In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa. Numerous examples, including many important semigroups from the literatur...

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...

We study the Bergman complex B(M) of a matroid M : a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of its lattice of flats. In addition, we show that the Bergman fan B̃(Kn) of the graphical matroid of the complete graph Kn is homeo...

and Applied Analysis 3 Let K z, z be the Bergman kernel function. Then the Bergman metric Hz β, β can be defined as

The fundamental boundary value problem in the function theory of several complex variables is the ∂-Neumann problem. The L2 existence theory on bounded pseudoconvex domains and the C∞ regularity of solutions up to the boundary on smooth, bounded, strongly pseudoconvex domains were proved in the 1960s. On the other hand, it was discovered quite recently that global regularity up to the boundary ...

In this paper we introduce techniques from complex harmonic analysis to prove a weaker version of the Geometric Arveson-Douglas Conjecture on the Bergman space for a complex analytic subset that is smooth on the boundary of the unit ball and intersects transversally with it. In fact, we prove that the projection operator onto the corresponding quotient module is in the Toeplitz algebra T (L∞), ...

Let be a pseudoconvex domain with C2 boundary in CP, n ≥ 2. We prove that the ∂̄-Neumann operator N exists for square-integrable forms on . Furthermore, there exists a number 0 > 0 such that the operators N , ∂̄∗N , ∂̄N and the Bergman projection are regular in the Sobolev space W ( ) for < 0. The ∂̄-Neumann operator is used to construct ∂̄-closed extension on for forms on the boundary b . This give...

In this paper, we shall study differential geometric properties of bounded domains in Cn. Here is the summary of our results. We consider an w-dimensional complex manifold M and the Hilbert space of square integrable holomorphic re-forms on M. After Bergman [3; 4; 5], we define the kernel form on M (instead of the kernel function) and, under certain assumptions, we define the invariant metric o...