Computations of instanton invariants

Abstract

Motivated by newly discovered properties of instantons on non-compact spaces, we realised that certain analytic invariants of vector bundles detect fine geometric properties. We present numerical algorithms, implemented in Macaulay 2, to compute these invariants. Precisely, we obtain the direct image and first derived functor of the contraction map $\pi \colon Z \to X$, where $Z$ is the total space of a negative bundle over $\mathbb{P}^1$ and $\pi$ contracts the zero section. We obtain two numerical invariants of a rank-2 vector bundle $E$ on $Z$, the width $h^0\bigl(X; (\pi_*E)^{\vee \vee} \bigl/ \pi_*E\bigr)$ and the height $h^0\bigl(X; R^1 \pi_*E \bigr)$, whose sum is the local holomorphic Euler characteristic $\chi^\text{loc}(E)$.