Superconnection and family Bergman kernels

We establish an asymptotic expansion for families of Bergman kernels. The key idea is to use the superconnection as in the local family index theorem. Superconnexion et noyaux de Bergman en famille Résumé. Nous annonçons des résultats sur le développement asymptotique du noyau de Bergman en famille. Let W,S be smooth compact complex manifolds. Let π : W → S be a holomorphic submersion with compact fiber X and dimC X = n. We will add a subscript R for the corresponding real objects. Thus TX is the holomorphic relative tangent bundle of π, and TRX is the corresponding real vector bundle. Let JR be the complex structure on TRX. Let E be a holomorphic vector bundle on W . Let L be a holomorphic line bundle on W . Let h, h be Hermitian metrics on L,E. Let ∇L,∇E be the holomorphic Hermitian connections on (L, h), (E, h) with their curvatures R, R respectively. Set ω = √ −1 2π R. (0.1) Then ω is a smooth real 2-form of complex type (1, 1) on W . We suppose that ω defines a fiberwise Kähler form along the fiber X, i.e. gR(u, v) = ω(u, JRv) (0.2) defines a (fiberwise) Riemannian metric on TRX. We denote by h TX the corresponding Hermitian metric on TX. Let dvX be the Riemannian volume form on (X, g R). By the Kodaira vanishing theorem, there exists p0 ∈ N such that H(X, (L ⊗ E)|X) forms a vector bundle, denoted by H(X,L ⊗ E), on S for p > p0. From now on, we always assume p > p0. By the Grothendieck-Riemann-Roch Theorem, for p > p0, we have