A Family Resolvent Cocycle,Higher Spectral Flow And Bergman Kernal

In this paper,we introduce a family resolvent cocycle and express the Chern Character of Dai-Zhang higher spectral flow as a pairing of a family resolvent cocycle and the odd Chern character of a unitary matrix,which generalize the odd index formula of Carey A L et al.to the family case.We establish the cancellation of the first[2j—q|-terms in the diagonal asymptotic expansion of the restriction to the(0,2j)-forms of the Bergman kernel associated to the modified Spinc Dirac operator on high tensor powers of a line bundle with mixed curvature twisted by a(non necessarily holomorphic)complex vector bundle,over a compact symplectic manifold.Moreover,we give a local formula for the first and the second(non-zero)leading coefficients which generalizes the Puchol-Zhu's results.