Vertical Morse-Bott-Smale flows and characteristics forms (Q2833066)

This long paper is a natural continuation of the author's paper [Indiana Univ. Math. J. 65, No. 1, 93-169 (2016); Zbl 1341.58006] from the point of view of applications. Its main result is NEWLINENEWLINETheorem 6.7: Let \(E\rightarrow B\) be a Hermitian bundle endowed with a metric compatible connection \(\nabla \). Let \(A\in \mathrm{Sym}(E)\) be an endomorphism of self-adjoint operators. If \(E\) is \(\mathbb{Z}_2\)-graded suppose moreover that the \(\mathbb{Z}_2\)-grading involution is parallel with respect to \(\nabla \) and that \(A\) is odd. Let \(ch(A_t)\) be the Chern character form associated with the superconnection \(\nabla +tA\). Suppose that \(A\) is s-normal. Then we have the following: a) In the ungraded case, \(\lim_{t\rightarrow \infty }ch(A_t)=\sum _{k\geq 1}\mathrm{Res}_k^{\mathrm{odd}}\cdot [A^{-1}(S(k))]\). b) In the \(\mathbb{Z}_2\)-graded case, \(\lim _{t\rightarrow \infty }ch(A_t)=\sum _{k\geq 1}\mathrm{Res} _k^{\mathrm{even}}\cdot [A^{-1}(\Sigma _k)]\).