# Computations of instanton invariants - Mathematics > Commutative Algebra

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Abstract: Motivated by newly discovered properties of instantons on non-compact spaces,we realised that certain analytic invariants of vector bundles detect finegeometric properties. We present numerical algorithms, implemented in Macaulay2, to compute these invariants.Precisely, we obtain the direct image and first derived functor of thecontraction map $\pi \colon Z \to X$, where $Z$ is the total space of anegative bundle over $\mathbb{P}^1$ and $\pi$ contracts the zero section. Weobtain two numerical invariants of a rank-2 vector bundle $E$ on $Z$, the width$h^0\biglX; \pi *E^{\vee \vee} \bigl- \pi *E\bigr$ and the height$h^0\biglX; R^1 \pi *E \bigr$, whose sum is the local holomorphic Eulercharacteristic $\chi^\text{loc}E$.

Autor: Thomas Köppe

Fuente: https://arxiv.org/