Superconnection and family Bergman kernels

This paper is the second part of a series of papers on noncommutative geometry and conformal geometry. In this paper, we compute explicitly the Connes–Chern character of an equivariant Dirac spectral triple. The formula that we obtain is used in the first paper of the series. The computation has two main steps. The first step is the justification that the CM cocycle represents the Connes–Chern character. The second step is the computation of the CM cocycle as a byproduct of a new proof of the local equivariant index theorem of Donnelly–Patodi, Gilkey and Kawasaki. The proof combines the rescaling method of Getzler with an equivariant version of the Greiner–Hadamard approach to the heat kernel asymptotics. Finally, as a further application of this approach, we compute the short-time limit of the JLO cocycle of an equivariant Dirac spectral triple.

We report on the construction of secondary invariants in connection with the Atiyah-Singer index theorem for families, and the Theorem of Riemann-Roch-Grothendieck. The local families index theorem plays an important role in the construction. In complex geometry, the corresponding objects are the analytic torsion forms and the analytic torsion currents. These objects exhibit natural functorial properties with respect to composition of maps. Gillet and Soulé have used these objects to prove a Riemann-Roch theorem in Arakelov geometry. Also we state a Riemann-Roch theorem for flat vector bundles, and report on the construction of corresponding higher analytic torsion forms.

We intend to produce a theory which generalises topological spaces; we call it non-commutative topology. In non-commutative differential geometry the basic homology theory is the periodic cyclic homology, based on the bi-complex (b, B). In non-commutative topology this structure will be replaced by the bi-complex \((\tilde {b}, d)\); the boundary \(\tilde {b}\) is called topological Hochschild boundary. These ideas combine Connes’ work (Noncommutative geometry, Academic Press, 1994) with ideas of the articles by Teleman and Teleman (St Cerc Math Tom 18(5):753–762, Bucarest, 1966; Rev Roumaine Math Pures Applic 12:725–731, 1967; Central European journal of mathematics. C*-algebras and elliptic theory II. Trends in mathematics, 251–265, Birkhauser, 2008; Local3 index theorem. arXiv: 1109.6095v1, math.KT, 28 Sep. 2011; textitLocal Hochschild homology of Hilbert-Schmidt operators on simplicial spaces. arXiv hal-00707040, Version 1, 11 June 2012; Local algebraic K-theory, 26 Aug 2013 - arXiv.org > math > arXiv:1307.7014, 2013; The local index theorem, HAL-00825083, arXiv 1305.5329, 22 May 2013; \(K_ {i}^{loc}(\ mathbb {C}) \), i = 0, 1 10 set 2013 - arXiv:1309.2421v1 [math.KT] 10 Sep 2013).

Local atiyah-singer index theorem for families of dirac operators

In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed’s odd dimensional index formula. Then we extend this theorem to the noncommutative geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.

The noncommutative geometry of graph [formula omitted]-algebras I: The index theorem

In this paper, we prove a local equivariant index theorem for sub-signature operators which generalizes Weiping Zhang’s index theorem for sub-signature operators.

Superconnections, anomalies and non-BPS brane charges

In this paper, we give a local Riemann-Roch theorem for certain almost Hermitian manifolds by a canonical connection and the theory of almost Hermitian structure of null eccentricity. In particular, we obtain a local Riemann-Roch theorem for all closed almost complex manifolds of dimension 4. Also, by using the heat equation method, we give an analytic approach to a local index theorem for generalized Dirac operators.

We show that one can define (p,∞)-summable spectral triples using degenerate metrics on smooth manifolds. Furthermore, these triples satisfy Connes–Moscovici's discrete and finite dimension spectrum hypothesis, allowing one to use the Local Index Theorem [1] to compute the pairing with K-theory. We demonstrate this with a concrete example.

Author(s): Seto, Shoo | Advisor(s): Lu, Zhiqin | Abstract: Let $(L,h) \to (M,\omega)$ be a polarized K quot;ahler manifold. We define the Bergman kernel for $H^0(M,L^k)$, holomorphic sections of the high tensor powers of the line bundle $L$. In this thesis, we will study the asymptotic expansion of the Bergman kernel. We will consider the on-diagonal, near-diagonal and far off-diagonal, using $\mathcal{L}^2$ estimates to show the existence of the asymptotic expansion and computation of the coefficients for the on and near-diagonal case, and a heat kernel approach to show the exponential decay of the off-diagonal of the Bergman kernel for noncompact manifolds assuming only a lower bound on Ricci curvature and $C^2$ regularity of the metric.

Brownian Chen series and Atiyah–Singer theorem

Let Ω be a bounded pseudoconvex domain in C N , φ, ψ two positive functions on Ω such that − log ψ, − log φ are plurisubharmonic, and z∈Ω a point at which − log φ is smooth and strictly plurisubharmonic. We show that as k→∞, the Bergman kernels with respect to the weights φ k ψ have an asymptotic expansion for x,y near z, where φ(x,y) is an almost-analytic extension of &\phi;(x)=φ(x,x) and similarly for ψ. Further, . If in addition Ω is of finite type, φ,ψ behave reasonably at the boundary, and − log φ, − log ψ are strictly plurisubharmonic on Ω, we obtain also an analogous asymptotic expansion for the Berezin transform and give applications to the Berezin quantization. Finally, for Ω smoothly bounded and strictly pseudoconvex and φ a smooth strictly plurisubharmonic defining function for Ω, we also obtain results on the Berezin–Toeplitz quantization.

In this paper, we survey recent results in [HMA12] about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of the Kodaira embedding theorem.KeywordsBergman kernel asymptoticsThe Kodaira embedding theorem

Asymptotic inversion of Toeplitz matrices with one singularity in the symbol