From Sheaf Cohomology to the Algebraic de Rham Theorem

Let X be a smooth complex algebraic variety with the Zariski topology, and let Y be the underlying complex manifold with the complex topology. Grothendieck's algebraic de Rham theorem asserts that the singular cohomology of Y with complex coefficients can be computed from the complex of sheaves of algebraic differential forms on X. This article gives an elementary proof of Grothendieck's algebraic de Rham theorem, elementary in the sense that we use only tools from standard textbooks as well as Serre's FAC and GAGA papers.Comment: 53 pages; this version replaces an earlier version submitted in August 2013. Some misprints have been correcte

Gysin formulas and equivariant cohomology

Under Poincar\'e duality, a smooth map of compact oriented manifolds induces a pushforward map in cohomology, called the "Gysin map." It plays an important role in enumerative geometry. Using the equivariant localization formula, the author gave in 2017 a general formula for the Gysin map of a fiber bundle with equivariantly formal fibers. Equivariantly formal manifolds include all manifolds with cohomology in only even degrees such as complex projective spaces, Grassmannians, and flag manifolds as well as $G/H$, where $G$ is a compact Lie group and $H$ is a closed subgroup of maximal rank. This article is a simplified exposition using the example of a projective bundle to illustrate the algorithm.Comment: To be published in "Group Actions and Equivariant Cohomology", Contemporary Mathematics, AM

Computing Topological Invariants Using Fixed Points

When a torus acts on a compact oriented manifold with isolated fixed points, the equivariant localization formula of Atiyah--Bott and Berline--Vergne converts the integral of an equivariantly closed form into a finite sum over the fixed points of the action, thus providing a powerful tool for computing integrals on a manifold. An integral can also be viewed as a pushforward map from a manifold to a point, and in this guise it is intimately related to the Gysin homomorphism. This article highlights two applications of the equivariant localization formula. We show how to use it to compute characteristic numbers of a homogeneous space and to derive a formula for the Gysin map of a fiber bundle.Comment: 13 pages. This article is based on a talk given at the Sixth International Congress of Chinese Mathematicians, Taipei, Taiwan, in 201

Equivariant Characteristic Classes

This is a commentary on Raoul Bott and Loring Tu's joint article "Equivariant characteristic classes in the Cartan model," which appeared in "Geometry, Analysis, and Applications (Varanasi, 2000)," World Scientif Publishing, River Edge, NJ, 3--20. The article is also included in "Raoul Bott: Collected Papers," Vol. 5. The commentary discusses the genesis of the article, its influence, and the current state of the problem concerning equivariant characteristic classes