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Let $\pi:E\to X$ be a holomorphic vector bundle over a complex manifold. Denote by $\tilde{E}=\pi^*E\to E$ the pullback of $E$ over itself. There exists a tautological line bundle $L\subset \tilde{E}$ ...

Given a short exact sequence of holomorphic Hermitian vector bundles $$0\rightarrow F\rightarrow E\rightarrow G\rightarrow 0,$$ the second fundamental form measures the obstruction of $E\simeq F\oplus ...

Let $E\rightarrow \mathbb{P}^1$ be a complex vector bundle and let $a_{(0,0)},a_{(1,0)},a_{(0,1)},a_{(1,1)}$ be differential forms such that $a_{(i,j)}\in\Omega^{i,j}(\mathbb{P}^1,End(E))$. I would ...

The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" ...

A line bundle over a complex manifold is called positive is if its Chern class is the fundamental form of a Kaehler manifold. For vector bundles of higher rank, the Chern class is no longer in general ...