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In this paper we study an integral invariant which obstructs the existence ona compact complex manifold of a volume form with the determinant of its Ricciform proportional to itself, in particular obstructs the existence of aK\"ahler-Einstein metric, and has been studied since 1980's. We study thisinvariant from the view point of locally conformally K\"ahler geometry. Wefirst see that we can define an integral invariant for coverings of compactcomplex manifolds with automorphic volume forms. This situation typicallyoccurs for locally conformally K\"ahler manifolds. Secondly, we see that thisinvariant coincides with the former one. We also show that the invariantvanishes for any compact Vaisman manifolds.

An integral invariant from the view point of locally conformally K\"ahler geometry