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On connected automorphism groups of algebraic varieties

Let $X$ be a normal projective algebraic variety, $G$ its largest connected automorphism group, and $A(G)$ the Albanese variety of $G$. We determine the isogeny class of $A(G)$ in terms of the geometry of $X$. In characteristic 0, we show that the dimension of $A(G)$ is the rank of every maximal trivial direct summand of the tangent sheaf of $X$. Also, we obtain an optimal bound for the dimension of the largest anti-affine closed subgroup of $G$ (which is the smallest closed subgroup that maps onto $A(G)$).