surface Z. We prove non-existence of sheaves with certain prescribed numerical invariants. Non-existence of instantons on Z with certain charges follows, and we conclude that ℓ 2 poses an obstruction to instanton decay. A Macaulay 2 algorithm to compute χ is made available at Proposition 4.1. Let E 1 and E 2 be sl(2, C)-bundles over Z k with splitting types j 1 and j 2 , respectively. There exists an isomorphismIn particular, E 1 can decay totally over Z k if and only if j 1 ≡ 0 mod k.This paper consists of applications of the local holomorphic Euler characteristic to problems of existence and decay of instantons. We also discuss the Kobayashi-Hitchin correspondence over Z k . We obtain, via discussion of the physical consequences and an ad hoc definition of stability (Definition 5.2), the following conclusions:Proposition 5.4. There is a one-to-one correspondence between framed SU (2)-instantons on Z k with local charge n and framed-stable sl(2, C)-bundles on Z k with χ loc = n.Corollary 5.5. An sl(2, C)-bundle over Z k represents an instanton if and only if its splitting type is a multiple of k.Theorem 6.8. The minimal local charge of a nontrivial SU (2)-instanton on Z k is χ min k = k−1. The local moduli space of (unframed) instantons on Z k having fixed local charge χ min k has dimension k − 2.

Classical deformations of noncompact surfaces and their moduli of instantons

“…Therefore, we describe holomorphic bundles on the deformed surfaces to obtain the corresponding information about instantons. The Kobayashi-Hitchin correspondence for the surfaces Z k was shown in [16,Prop. 5.3]; the proof uses compactification and appeals to the compact version of the correspondence as described in [27].…”

“…The noncompact surfaces Z k admit a rich structure of moduli spaces of instantons. Some properties of these moduli are described in [16], where it is shown that such moduli spaces are quasi-projective varieties whose dimensions increase with the charge. Here we show that after a deformation of the complex structure, the moduli of instantons on the deformation are empty, i.e.…”

Sheaves on singular varieties

“…Our results showed that over the spaces Z k with k ≥ 3 there do not exist any 1-instantons, nevertheless higher charge instantons do exist (of course we mean mathematical existence proofs). In [GKM,Proposition 54] we studied the Kobayashi-Hitchin correspondence for the spaces Z k : We showed that an SU (2)-instanton on Z k of charge n corresponds to a holomorphic SL(2)-bundle E on Z k with χ(ℓ, E) = n together with a trivialization of…”

“…A simple observation [GKM,Proposition 4.1] shows that there exists a trivialization of E| Z • k if and only if n = 0 mod k. This restricts the splitting type of an instanton bundle over Z k to be of the form nk and lead us to the following existence/non-existence result:…”

“…[GKM,Theorem 6.8] The minimal local charge of a non-trivial SU (2)-instanton on Z k is χ min k = k − 1. The local moduli space of (unframed) instantons on Z k with fixed local charge χ min k…”