2,841 research outputs found
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
invariant of (twisted) Dirac operators and the topological index is defined
through -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
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