James B. Carrell

A decomposition theorem for the integral homology of a variety

This homology basis formula (1) was motivated by a theorem of Frankel [7]. In [5], also motivated by [7], it was shown that (1) is valid for integer coefficients if X is a compact Kaehler manifold with holomorphic C* action having fixed points. In this paper we further extend (1) by allowing X to be a (singular) compact complex space admitting a so called good decomposition, which is a generalization of the Bialynicki-Birula decomposition of a smooth complex projective variety with algebraic C* action.

A remark on the Grothendieck residue map

The purpose of this note is to give a direct proof that a global integral over a compact complex manifold X can be evaluated on the zero set of a meromorphic vector field on X with isolated zeros via a Grothendieck residue morphism. A special case of this evaluation is the meromorphic vector field theorem of Baum and Bott [1]. The present proof suggests some complements of the M.V.F. Theorem which are contained in Theorem 2.

Vector fields, residues and cohomology

1. Residue formulas. Let E be a holomorphic vector bundle on a compact complex manifold X of dimension n with structure sheaf OX , and let E be the locally free sheaf of OX -modules (or briefly, an OX -sheaf) canonically associated to E. A residue formula for E expresses the Chern numbers of E as finite sums of residues. Recall that a Chern number is associated to a symmetric OX -linear map p : EndOX (E)⊗n → OX as follows. Letting c(E) ∈ H(X,EndOX (E)⊗ΩX)) denote the Chern class of E in the sense of [At], one may apply p to c(E) to obtain a class p(E) = p(c(E)) ∈ H(X,ΩX) which may be evaluated on the fundamental cycle of X. The number (2πi)−n ∫ X p(E) is the associated Chern number of X; we will discuss computing these numbers as sums of residues. Let ΘX denote the sheaf of sections of the holomorphic tangent bundle W of X. Let L be an invertible OX -sheaf, and assume V ∈ H(X,ΘX ⊗ L) is a section that has only isolated zeros. The zero set of V can be given the structure of a possibly unreduced scheme Z, called the zero scheme of V . Namely, Z is the finite subscheme of X defined by the sheaf of ideals IZ = i(V )(ΩX ⊗ L−1) ⊂ OX , where, i(V ) : ΩX ⊗ L−1 → OX denotes the canonical contraction operator defined by viewing V as an operator V : ΩX → L (so i(V ) = V ⊗ 1). The structure sheaf of Z is denoted by OZ . Thus OZ := OX/IZ . Letting LZ := L ⊗ OZ denote the pull back to Z of L, there is a canonical C-linear map ResV : H(Z,LZ) → C called the Grothendieck residue morphism [C1], [CL2], which is based on [Be], [H], [V] (also see [L]). A residue formula for a pair (p,E) as above will consist of using V to associate a natural class pZ(E) ∈ H(Z,LZ) to p(E) (the localization to Z) such that

Vector Fields and Cohomology of G/B

The topic I will discuss today is one which arose from a question which I believe Professor Matsushima originally asked: namely, if one is given a holomorphic vector field V on a projective manifold X, is it true that X has no nontrivial holomorphic p-forms if p > dimC zero (V)? Alan Howard answered this question affirmatively in [H] and later, D. Lieberman and I discovered other relationships between zeros of holomorphic vector fields and topology. Perhaps the most interesting of these is that if one has a holomorphic vector field V on a compact Kaehler manifold X with isolated zeros, then the whole cohomology ring of X can be calculated on the zeros of V. Although holomorphic vector fields with isolated zeros are not abundant, they do exist on a fundamental class of spaces, namely the algebraic homogeneous spaces. In the one example that has been carefully analyzed, the Grassmannians, the calculation of the cohomology ring on the zeros of V gives a new insight on the connection between Schubert calculus and the theory of symmetric functions [C].

The Structure Theory of Linear Mappings

Throughout this chapter, V will be a finite-dimensional vector space over \({\mathbb F}\). Our goal is to prove two theorems that describe the structure of an arbitrary linear mapping \(T:V\rightarrow V\) having the property that all the roots of its characteristic polynomial lie in \({\mathbb F}\). To describe this situation, let us say that \({\mathbb F}\) contains the eigenvalues of T. Recall that a linear mapping \(T:V\rightarrow V\) is also called an endomorphism of V, and in this chapter, we will usually use that term.

Linear Algebraic Groups: an Introduction

The purpose of this section is to give a brief informal introduction, with very few proofs, to the subject of linear algebraic groups, a far-reaching generalization of matrix theory and linear algebra.

Theorems on Group Theory

Our treatment of matrix theory and the theory of finite-dimensional vector spaces and their linear mappings is finished, and we now turn to the theory of groups.

An Introduction to the Theory of Determinants

In this chapter, we will introduce and study a remarkable function called the determinant, which assigns to an \(n\times n\) matrix A over a field \({\mathbb F}\) a scalar \(\det (A)\in {\mathbb F}\) having two remarkable properties: \(\det (A)\ne 0\) if and only if A is invertible, and if B is also in \({\mathbb F}^{n\times n}\), then \(\det (AB)=\det (A)\det (B)\). The latter property is referred to as the product formula.

Matrix Inverses, Matrix Groups and the \({ LPDU}\) Decomposition

In this chapter we continue our introduction to matrix theory, beginning with the notion of a matrix inverse and the definition of a matrix group. For now, the main example of a matrix group is the group \(GL(n,\mathbb {F})\) of invertible \(n\times n\) matrices over a field \(\mathbb {F}\) and its subgroups. We will also show that every matrix \(A\in \mathbb {F}^{n\times n}\) can be factored as a product \({ LPDU}\), where each of L, P, D, and U is a matrix in an explicit subset of \(\mathbb {F}^{n\times n}\). For example, P is a partial permutation matrix, D is diagonal, and L and U are lower and upper triangular respectively.

Matrix Inverses, Matrix Groups and the

In this chapter we continue our introduction to matrix theory, beginning with the notion of a matrix inverse and the definition of a matrix group. For now, the main example of a matrix group is the group \(GL(n,\mathbb {F})\) of invertible \(n\times n\) matrices over a field \(\mathbb {F}\) and its subgroups. We will also show that every matrix \(A\in \mathbb {F}^{n\times n}\) can be factored as a product \({ LPDU}\), where each of L, P, D, and U is a matrix in an explicit subset of \(\mathbb {F}^{n\times n}\). For example, P is a partial permutation matrix, D is diagonal, and L and U are lower and upper triangular respectively.