2,342 research outputs found

    A note on the Lichnerowicz vanishing theorem for proper actions

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    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

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    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

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    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

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    We establish a mod 2 index theorem for real vector bundles over 8k+2 dimensional compact pin^- manifolds. The analytic index is the reduced η\eta invariant of (twisted) Dirac operators and the topological index is defined through KOKO-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

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    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

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    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

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    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

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

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    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

    Dirac operators on foliations: the Lichnerowicz inequality

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    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

    Eta-invariants, Torsion forms and Flat vector bundles

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    We present a new proof, as well as a C/Q{\bf C/Q} extension, of the Riemann-Roch-Grothendieck theorem of Bismut-Lott for flat vector bundles. The main techniques used are the computations of the adiabatic limits of η\eta-invariants associated to the so-called sub-signature operators. We further show that the Bismut-Lott analytic torsion form can be derived naturally from the transgression of the η\eta-forms appearing in the adiabatic limit computations.Comment: 42 page
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