Sections of the canonical bundle This is maybe a stupid question.
Let $M$ be a simply-connected complex (kahler?) manifold, is it true that the canonical bundle $K_M$ has always (global) sections?
For example, we know that an Enriques surface is not simply-connected, but its canonical bundle has no sections. On the other hand, a K3 surface is simply-connected and its canonical has sections.
I was wondering if this is always the case. My guess is: yes! but I still cannot prove it. If anyone can give me an hint it will be much appreciated.
Thank you very much!
 A: You can find the several interesting results about existence of global holomorphic section on P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York (1978).
If $L$ is a holomorphic line bundle over a compact Kahler manifold $M$ such that $c_1(L) < 0$,
then there are no nonzero global holomorphic sections of $L$. For proof see this survey paper
Let $L$ be a holomorphic line bundle over a compact, Kahler manifold $M$ with $c_1(L) = 0.$
If $L$ is not the trivial bundle, then $H^0(M, L) = 0.$ See Proposition 10.4
Reid conjecture: If $X$ is a $\mathbb Q$-Fano 3-fold, then $\mathcal O_X(-K_X)$ has a global section.
Lemma 16.3: Let $X_d$ in $\mathbb P(a_0,...,a_4)$ be a family of $\mathbb Q$-Fano 3-folds with only isolated terminal singularities. Suppose also that $a_0\leq a_1\leq ...\leq a_4$ and $a_4\nmid d$, then $\omega_X^{-1}$, has a global section.
The space of global sections of $\mathcal O_{\mathbb P_k^n}(d)$ is $0$ for $d < 0$ and isomorphic to the vector
space of homogeneous polynomials of degree $d$ in $n + 1$ variables for $d ≥ 0$. See p.13
