Complex Valued Bismut-Lott Index Theorem
- PDF / 222,768 Bytes
- 11 Pages / 510.236 x 737.008 pts Page_size
- 28 Downloads / 181 Views
Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Complex Valued Bismut–Lott Index Theorem Guang Xiang SU Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, P. R. China E-mail : [email protected] Abstract In this paper, assuming there is a fiberwise Morse function, we extend Bismut–Lott index theorem to the complex valued case. Keywords
Bismut–Lott index theorem, Morse function, Witten deformation
MR(2010) Subject Classification
1
58J20
Introduction
Let M be a smooth manifold, let E be a flat vector bundle on M , equipped with a flat connection ∇E . Let g E be a Hermitian metric on E. Set ω(∇E , g E ) = (g E )−1 ∇E g E .
(1.1)
Then ω(∇E , g E ) is a 1-form with values in End(E). Let ϕ : Λ• (T ∗ M ) → Λ• (T ∗ M ) be given by ϕα = (2iπ)−degα/2 α. If h is a holomorphic odd function, put h(∇E , g E ) = (2iπ)1/2 ϕTr[h(ω(∇E , g E )/2)].
(1.2)
Then by [1], the form h(∇E , g E ) is closed, its cohomology class h(∇E ) does not depend on g E . π → S be a smooth fiber bundle with base S which is a compact smooth manifold Let X → M − and connected closed fibers Xs = π −1 (s). Let F → M be a flat complex vector bundle on M . Let H • (X; F |X ) denote the flat complex vector bundle on S whose fiber over s ∈ S is isomorphic • to the cohomology group H • (Xs , F |Xs ). Let ∇H (X,F |X ) be the flat Gauss–Manin connection on H • (X, F |X ). Let T X be the vertical tangent bundle of the fiber bundle and let o(T X) be its orientation bundle, a flat real line bundle on M . Let e(T X) ∈ H dimX (M ; o(T X)) be the Euler class of T X (cf. [8]). In [1, Theorem 3.17], Bismut and Lott proved a Riemann–Roch–Grothendieck formula for such classes. Namely, H • (X,F |X ) h(∇ )= e(T X)h(∇F ) in H odd (S, R). (1.3) X
In order to write the formula (1.3) in the differential form level, Bismut and Lott introduced the analytic torsion form which extended the Ray–Singer analytic torsion ([6]) to the family case. Received July 11, 2019, accepted June 5, 2020 Supported by NSFC (Grant No. 11931007)
Su G. X.
1222
In [3–5], Burghelea and Haller defined the complex valued Ray–Singer torsion under the condition that there exists a non-degenerate symmetric bilinear form on the flat complex vector bundle. So it is natural to extend the Bismut–Lott index theorem to this case. Since the kernel of the family non-self-adjoint Laplacian in general does not form a vector bundle, so in general it is difficult to consider this problem. In [2], Bismut and Goette proved the Cheeger–M¨ uller theorem for the equivariant Bismut– Lott analytic torsion form assuming there exists an equivariant fiberwise Morse function. They used the Witten deformation in [2]. In particular, for T ≥ 0 large enough, the small eigenvalues [0,1] form a vector bundle FT . Inspired by this, if we consider the Witten deformation of the [0,1] Burghelea–Haller Laplacian, then the small eigenvalues also form a vector bundle Fb,T for T ≥ 0 large enough. Therefore we extend [1, Theorem 3.17] to the following theorem. T
Data Loading...
