George Marinescu

Toeplitz Operators on Symplectic Manifolds

We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established.

ASYMPTOTICS OF SPECTRAL FUNCTION OF LOWER ENERGY FORMS AND BERGMAN KERNEL OF SEMI-POSITIVE AND BIG LINE BUNDLES

In this paper we study the asymptotic behaviour of the spectral function corre- sponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degenerate. As application we obtain the Bergman kernel asymptotics for adjoint semi-positive line bundles over complete Khler manifolds, on the set where the curvature is positive. We also prove the asymptotics for big line bundles endowed with singular Hermitian metrics with strictly positive curvature current. In this case the full asymptotics holds outside the singular locus of the metric.

The spin

Abstract. We study the asymptotic of the spectrum of the

^{\bf c}

{\rm spin^c}

Dirac operator on high tensor powers of a line bundle

Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin–Uribe [13, Theorem 2], which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin [10].

Embeddability of some strongly pseudoconvex cr manifolds

We obtain an embedding theorem for compact strongly pseudoconvex CR manifolds which are boundaries of some complete Hermitian manifolds. We use this to compactify some negatively curved Kahler manifolds with compact strongly pseudoconvex boundary. An embedding theorem for Sasakian manifolds is also derived.

Quantum Hall Effect and Quillen Metric

We study the generating functional, the adiabatic curvature and the adiabatic phase for the integer quantum Hall effect (QHE) on a compact Riemann surface. For the generating functional we derive its asymptotic expansion for the large flux of the magnetic field, i.e., for the large degree k of the positive Hermitian line bundle Lk. The expansion consists of the anomalous and exact terms. The anomalous terms are the leading terms of the expansion. This part is responsible for the quantization of the adiabatic transport coefficients in QHE. We then identify the non-local (anomalous) part of the expansion with the Quillen metric on the determinant line bundle, and the subleading exact part with the asymptotics of the regularized spectral determinant of the Laplacian for the line bundle Lk, at large k. Finally, we show how the generating functional of the integer QHE is related to the gauge and gravitational (2+1)d Chern–Simons functionals. We observe the relation between the Bismut-Gillet-Soulé curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. We then obtain the geometric part of the adiabatic phase in QHE, given by the Chern–Simons functional.

Exponential Estimate for the asymptotics of Bergman kernels

We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. Further, we give Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line bundles.

Szegö kernel asymptotics and Morse inequalities on CR manifolds

Let X be an abstract compact orientable CR manifold of dimension