A note on the Lichnerowicz vanishing theorem for proper actions

We prove a Lichnerowicz type vanishing theorem for non-compact spin manifolds admiting proper cocompact actions. This extends a previous result of Ziran Liu who proves it for the case where the acting group is unimodular.Comment: 3 page

Holomorphic quantization formula in singular reduction

We show that the holomorphic Morse inequalities proved by Tian and the author [TZ1, 2] are in effect equalities by refining the analytic arguments in [TZ1, 2].Comment: Only the abstract and the introduction are changed, in which the incorrect comments regarding Teleman's work are now correcte

Circle actions and Z/k-manifolds

We establish an S^1-equivariant index theorem for Dirac operators on Z/k-manifolds. As an application, we generalize the Atiyah-Hirzebruch vanishing theorem for S^1-actions on closed spin manifolds to the case of Z/k-manifolds.Comment: 6 pages. Minor changes for the published versio

A mod 2 index theorem for pin−^- manifolds

We establish a mod 2 index theorem for real vector bundles over 8k+2 dimensional compact pin$^-$ manifolds. The analytic index is the reduced $\eta$ invariant of (twisted) Dirac operators and the topological index is defined through $KO$-theory. Our main result extends the mod 2 index theorem of Atiyan and Singer to non-orientable manifolds.Comment: 21 pages. MSRI Preprint No. 053-9

Positive scalar curvature on foliations: the enlargeability

We generalize the famous result of Gromov and Lawson on the nonexistence of metric of positive scalar curvature on enlargeable manifolds to the case of foliations, without using index theorems on noncompact manifolds.Comment: 7 pages, accepted versio

Positive scalar curvature on foliations

We generalize classical theorems due to Lichnerowicz and Hitchin on the existence of Riemannian metrics of positive scalar curvature on spin manifolds to the case of foliated spin manifolds. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus, which generalizes the famous theorem of Schoen-Yau and Gromov-Lawson on the non-existence of metrics of positive scalar curvature on torus to the case of foliations. Moreover, our method, which is partly inspired by the analytic localization techniques of Bismut-Lebeau, also applies to give a new proof of the celebrated Connes vanishing theorem without using noncommutative geometry.Comment: 28 pages. Title changed. Revised version to appear in Annals of Mathematics. arXiv admin note: text overlap with arXiv:1204.222

The Mathematical Work of V. K. Patodi

We give a brief survey on aspects of the local index theory as developed from the mathematical works of V. K. Patodi. It is dedicated to the 70th anniversary of Patodi.Comment: 28 pages. To appear in Communications in Mathematics and Statistic

Heat kernels and the index theorems on even and odd dimensional manifolds

In this talk, we review the heat kernel approach to the Atiyah-Singer index theorem for Dirac operators on closed manifolds, as well as the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. We also discuss the odd dimensional counterparts of the above results. In particular, we describe a joint result with Xianzhe Dai on an index theorem for Toeplitz operators on odd dimensional manifolds with boundary

Dirac operators on foliations: the Lichnerowicz inequality

We construct Dirac operators on foliations by applying the Bismut-Lebeau analytic localization technique to the Connes fibration over a foliation. The Laplacian of the resulting Dirac operators has better lower bound than that obtained by using the usual adiabatic limit arguments on the original foliation. As a consequence, we prove an extension of the Lichnerowicz-Hitchin vanishing theorem to the case of foliations.Comment: 53 pages. Title, abstract and the main results changed. The vanishing consequence is not as strong as originally claimed. The originally claimed vanishing results will be dealt with in a separate pape

A Poincar\'e-Hopf type formula for Chern character numbers

For two complex vector bundles admitting a homomorphism with isolated singularities between them, we establish a Poincar\'e-Hopf type formula for the difference of the Chern character numbers of these two vector bundles. As a consequence, we extend the original Poincar\'e-Hopf index formula to the case of complex vector fields (to appear in Mathematische Zeitschrift)Comment: 10 page