Tatsuo Suwa

RESIDUES FOR SINGULAR PAIRS AND DYNAMICS OF BIHOLOMORPHIC MAPS OF SINGULAR SURFACES

We prove the existence of a parabolic curve for a germ of biholomorphic map tangent to the identity at an isolated singular point of a surface under some conditions. For this purpose, we present a Camacho–Sad type index theorem for fixed curves of biholomorphic maps of singular surfaces and develop a local intersection theory of curves in singular surfaces from an analytic approach by means of Grothendieck residues.

Localization of Atiyah classes

We construct the Atiyah classes of holomorphic vector bundles using (1, 0)-connections and developing a Chern-Weil type theory, allowing us to effectively compare Chern and Atiyah forms. Combining this point of view with the Cech-Dolbeault cohomology, we prove several results about vanishing and localization of Atiyah classes, and give some applications. In particular, we prove a Bott type vanishing theorem for (not necessarily involutive) holomorphic distributions. As an example we also present an explicit computation of the residue of a singular distribution on the normal bundle of an invariant submanifold that arises from the Camacho-Sad type localization.

Localized intersection of currents and the Lefschetz coincidence point theorem

We introduce the notion of a Thom class of a current and define the localized intersection of currents. In particular, we consider the situation where we have a

Indices of Vector Fields and Residues of Singular Holomorphic Foliations

C^infty

RESIDUES OF CHERN CLASSES ON SINGULAR VARIETIES

C∞ map of manifolds and study localized intersections of the source manifold and currents on the target manifold. We then obtain a residue theorem on the source manifold and give explicit formulas for the residues in some cases. These are applied to the problem of coincidence points of two maps. We define the global and local coincidence homology classes and indices. A representation of the Thom class of the graph as a Čech–dexa0Rham cocycle immediately gives us an explicit expression of the index at an isolated coincidence point, which in turn gives explicit coincidence classes in some non-isolated components. Combining these, we have a general coincidence point theorem including the one by S. Lefschetz.

Dual Class of a Subvariety

1. Analytic functions of one complex variable 2. Analytic functions of several complex variables 3. Germs of holomorphic functions 4. Complex manifolds and analytic varieties 5. Germs of varieties 6. Vector bundles 7. Vector fields and differential forms 8. Chern classes of complex vector bundles 9. Divisors 10. Complete intersections and local complete intersections 11. Grothendieck residues 12. Residues at an isolated zero 13. Examples 14. Sheaves and cohomology 15. de Rham and Dolbeault theorems 16. Poincaré and Kodaira-Serre dualities 17. Riemann-Roch theorem Glossary Bibliography Biographical Sketch