3 problems

Let $X$ be a compact complex threefold satisfying the ordinary $\partial\bar\partial$-lemma, and let $E$ be a numerically flat holomorphic vector bundle of rank $2$ on $X$. Must…

Let $M$ be a compact, connected Riemann surface, let $E\rightarrow M$ be a holomorphic vector bundle with a smooth Hermitian metric $h$, and let $dV$ be a smooth volume form. For a…

Let $M$ be an algebraic manifold with $\dim_{\mathbb{C}}M\geq3$, let $L$ be a very ample line bundle on $M$, and let $\omega\in K(M,L,x\,dy-y\,dx)$ be a foliation with normal bundl…