Recent and new results on octonionic Bergman and Szegö kernels

Theta Functions and Szegö Kernels

We study relations between two fundamental constructions associated to vector bundles on a smooth complex projective curve: the theta function (a section of a line bundle on the moduli space of vector bundles) and the Szegö kernel (a section of a vector bundle on the square of the curve). Two types of relations are demonstrated. First, we establish a higher–rank version of the prime form, descr...

Weighted Bergman kernels on orbifolds

We describe a notion of ampleness for line bundles on orbifolds with cyclic quotient singularities that is related to embeddings in weighted projective space, and prove a global asymptotic expansion for a weighted Bergman kernel associated to such a line bundle.

Moment Maps and Equivariant Szegö Kernels

Let M be a connected n-dimensional complex projective manifold and consider an Hermitian ample holomorphic line bundle (L, hL) on M . Suppose that the unique compatible covariant derivative ∇L on L has curvature −2πiΩ, where Ω is a Kähler form. Let G be a compact connected semisimple Lie group and μ : G×M → M a holomorphic Hamiltonian action on (M,Ω). Let g be the Lie algebra of G, and denote b...

Weighted Bergman Kernels and Quantization

Let Ω be a bounded pseudoconvex domain in C N , φ, ψ two positive functions on Ω such that − logψ,− log φ are plurisubharmonic, z ∈ Ω a point at which − log φ is smooth and strictly plurisubharmonic, and M a nonnegative integer. We show that as k → ∞, the Bergman kernels with respect to the weights φkψM have an asymptotic expansion KφkψM (x, y) = kN πNφ(x, y)kψ(x, y)M ∞ ∑ j=0 bj(x, y) k −j , b0...

WEIGHTED BERGMAN KERNELS AND QUANTIZATIONMiroslav