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Equivariant Genera of Complex Algebraic Varieties

For smooth manifolds, Atiyah and Meyer studied contributions of monodromy to usual signatures. In this note we obtain Atiyah-Meyer type formulae for equivariant Hodge-theoretic genera of complex algebraic varieties. Equivariant Hirzebruch genera y(X;g) of a quasi-projective variety X acted upon by a nite group of algebraic automorphisms are dened by combining the group action with the information encoded by the Hodge ltration of the mixed Hodge structure in cohomology. While for a projective algebraic manifold y(X;g) can by computed by the Atiyah- Singer holomorphic Lefschetz theorem, we derive a Atiyah-Meyer type formula for y(X;g) in the case when X is not necessarily smooth or compact, but just bers equivariantly (in the complex topology) over a compact algebraic manifold. These results apply to computing Hodge-theoretic invariants of orbit spaces. We also obtain some results comparing equivariant Hodge-theoretic genera of the range and domain of an equivariant algebraic map in terms of its singularities.