Integral representation formulas on analytic varieties

Abstract

Integral representation formulas for holomorphic functions on analytic subvarieties of domains of C" are derived. These formulas generalize the Cauchy-Fantappie formula and the Weil formula for analytic polyhedra. The kernels we obtain are explicitly defined.Introduction. In recent years integral formulas and their applications have attracted a lot of attention in several complex variables; see for example [4,5,7,8,9, 10] and the most relevant to our work papers of Stout [15], Palm [11] and Henkin and Leiterer [6].In this paper we develop analogues of Cauchy-Fantappie Kernels for analytic subvarieties of domains of C". First let us recall the CauchyFantappie formula. Let ΰcC" be a bounded domain with smooth boundary and let γ: (dD) X D -> C" be a smooth function so that Some of the interesting features of Theorem I.I is on the one hand the explicit form of the kernel K^ξ, z) and on the other hand the fact that M is allowed to have, finitely many, singular points.The case m = 0 of Theorem LI is the formula (2) and the case m = 1 of it was obtained by Stout [15]. In fact Stout's paper was the starting point of this work. The kernel obtained by Stout coincides with ours in the case m = 1; this is not immediate however and in §11 we show that they are indeed the same.The proof of Theorem I.I (given in this paper) is an extension of Stout's proof from the case m = 1 to the general case. A second proof of Theorem I.I is contained in Hatziafratis [3].In §111 we develop a Weil type integral formula for analytic polyhedra on analytic varieties. The main result of this section is Theorem III.l which generalizes the Weil integral formula for analytic polyhedra in C" (see [14, 16]). To obtain this result we combine results from §1 together with some standard techniques contained for example in Range-Siu [12].As we pointed out before the results in this paper are related to those of Palm [11] and Henkin and Leiterer [6]. The setting of Henkin and Leiterer [6] is more general than ours (we allow, however, finitely many singular points on M). On the other hand our results are more explicit than theirs. In fact we do not know the relation between our results and those of Henkin-Leiterer and Palm.Acknowledgment. I would like to express my warmest thanks to Professor Alexander Nagel for very helpful discussions.

An analytic Grothendieck Riemann Roch theorem

“…When Ω I is smooth and intersects with ∂B m transversely, Beatrous [7] proved that R maps L 2 a (B m ) continuously onto L 2 a,M (Ω I ). Using the developments in complex analysis [28], [35], we have the following generalization of Beatrous' result. …”

“…3.5] holds on Ω I and states that every holomorphic function on Ω 0 I is holomorphic on Ω I . Furthermore, Assumption 1.1 plays a key role in the integral formula obtained in [28]. Hence Assumption 1.1 is crucial in Theorem 1.4 for R to be surjective.…”

An integral formula on submanifolds of domains of $\Bbb C^n$

“…Then we have to show that (1 .2) holds; and this is done along the same lines as in [3]] . Here is an outline of this proof in the case u is a holomorphic function on M. In this case we have to show that f CEaU, which holds by the result of [3] (or [2] for that case) since Ko = Kó in Uv n M. Our result gives, in particular, integral formulas for domains D CC X, if X is a Stein manifold, via the theorem that Stein manifolds admit embedding in some ON.…”

General Integral Representation of the Holomorphic Functions on the Analytic Subvariety

“…Sergeev and Henkin t5] also obtained an integral representation for the strictly pseudoconvex ployhedra. Stout [6] and Hatziafratis [7] have respectively proved integral formulas for strictly pseudoconvex domains in codimension-one and codimension-m complex submanifolds of C n . The formula which was given by Stout is valid not only for nonsingular hyper surf aces, but also for certain subvarieties which may possess sufficiently restricted singular points.…”

“…integral formulas for holomorphic functions. The papers of Stout [6] , Hatziafratis [7] and the author [8] are most relevant references to this work. §2.…”