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I'm reading Yang–Mills connections and Einstein–Hermitian metrics by Itoh and Nakajima. On definition 1.8 they define a notion for an Einstein–Hermitian connection $A$ by $$K_A = \lambda(E)\mathrm{id}...

I'm confused with working out the Chern character in the following special setting. Let $E$ be a spinor bundle $$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$ over sphere $S^{2n}$, where $\rho$ ...

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 ...

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 ...

As the question suggests, what is the intuition behind the Kodaira Vanishing Theorem? The Kodaira Vanishing Theorem says that the cohomology groups $H^q(M, L \otimes K_M)$ vanish for $q \ge 1$ when $L$...