Let $X$ be a compact Kahler manifold of complex dimension $n$. Fix a nonzero class $u \in H^1(X,T\_X)$. This gives a linear morphism
$$
\phi\_u : H^0(X,\Omega^n) \to H^{1}(X,\Omega^{n-1}),
\quad \sigma \mapsto u \cup \sigma.
$$
Is $\phi\_u$ injective?


It is so for manifolds with $\Omega^n\_X = \mathcal O\_X$; the proof I've got is not hard but uses Ricci-flat Kahler metrics and the hard Lefschetz theorem so it cannot generalize to other situations. In the examples I know (curves, hypersurfaces in $\mathbb P^n$) we have $h^{n,0} \leq h^{n-1,1}$, so I haven't stumbled upon an obvious counterexample yet.



Is the cup product of holomorphic $n$-forms with a fixed class injective?
The answer to this question is negative in dimensions $\ge 3$. For example, take a quintic in $\mathbb CP^4$ and consider its blow up $X$ in $10^{100}$ points (just to be safe). Then the space $H^1(X, T\_X)$ will be huge, since it parametrises deformations of the blown up variety and you can move points as you wish. So there will be non-zero $u\in H^1(X, T\_X)$ so that you map is trivial. 


Note that when you blow up the quintic $H^{2,1}$ does not change. 


Also, this trick with blow ups will not work for Kahler surfaces as is explained for example, in appendix 1 in <http://arxiv.org/pdf/1301.0478.pdf>


