Analytic continuation of vector bundles with L^p-curvature

Research Report MRR98-050

Analytic continuation of vector bundles with L^p-curvature

Adam Harris and Yoshihiro Tonegawa

Abstract: This article addresses the problem of removable singularities for a Hermitian--holomorphic vector bundle ${\Cal E}$, defined on the complement of an analytic set $A$ of complex codimension at least two in a complex $n$--dimensional manifold $X$. In particular it is shown here that there exists a unque holomorphic bundle $\hat{\Cal E}$ on $X$, such that $\hat{\Cal E}\mid_{X\setminus A}\cong {\Cal E}$, when the curvature of ${\Cal E}$ belongs to $L^{n} (X\setminus A)$. This result is in fact sharp, as counterexamples exist for the extensibility of ${\Cal E}$ with curvature in $L^{p}, \ p< n$. Extension across general closed subsets of finite $(2n-4)$--dimensional Hausdorff measure then follows directly from a slicing theorem of Bando and Siu.

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Analytic continuation of vector bundles with Lp-curvature

Adam Harris and Yoshihiro Tonegawa

Analytic continuation of vector bundles with L^p-curvature