On adelic Chern forms and the Bott residue formula Academic Article uri icon

abstract

  • 0. Introduction. The Bott Residue Formula gained renewed attention recently due to its use in enumerative algebraic geometry (cf. [ES], [Ko]). If X is a smooth projective variety over a field k of characteristic 0, then Bott’s formula makes sense purely algebraically, with the Chern classes taken in the algebraic De Rham cohomology H·DR(X/k). In this paper we survey an algebraic proof of the formula using Beilinson adeles, which was discovered by R. Hubl and the author (see [HY]). Suppose v ∈ Γ(X, TX) is a global vector field with isolated, simple, k-rational zeroes (see Remark 3.2 for generalizations). Let E1, . . . , Em be locally free OX -modules. Suppose Λi is an action of v on Ei, i.e. a differential operator Λi : Ei → Ei satisfying Λi(ae) = v(a)e + aΛi(e) for local sections a ∈ OX , e ∈ Ei. Suppose Q(ti,j) is a homogeneous polynomial of degree n = dimX in the variables ti,j (i = 1, . . . ,m; j = 1, . . . , ri; ri := rank Ei) which have degrees deg ti,j = j. For a zero z of v let us denote by Λi|z the restriction of Λi to Ei|z := Ei ⊗ k(z), which is a k-linear endomorphism. Also let us denote by ad v|z the restriction of ad v to TX ⊗ k(z); this is invertible. We let Pi denote the ith conjugation-invariant polynomial on matrices (of unspecified size). Finally let ∫

publication date

  • January 1, 1996