Let $QProj_{k}$ stand for the category of quasi-projective varieties over a field $k$. Consider a Galois extension of fields $L\subset K$. From Galois descent theory we know that to give an object $X\in QProj_{K}$ is equivalent to giving an object $X_L\in QProj_{L}$ and a family of isomorphisms $\rho(g):g_*X_L\to X_L$ for each $g\in\textrm{Gal}(L/K)$ which agree, that is $\rho(gh)=\rho(g)g_*\rho(h)$.
Now suppose an $L$-variety $X_L$ is Galois-invariant in the sense that is there are isomorphisms $g_*X_L\to X_L$ for each $g\in\textrm{Gal}(L/K)$ without the agreement condition. Then it can be the case that $X_L$ is not defined over $K$. If $X_L$ has no nontrivial automorphisms then the agreement condition is obviously satisfied and $X_L$ is defined over $K$. Otherwise can we construct some obstruction in certain Galois cohomology group?
I am particularly interested in the case of complete curves where we can pass to function fields, but I am not sure how to formulate descent theory in these terms.
Thank you for any useful references.