Computing the Gysin Map Using Fixed Points

Abstract

ABSTRACT. The Gysin map of a map between compact oriented manifolds is the map in cohomology induced by the push-forward map in homology. In enumerative algebraic geometry, formulas for the Gysin map of a flag bundle play a vital role. These formulas are usually proven by algebraic or combinatorial means. This article shows how the localization formula in equivariant cohomology provides a systematic method for calculating the Gysin homomorphism in the ordinary cohomology of a fiber bundle. As examples, we recover classical pushforward formulas for generalized flag bundles. Our method extends the classical formulas to fiber bundles with equivariantly formal fibers.In enumerative algebraic geometry, to count the number of objects satisfying a set of conditions, one method is to represent the objects satisfying each condition by cycles in a parameter space M and then to compute the intersection of these cycles in M. When the parameter space M is a compact oriented manifold, by Poincaré duality, the intersection of cycles can be calculated as a product of classes in the rational cohomology ring. Sometimes, a cycle B in M is the image f (A) of a cycle A in another compact oriented manifold E under a map f : E → M. In this case the homology class [B] of B is the image f * [A] of the homology class of A under the induced map f * : H * (E) → H * (M) in homology, and the Poincaré dual η B of B is the image of the Poincaré dual η A of A under the map H * (E) → H * (M) in cohomology corresponding to the induced map f * in homology. This map in cohomology, also denoted by f * , is called the Gysin map, the Gysin homomorphism, or the pushforward map in cohomology. It is defined by the commutative diagramwhere e and m are the dimensions of E and M respectively and the vertical maps are the Poincaré duality isomorphisms. The calculation of the Gysin map for various flag bundles plays an important role in enumerative algebraic geometry, for example in determining the cohomology classes of degeneracy loci ([22], [19], [15], and [13, Ch. 14]). Other applications of the Gysin map, for example, to the computation of Thom polynomials associated to ThomBoardman singularities and to the computation of the dual cohomology classes of bundles of Schubert varieties, may be found in [12].

The Strong Maximal Rank conjecture and higher rank Brill–Noether theory

“…Remark The map is known as the Lagrange–Sylvester symmetrizer (see also [35]). The attentive reader will notice that the denominator on the right‐hand side differs from that in the original formula of Pragacz by a power of .…”

Towards the Green–Griffiths–Lang conjecture via equivariant localisation

“…Smooth complex varieties are equivariantly formal, but we use more than that, namely the explicit descriptions of given above only apply for tower of Grassmannian and flag bundles such as . Note that the present paper was submitted before the first appearance of and our approach is independent of .…”

Universal Gysin formulas for flag bundles

“…-using Grothendieck residues (Akyildiz-Carrell [1], Damon [4], Quillen [22]); -using localization and residues at infinity (Bérczi-Szenes [2], Tu [23], Zielenkiewicz [25,24]); -using symmetrizing operators (Brion [3], the second author, e.g. [18,19] and Ratajski [21]); -using Schur functions and Grassmann extensions (Józefiak, Lascoux and the second author [13]; for supersymmetric functions see [18,9]); -using residues and Grassmann extensions (Kazarian [15,16]), which leads to formulas showing similarities with ours.…”