Reproducing kernel in \(\mathbb C^d\) and volume forms on toric varieties. (English. Russian original) Zbl 1105.14069

Shuplev, A. V.

Russ. Math. Surv. 60, No. 2, 373-375 (2005); translation from Usp. Mat. Nauk 60, No. 2, 179-180 (2005).

A finite set \(Z=\bigcup E_i\) of complex planes of arbitrary dimensions in \(\mathbb{C}^d\) is said to be atomic if the non-zero de Rham cohomology group of highest degree \(k\) of its complement is monogenic. A generating element \(\eta\) of the group \(H^k_{\text{DR}}(\mathbb C^d\setminus Z)\) is called a kernel for \(Z\). The theory of toric varieties delivers the class of sets \(Z\) that are atomic. Namely, any \(n\)-dimensional compact simplicial toric variety is representable as a quotient space \(X_{\Delta}:=(\mathbb{C}^d\setminus Z)/G\), where \(Z=Z_{\Delta}\) is a set of coordinate planes and \(G\) is a complex \((d-n)\)-dimensional torus, in which \(Z\) and \(G\) are encoded by an integral polyhedron \(\Delta\) in \(\mathbb{R}^n\) with \(d\) faces. The coordinates \(\xi=(\xi_1,\dots,\xi_d)\) in \(\mathbb{C}^d\) are considered as homogeneous coordinates of \(X_{\Delta}\). Let \(\omega(\xi)\) be a volume \((n,n)\)-form on \(X_{\Delta}\) written in homogeneous coordinates. The main result of this note states that the differential form \[ \omega(\xi)\wedge\frac{d\xi_{n+1}}{\xi_{n+1}}\wedge\dots\wedge\frac{d\xi_d}{\xi_d} \] on \(\mathbb{C}^d\) is a kernel for \(Z_{\Delta}\). The kernel of the classical Bochner-Martinelli integral representation is given as an example. Some further results on the volume form and integral representations are given.

Reviewer: Ivan V. Arzhantsev (Moskva)

MSC:

14M25	Toric varieties, Newton polyhedra, Okounkov bodies
32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)
14N20	Configurations and arrangements of linear subspaces
58A14	Hodge theory in global analysis
32A26	Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)

Keywords:

toric varieties; differential forms; coordinate planes; volume forms