Chern-weil Theory and Some Results on Classic Genera

In this survey article, we first review the Chern-Weil theory of characteristic classes of vector bundles over smooth manifolds. Then some well-known characteristic classes appearing in many places in geometry and topology as well as some interesting results on them related to elliptic genera which involve some of my joint work in [HZ1, 2], are briefly introduced. 1. Chern-Weil theory for characteristic classes The purpose of this section is to give a brief introduction to geometric aspects of the theory of characteristic classes, which was developed by Shiing-shen Chern and André Weil. This section is organized as follows: in a), we briefly review the de Rham cohomology theory; in b), we introduce the theory of connection and curvature on vector bundles over smooth manifolds; in c), the central theorem, Chern-Weil theorem, is introduced in a modern form; Chern classes and Pontrjagin classes are constructed in d); at last in e), we list the main properties of Chern classes and introduce the Chern root algorithm. We will basically follow [Z3] in a)-d). a). Review of the de Rham Cohomology Theory Let M be a smooth closed manifold. Let TM and T ∗M denote the tangent bundle and cotangent bundle of M respectively. Denote Ω∗(M) := Γ(Λ∗(T ∗M)) as the space of smooth sections of Λ∗(T ∗M), where Λ∗(T ∗M) is the complex exterior algebra bundle of T ∗M . Let d : Ω∗(M)→ Ω∗(M) be the God-given exterior differential operator on Ω∗(M), which satisfies d2 = 0. Then one finds that for any integer p such that 0 ≤ p ≤ dimM , dΩ(M) ⊂ kerd |Ωp+1(M), which leads the definition of de Rham complex as well as its associated cohomology: de Rham cohomology. Definition 1.1. The de Rham complex (Ω∗(M), d) is the complex defined by 0→ Ω(M) d → Ω(M) d → · · · d → Ω(M)→ 0. Definition 1.2. For any integer p such that 0 ≤ p ≤ dimM , the p-th de Rham cohomology of M with complex coefficients is defined by H dR(M,C) := kerd |Ωp(M) dΩp−1(M) . The total de Rham cohomology of M is defined as H∗ dR(M,C) := dimM ⊕