Equivariant Hirzebruch genera of a variety X acted upon by a finite group of algebraic automorphisms are defined by combining the group action with the information encoded by the Hodge filtration in cohomology. For smooth manifolds, Atiyah and Meyer studied contributions of monodromy to usual signatures. While for a projective manifold equivariant genera can by computed by the Atiyah-Singer holomorphic Lefschetz theorem, we derive a Atiyah-Meyer-type formula for such genera even when X is not necessarily smooth or compact, but just fibers equivariantly (in the complex topology) over an algebraic manifold. These results apply to computing Hirzebruch invariants of orbit spaces. We also obtain results comparing equivariant genera of the range and domain of an equivariant morphism in terms of its singularities.
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