I am just starting to study complex K3 surfaces and I'm trying to understand why the different definitions I found are actually equivalent.
Definition 1: a (complex) K3 surface is a compact, connected, Kahler, complex surface such that the canonical bundle is trivial and $H^1(X,\mathcal{O}_X)=0$.
Definition 2: a (complex) K3 surface is a compact, connected, Kahler, complex surface such that the canonical bundle is trivial and $\pi_1(X)=0$.
I've seen that (1) implies (2) on Huybrechts book "Lectures on K3 manifold" but I'm still wondering why the converse is true. I tried playing with the exponential sequence but I still got nothing.
That (2)$\Rightarrow$(1) follows from Hodge theory.
Hodge decomposition says $H^1(X,\mathbb{C})\cong H^{1,0}(X)\oplus H^{0,1}(X)$ and $\overline{H^{1,0}(X)}=H^{0,1}(X)$. Since $H^{0,1}(X)=H^1(X,O_X)$, we are done by $H^1(X,\mathbb{C})=\operatorname{Hom}(\pi_1(X),\mathbb{C})=0$.