Forms of algebraic varieties Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, such that there exists an isomorphism $\psi\colon X_L\to Y_L$ defined over $L$ (but not over $K$ in general).
The map $\psi$ induces a 1-co cycle from $Gal(L/K)$ to the group $Aut(X_L) $ of $L$-automorphisms, and it is easy to check that the image in $H^1 (Gal, Aut(X_L))$ only depends on the class of $Y$, up to $K$-isomorphisms. My question is the following: is every element of $H^1$ obtained? In other words, we have a map from the set of $L$-forms of $X$ to $H^1 (..)$, the map is injective but is it surjective?
If no, could it be true in some natural cases, if yes I would be happy to see a reference.
 A: CW answer, copied from the comments of Timo Keller and abx:
See J.P. Serre's “Galois Cohomology”, Chapter III, 1.3, Proposition 5.
A: In case you are interested in seeing a situation where the map is not surjective, consider the set of rational maps of degree $f:\mathbb{P}^1\to\mathbb{P}^1$, modulo the conjugation action of $\phi\in\text{PGL}_2$, i.e., $f^\phi=\phi\circ f\circ\phi^{-1}$. This is the natural action if one is interested in dynamics (iteration of the map $f$). The automorphism group of $f$ is $\text{Aut}(f)=\{\phi\in\text{PGL}_2:f^\phi=f\}$. Two maps $f_1$ and $f_2$ defined over $K$ are isomorphic over $\bar K$ if $f_1=f_2^\phi$ for some $\phi\in\text{PGL}_2(\bar K)$, and the set of twists of $f$ is the set of maps that are $\bar K$-isomorphic to $f$, modulo $K$-isomorphism. Just as in the case you're looking at, the set of twists injects into the cohomology group $H^1(G_{\bar K/K},\text{Aut}(f))$. But the image is not surjective. The image turns out to be exactly the elements of $H^1(G_{\bar K/K},\text{Aut}(f))$ that become trivial in $H^1(G_{\bar K/K},\text{PGL}_2(\bar K))$. You can read about this in the following places:
The Arithmetic of Dynamical Systems, Springer, Section 4.9, specifically Theorem 4.79.
The Field of Definition for Dynamical Systems, Compositio Math. 98 (1995), 269-304.
