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This question is an extension of this one. Consider a complex manifold $(M^{2n}, J)$. Fix $1 \leq p \leq n-1$, and suppose that the space of holomorphic sections of $\Lambda^{p,0}$ spans $\Lambda^{p,0}_x$ for all $x \in M$. (The referenced question above is the case $p=1$.) How wide a class of manifolds is this? Certainly complex tori satisfy this, but I imagine there are other examples. Of particular interest is the case $p=2$, where the question is at least in principle related to the existence of holomorphic Poisson structures.

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A very incomplete answer: if $c_1<0$ then there are no global sections of these tensor bundles: Kobayashi, First Chern class and holomorphic tensor fields, Nagoya Math. J., vol. 77, 1980, theorem A.

A compact complex manifold admitting a Kaehler metric immerses into a complex torus if and only if its cotangent bundle is spanned by global sections (https://arxiv.org/abs/1702.01701).

On the other hand, Robert Bryant, in his paper Rigidity and quasi-rigidity of Hermitian cycles in Hermitian symmetric spaces, says that

if a vector bundle $F \to M$ is generated by its global sections and $M$ is compact and Kaehler, then, as is well-known, $c_2(F) ≥ 0$. If equality holds, then either $F$ is the pullback to $M$ of a holomorphic bundle $F' → C$ over a curve $C$ via a holomorphic map $M → C$ or else $F = L ⊕ T$ where $L$ is a line bundle and $T$ is trivial. There is a similar (though more complicated) characterization when $c_3(F) = 0$.

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