arXiv research

Classifies simply-connected pluriclosed manifolds with parallel Bismut torsion.

The paper classifies Bismut Kähler-like manifolds in dimensions 4 and 5.

We define the geometric complex associated to a Morse-Bott-Smale vector field, cf. [Austin-Braam, 1995], and its associated spectral sequence. We prove an extension of the Bismut-Zhang theorem to Morse-Bott-Smale functions. The proof is based on the Bismut-Zhang theorem for Morse-Smale functions, see [Bismut-Zhang, 199…

Characterizes Hermitian manifolds with Bismut parallel torsion.

The study characterizes Hermitian manifolds with parallel Bismut-Strominger torsion.

In this paper we give a proof of an index theorem by Bismut. As a consequence we obtain another proof of the Grothendieck-Riemann-Roch theorem in differential cohomology.

The paper computes a residue density for a specific Laplacian on compact manifolds.

The paper compares two torsion invariants in complex vector bundles.

Paper extends variational formula for Bismut-Cheeger eta form, proving key theorems in K-theory.

Proves an equivariant version of index theorem for geometric families.

New homogeneous manifolds with invariant Bismut Ricci flat connections are constructed.

Paper compares higher torsions and removes fiberwise Morse function assumption.

Study on BAS manifolds with parallel torsion and curvature.

Bismut and Zhang computed the ratio of the Ray-Singer and the combinatorial torsions corresponding to non-unitary representations of the fundamental group. In this note we show that for representations which belong to a connected component containing a unitary representation the Bismut-Zhang formula follows rather easi…

The paper computes metrics and Einstein tensors on even-dimensional manifolds.

The paper classifies Hermitian manifolds with specific connection properties.

We show various vanishing theorems for the cohomology groups of compact hermitian manifolds for which the Bismut connection has (restricted) holonomy contained in SU(n) and classify all such manifolds of dimension four. In this way we provide necessary conditions for the existence of such structures on hermitian manifo…

Equivalence proven between two torsion invariants for flat vector bundles.

We present a new proof, as well as a ${\bf C/Q}$ 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 $η$-invariants associated to the so-called sub-signature operators. We further show that the Bismut-Lott a…

We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L^2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of …

Bismut Einstein metrics on complex manifolds are Kähler Einstein or Bismut Ricci flat.

Compact Riemannian manifolds with mostly positive curvature have finite fundamental groups.

Paper establishes loop space T-duality formulae and refines earlier work.

Study of orbifold Chern character using superconnections.

We define the Simons-Sullivan differential analytic index by translating the Freed-Lott differential analytic index via explicit ring isomorphisms between Freed-Lott differential K-theory and Simons-Sullivan differential K-theory. We prove the differential Grothendieck-Riemann-Roch theorem in Simons-Sullivan differenti…

Study SKT and CYT manifolds with parallel Bismut torsion.

In this paper, we provide a concrete interpretation of equivariant Reidemeister torsion and demonstrate that Bismut-Zhang's equivariant Cheeger-Müller theorem simplifies considerably when applied to locally symmetric spaces. In a companion paper, this allows us to extend recent results on torsion cohomology growth and …

Holonomy group of Bismut connection on Vaisman manifolds is studied.

Study on balanced Hermitian threefolds with parallel Bismut torsion.

Study Bismut-Griffiths-positivity in non-Kähler manifolds under Hermitian curvature flows.

Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.

We generalize a theorem of Bismut-Zhang, which extends the Cheeger-Mueller theorem on Ray-Singer torsion and Reidemeister torsion, to the case where the flat vector bundle over a closed manifold carries a nondegenerate symmetric bilinear form. As a consequence, we prove the Burghelea-Haller conjecture which gives an an…

New insights prevent certain types of metrics on compact spaces.

Study on Hermitian metrics on Lie algebras with specific ideals.

Study proves Kählerness criteria for Hermitian surfaces under specific curvature conditions.

Paper proves gluing formula for analytic torsions using Witten deformation for non-Morse functions.

We define exotic twisted $S^1$-equivariant cohomology for the loop space $LZ$ of a smooth manifold $Z$ via the invariant differential forms on $LZ$ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Cher…

Researchers classify invariant Hermitian structures on flag manifolds with parallel Bismut torsion.

In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. I…

New proof of Grauert's theorem using differential geometry.

New infinite families of flat spaces found from symmetric spaces.

The purpose of this paper is to give a proof of the real part of the Riemann-Roch-Grothendieck theorem for complex flat vector bundles at the differential form level in the even dimensional fiber case. The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac op…

For a complex flat vector bundle over a fibered manifold, we consider the 1-parameter family of certain deformed sub-signature operators introduced by Ma-Zhang. We compute the adiabatic limit of the Bismut-Freed connection associated to this family and show that the Bismut-Lott analytic torsion form shows up naturally …

This article surveys the relations among local and nonlocal invariants in Atiyah-Singer index theory. We discuss the local invariants that arise from the heat equation approach to the index theorem for geometric operators, as well as the nonlocal invariants (the eta invariant, the determinant of the Laplacian/analytic …

Formula for twisted orbital integrals using hypoelliptic Laplacian.

Given a smooth compact manifold with boundary, we show that the subcomplex of the deformed de Rham complex consisting of eigenspaces of small eigenvalues of the Witten Laplacian is canonically isomorphic to the Thom-Smale complex constructed by Laudenbach. Our proof is based on Bismut-Lebeau's analytic localization tec…

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 consequ…

In this paper an analytic proof of a generalization of a theorem of Bismut ([Bis1, Theorem 5.1]) is given, which says that, when $v$ is a transversal holomorphic vector field on a compact complex manifold $X$ with a zero point set $Y$, the embedding $j:Y\to X$ induces a natural isomorphism between the holomorphic equiv…