Computations of instanton invariants
Abstract
Description
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 $π\colon Z \to X$, where $Z$ is the total space of a negative bundle over $\mathbb{P}^1$ and $π$ contracts the zero section. We obtain two numerical invariants of a rank-2 vector bundle $E$ on $Z$, the width $h^0\bigl(X; (π_*E)^{\vee \vee} \bigl/ π_*E\bigr)$ and the height $h^0\bigl(X; R^1 π_*E \bigr)$, whose sum is the local holomorphic Euler characteristic $χ^\text{loc}(E)$.
24/25 pages (A4/letter); Revisions: v1 - as submitted for publication. Contains the actual Macaulay2-code and explanation as attachments
24/25 pages (A4/letter); Revisions: v1 - as submitted for publication. Contains the actual Macaulay2-code and explanation as attachments