Superconnection and family Bergman kernels
We establish an asymptotic expansion for families of Bergman kernels. The key idea is to use the superconnection as in the local family index theorem.
math.DG↗
arXiv subjects
Xiaonan Ma
Publications and source records attributed to Xiaonan Ma.
η-invariant and flat vector bundles II
We first apply the method and results in the previous paper to give a new proof of a result (hold in $ {\bf C}/{\bf Z}$) of Gilkey on the variation of h-invariants associated to non self-adjoint Dirac type operators. We then give an explicit local expression of certain h-invariant appearing in recent papers of Braverman-Kappeler on what they call refined analytic torsion, and propose an alternate formulation of their definition of the refined analytic torsion. A refinement in ${\bf C}$ of the above variation formula is also proposed.
math.DG↗
A remark on 'Some numerical results in complex differential geometry'
In this note we verify certain statement about the operator $Q\_K$ constructed by Donaldson in [3] by using the full asymptotic expansion of Bergman kernel obtained in [2] and [4].
math.DG↗
The first coefficients of the asymptotic expansion of the Bergman kernel of the spin^c Dirac operator
We establish the existence of the asymptotic expansion of the Bergman kernel associated to the spin-c Dirac operators acting on high tensor powers of line bundles with non-degenerate mixed curvature (negative and positive eigenvalues) by extending the paper " On the asymptotic expansion of Bergman kernel " (math.DG/0404494) of Dai-Liu-Ma. We compute the second coefficient b_1 in the asymptotic expansion using the method of our paper "Generalized Bergman kernels on symplectic manifolds" (math.DG/0411559).
math.CV↗
Generalized Bergman kernels on symplectic manifolds
We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them. As consequence, we calculate the density of states function of the Bochner-Laplacian and establish a symplectic version of the convergence of the induced Fubini-Study metric. We also discuss generalizations of the asymptotic expansion for non-compact or singular manifolds as well as their applications. Our approach is inspired by the analytic localization techniques of Bismut-Lebeau.
math.DG↗
On the asymptotic expansion of Bergman kernel
We study the asymptotic of the Bergman kernel of the spin$^c$ Dirac operator on high tensor powers of a line bundle.
math.DG↗
Eta-invariants, Torsion forms and 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 analytic torsion form can be derived naturally from the transgression of the $η$-forms appearing in the adiabatic limit computations.
math.DG↗
$K$-theory associated to vertex operator algebras
We introduce two $K$-theories, one for vector bundles whose fibers are modules of vertex operator algebras, another for vector bundles whose fibers are modules of associative algebras. We verify the cohomological properties of these $K$-theories, and construct a natural homomorphism from the VOA K-theory to the associative algebra K-theory.
math.DG↗
Elliptic genus and vertex operator algebras
We construct bundles of modules of vertex operator algebras, and prove the rigidity and vanishing theorem for the Dirac operator on loop space twisted by such bundles. This result generalizes many previous results.
math.DG↗
The SPIN$^c$ Dirac operator on high tensor powers of a line bundle
We study the asymptotic of the spectrum of the \spin Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin-Uribe, which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin.
math.DG↗
On orbifold elliptic genus
We define orbifold elliptic genus for general orbifolds which generalizes the definition of Borisov and Libgober, and prove their rigidity property.
math.DG↗
Rigidity and Vanishing Theorems in K-Theory II
We extend our family rigidity and vanishing theorems in [{\bf LiuMaZ}] to the Spin^c case. In particular, we prove a K-theory version of the main results of [{\bf H}], [{\bf Liu1}, Theorem B] for a family of almost complex manifolds.
math.KT↗
On Elliptic Genera and Foliations
We prove several vanishing theorems for a class of generalized elliptic genera on foliated manifolds, by using classical equivariant index theory. The main techniques are the use of the Jacobi theta-functions and the construction of a new class of elliptic operators associated to foliations.
math.DG↗
Rigidity and Vanishing Theorems in K-Theory I
In this paper, we first establish a K-theory version of the equivariant family index theorem for a circle action, then use it to prove several rigidity and vanishing theorems on the equivariant K-theory level.
math.KT↗