Chin-Yu Hsiao

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.

Szegö kernel asymptotics and Morse inequalities on CR manifolds

Let X be an abstract compact orientable CR manifold of dimension

On the singularities of the Szegő projections on lower energy forms

{2n-1, n\,\geqslant\,2}

Berezin-Toeplitz quantization for lower energy forms

, and let Lk be the k-th tensor power of a CR complex line bundle L over X. We assume that condition Y(q) holds at each point of X. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in Lk, for large k. After integration, this gives weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities. We apply the strong Morse inequalities to the embedding of some convex–concave manifolds.

Extremal metrics for the Q′-curvature in three dimensions

Let X be a compact connected CR manifold of dimension

On the stability of equivariant embedding of compact CR manifolds with circle action

2n-1, n\ge 2