Let $X$ be a smooth projective variety over a field $k$ of characteristic zero equipped with a point $e\in X(k)$. There is Albanese morphism $a:X\to \mathrm{Alb}\,X$ which is initial among pointed maps to abelian varieties.
Question. Is there a purely algebraic proof of the fact that the induced map $$a^*:H^0(\mathrm{Alb}\, X,\Omega^1)\to H^0(X,\Omega^1)$$ is an isomorphism?
This map is always an injection (in arbitrary characteristic this is a theorem of Igusa, but in characteristic zero this can be proven easily by further restricting 1-forms to сurves in $X$ obtained as hyperplane sections).
The dimension of $H^0(\mathrm{Alb}\, X,\Omega^1)$ is equal to $\dim \mathrm{Alb}X$. Since $\mathrm{Alb}X$ is dual to the Picard variety, $\dim \mathrm{Alb}X=\dim \mathrm{Pic}^0 X=\dim_k H^1(X,\mathcal{O})$ where the last equality comes from the fact that $T_0\mathrm{Pic}^0X\simeq H^1(X,\mathcal{O})$ by deformation theory. Therefore, the Hodge symmetry $\dim_k H^0(X,\Omega^1)=\dim_k H^1(X,\mathcal{O})$ would imply that the map is an isomorphism by comparing dimensions. Technically, there is a "purely algebraic" proof of Hodge symmetry using $p$-adic Hodge theory, but I'm hoping that there is a simpler geometric argument that answers the original question.
