We give a necessary and a sufficient condition for the boundedness of the Toeplitz product TF TG∗ on the vector valued Bergman space L 2 a(C ), where F and G are matrix symbols with scalar valued Bergman space entries. The results generalize those in the scalar valued Bergman space case [4]. We also characterize boundedness and invertibility of Toeplitz products TF TG∗ in terms of the Berezin t...

We give an elementary introduction to CR and pseudohermitian geometry, starting from H. Lewy’s legacy (cf. [20]) i.e. tangential Cauchy-Riemann equations on the boundary of the Siegel domain. In this context we describe fundamental objects, such as contact structures, Levi forms, the Tanaka-Webster connection and the Fefferman metric (cf. e.g. [4]). Also naturally arising Hörmander systems of v...

In this note, we explain that Ross–Thomas’ result [4, Theorem 1.7] on the weighted Bergman kernels on orbifolds can be directly deduced from our previous result [1]. This result plays an important role in the companion paper [5] to prove an orbifold version of Donaldson theorem. In two very interesting papers [4, 5], Ross–Thomas describe a notion of ampleness for line bundles on Kähler orbifold...

Another facet of geodesics is GAT or “geometric approximation theory”. It is very difficult to work directly with infinite dimensional geometry and GAT is a special method in projective K’́ahler geometry to make fiinite dimensional approximations of the infinite dimensional locally symmetric space Hω by genuine finite dimensional symmetric spaces of type GC/G with G = SU(N). The compact group SU...

The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested se...

Estimates of the Bergman kernel and the Bergman and Kobayashi metrics on pseudoconvex domains near boundaries with constant Levi ranks are given. Mathematics Subject Classification (2000): 32F45; 32T27.

where the coefficients are entire functions. In [8], equations of the form (1) with coefficients in weighted Bergman or Hardy spaces are studied. The direct problem is proved, that is, if the coefficients aj(z), j = 0, ..., k − 1 of (1) belong to the weighted Bergman space, then all solutions are of finite order of growth and belong to weighted Bergman space. The inverse problem is also investi...

Given a matroid M on the ground set E, the Bergman fan B̃(M), or space of M -ultrametrics, is a polyhedral complex in RE which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics. Motivated by the phylogenetic situation, we study the following problem: Given a point ω in RE , we wish to find an M -ultrametric which is closest to it in the ...

In this article, we construct the canonical semipositive current or the canonical measure (= the potential of the canonical semipositive current) on a smooth projective variety with nonnegative Kodaira dimension in terms of a dynamical system of Bergman kernels. This current is considered to be a generalization of a Kähler-Einstein metric and coincides the one considered independently by J. Son...