Polystable bundles and representations of their automorphisms (Q2119247)

This article studies the relationship between stability for holomorphic vector budles on a compact Kähler manifold and Geometric Invariant Theory. Let \(X\) be a compact complex manifold and let \(E_0\) be a holomorphic vector bundle on \(X\). Let \(H^1(X, \mathrm{End} E_0)\) be the space of infinitesimal deformations. After preliminaries on connections on bundles in section 1, section 2 constructs a semi-universal deformation parameterized by the space \(\Psi^{-1}(0)\), where \(\Psi:\mathcal{N}\longrightarrow H^2(X, \mathrm{End} E_0)\) with \(\mathcal{N}\subset H^1(X, \mathrm{End} E_0)\). Sections 3 and 4 study the neighborhood \(\mathcal{N}\) in the case \(E_0\) is semistable and polystable, respectively, showing that, in the polystable case, \(\Psi\) is equivariant with respect to the action of \(\mathrm{Aut}(E_0)\) (c.f. Corollary 4.3) and that two bundles in \(\Psi^{-1}(0)\) are isomorphic if and only if they lie in the same orbit under \(\mathrm{Aut}(E_0)\) (c.f. Corollary 4.9) Section 5 deals with the relationship between the GIT problem of \(\Psi^{-1}(0)\) under the action of \(\mathrm{Aut}(E_0)\) (\(\Gamma\)-stability, with \(\Gamma=\mathrm{Aut} E_0\)) and stability with respect to the Kähler form (\(\omega\)-stability), showing that \(\Psi^{-1}(0)\) is equivariant under \(\Gamma\) (completing Theorem 1). Theorem 5.4 proves, restricting the neighborhood if necessary, one direction of a Hitchin-Kobayashi correspondence for these objects, i.e. \(\omega\)-(poly)stability of the holomorphic bundles implies \(\Gamma\)-(poly)stability in the sense of GIT (c.f. Theorems 2 and 3). Section 6 studies the central component of the curvature in a neighborhood of an hermitian connection. Using the existence of an Hermite-Einstein connection for \(E_0\) and the fact that PDEs are elliptical, the action of the complexified gauge group \(\mathcal{G}\) on the space of integrable semi-connections is reduced to an action of \(\Gamma\). Sections 7 and 8 show the other implication of the Hitchin-Kobayashi correspondence giving a condition for a connection \(\alpha\) near \(d_0\) (inducing the holomorphic structure \(E_0\)) having another connection with central component of the curvature equal to a scalar multiple of the identity in its orbit under \(\mathcal{G}\) (c.f. Theorem 7.2, with dependence on \(\alpha\) itself and Theorem 8.1, with dependence on \(\|\alpha\|\), completing Theorem 3). Finally, section 9 deals with the case when \(E\) is semistable not polystable with the polystable \(E_0\) in the closure of their orbit under \(\mathcal{G}\). Theorem 9.2 (Theorem 4) shows that \(E_0\) is isomorphic to the graded object of a Jordar-Hölder filtration of \(E\), as classical GIT says.