The index of projective families of elliptic operators

Projective elliptic genera and elliptic pseudodifferential genera

In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in Mathai-Melrose-Singer (2006) without requiring spin condition. We prove the modularity properties of these projective elliptic genera. As an application, we construct elliptic pseudodifferential genera for any elliptic pseudodifferential operator. This suggests the existence of …

Equivariant and fractional index of projective elliptic operators

In this note the fractional analytic index, for a projective elliptic operator associated to an Azumaya bundle, of DG/0402329 is related to the equivariant index of Atiyah and Singer for an associated transversally elliptic operator.

The index of projective families of elliptic operators: the decomposable case

An index theory for projective families of elliptic pseudodifferential operators is developed when the twisting, i.e. Dixmier-Douady, class is decomposable. One of the features of this special case is that the corresponding Azumaya bundle can be realized in terms of smoothing operators. The topological and the analytic index of a projective family of elliptic operators both take values in the twisted K-theory of the parameterizing space. The main result is the equality of these two notions of index. The twisted Chern character of the index class is then computed by a variant of Chern-Weil theo…

On abelian families and holomorphic normal projective connections

We prove Chern class equalities for abelian families with a holomorphic normal projective connection.

Index of projective elliptic operators

Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encompasses the fractional index formula of projective elliptic operator by Mathai-Melrose-Singer.

从椭圆微分算子的角度研究射影映射

Projective Maps from the Perspective of Elliptic Differential Operators

本文通过构建椭圆微分算子，从分析角度研究射影映射的几何特性，并拓展了微分几何中的经典结论。