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This paper discusses the role played by perturbed Dolbeault operators in relating the coherent sheaf and elliptic operator perspectives on the K homology of projective varieties. Among the consequences are index formulas for perturbed Dolbeault operators. In this paper we make some observations about the index theory of perturbed Dolbeault operators, with particular attention to the role of the homology Todd class [2], [3] in this index theory. A perturbed Dolbeault operator is the sum of a Dolbeault operator and a vector-bundle map (the perturbation) that is invertible off some compact subset of the underlying manifold. We focus on the following setting. Let s be a meromorphic section of a holomorphic vector bundle over a smooth projective algebraic variety V . Assume that the closure of the section’s zero locus has empty intersection with the closure of the section’s polar locus. Let M be the complement in V of the section’s polar locus. Over M use the section and all exterior powers of the holomorphic vector bundle’s dual to form a Koszul complex. Assemble this complex into a two-term complex consisting of a vectorbundle map from the direct sum of exterior powers of even degree to the direct sum of exterior powers of odd degree. This vector-bundle map is invertible away from the original section’s zero locus. We use it to provide a perturbation for the Dolbeault operator on M . In the first section we present conditions, for the most part involving the interaction of the perturbation with the metric chosen for M , that ensure that a perturbed Dolbeault operator represents a K homology class for V and that this K homology class is the Kasparov product of a K theory class represented by the perturbation and a K homology class represented by the Dolbeault operator. This product result permits the use of standard characteristic class techniques in calculating the homology Chern character and hence the index of the perturbed Dolbeault operator. If the section s is regular [7], the perturbed Dolbeault operator’s K homology class equals that determined by the structure sheaf of s’s zero scheme. It follows that the perturbed Dolbeault operator’s homology Chern character is the homology Todd Received by the editors February 4, 1999. 2000 Mathematics Subject Classification. Primary 58J20, 19L10, 19K35.

Perturbed Dolbeault operators and the homology Todd class