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I think if you impose the condition of "positivity" on a holomorphic vector bundle (meaning that the curvature is positive definite on vector-valued forms), you can find global holomorphic sections th …

Is every holomorphic vector bundle a direct summand of a trivial vector bundle on submanifolds of C^n? What about projective varities? I believe Swan's theorem says something about the first question. …

In the original paper of Mehta-Seshadri, it seems like they treat the case of zero parabolic degree (i.e. they prove that zero parabolic degree stable parabolic bundles correspond to irreps of the fun …

I'd like to know if there exists a holomorphic rank 2 sub-bundle of $T\mathbb{P}^3$ which, when restricted to a given line is $\mathcal{O}(-a)\oplus \mathcal{O}(a)$, but is trivial when restricted to …

I wish to know if there is a rank 2 vector bundle $E$ on $\mathbb{P}^1 \times \mathbb{P}^1$ such that $\mathbb{P}(E)$ when restricted to $\mathbb{P}^1 \times [0:1]$ is the $n$th Hirzebruch surface and …

This might be standard if true (if so, I shall be grateful if provided with a reference). Given a smooth map from a Stein manifold $X$ to $\operatorname{Gr}(k,n)$ (the Grassmannian of $k$ planes in $\ …