Let $F$ be a Enriques surface over $\Bbb C$. I have a question about a detail in the proof of Proposition 2.1. from Dolgachev's On automorphisms of Enriques surfaces.
This 2.1. Proposition. states that if $F$ is an Enriques surface, then $H_F:= \operatorname{Pic}(F)/\operatorname{Tors}$ is an even unimodular lattice of rank 10 with certain signature.
Now the proof argues as follows:
By the Hodge decomposition, we have $\operatorname{Pic}(F)=H^2(F,\Bbb Z), H^1(F,\Bbb Z)=\operatorname{Tors}$. By the formula $12(1-q+p_g)=K_F^2+c_2$, we get $H^2(F,\Bbb Z)/\operatorname{Tors}=\Bbb Z^{10}$ By Lefschetz, $\operatorname{Tors}(\operatorname{Pic}(F))= \operatorname{Tors}(H^2(F,\Bbb Z)$. By Poincare, $H_F= H^2(F,\Bbb Z)/\operatorname{Tors}$ is a unimodular lattice.
Question: The Lefschetz argument I not understood.
I'm not completely sure, but presumably Dolgachev refers to this Lefschetz theorem which states that the map $ \operatorname{Pic}(F) \to H^2(F,\Bbb Z) \to H^2(F,\Bbb C)$ obtained as composition first Chern map $c_1$ together with canonical inclusion $\Bbb Z \subset \Bbb C$ of constant sheaves, factors surjectively onto $H^2(F,\Bbb Z) \cap H^{1,1}(F)$ where the latter term comes from Hodge decompotion $H^2(F,\Bbb C)=H^{2,0} \oplus H^{1,1} \oplus H^{0,2} $.
Why from this follows $\operatorname{Tors}(\operatorname{Pic}(F))= \operatorname{Tors}(H^2(F,\Bbb Z))$?
(That's the thing, the question is not about why that holds, but why it's a consequence of Lefschetz as Dolgachev stated there)
By the way, Enriques condition implies $H^1(F, \mathcal{O}_F)=H^2(F, \mathcal{O}_F)=0$, so by complex exponential sequence we obtain already $\operatorname{Pic}(F)=H^2(F,\Bbb Z)$ as Abelian groups, so isn't Lefschetz application there redundant as we can just take torsion of both groups?
If not and there is more involved, any idea to what Dolgachev refering to there by application of Lefschetz to get this statementon torsions?
