# Projective variety of general type such that $S^m \Omega_X^1$ is globally generated - Part II

This is a follow-up to my previous question MO412306.

Let $$X$$ be a smooth complex projective surface of general type (this is the case I am mostly interested in, but one could ask the question in every dimension) and assume that for some $$m \geq 1$$ the vector bundle $$S^m \Omega_X^1$$ is generated by global sections, namely, the evaluation map $$H^0(X, \, S^m \Omega_X^1) \otimes \mathcal{O}_X \to S^m \Omega^1_X$$ is surjective.

The nice answer to the aforementioned question shows that in this situation $$K_X$$ is ample, essentially because our assumption on $$\Omega^1_X$$ implies that $$X$$ does not contain any smooth rational curve. So, let me now ask the following

Question. Is $$|K_X|$$ base-point free? If not, what is a counterexample?

Remark. The answer is yes when $$m=1$$. In fact, if $$\Omega_X$$ is globally generated then, for every $$x \in X$$, we can find two global sections $$a, \, b \in H^0(X, \, \Omega_X^1)$$ such that $$a(x), \, b(x)$$ generate the fibre $$\Omega^1_{X, \, x}$$. Thus $$a(x) \wedge b(x)$$ generate $$\wedge^2 \Omega^1_{X, \, x}=\omega_{X, \, x}$$, namely, $$a \wedge b$$ is a global section of $$K_X$$ that does not vanish at $$x$$.