A Ronkin type function for coamoebas

The Ronkin function plays a fundamental role in the theory of amoebas. We introduce an analogue of the Ronkin function in the setting of coamoebas. It turns out to be closely related to a certain toric arrangement known as the shell of the coamoeba and we use our Ronkin type function to obtain some properties of it

A Ronkin type function for coamoebas

The Ronkin function plays a fundamental role in the theory of amoebas. We introduce an analogue of the Ronkin function in the setting of coamoebas. It turns out to be closely related to a certain toric arrangement known as the shell of the coamoeba and we use our Ronkin type function to obtain some properties of it

Regularizations of products of residue and principal value currents

Let f 1 and f 2 be two functions on some complex n-manifold and let φ be a test form of bidegree (n, n - 2). Assume that (f 1 , f 2 ) defines a complete intersection. The integral of φ / (f 1 f 2 ) on {| f 1 | 2 = ε{lunate} 1 , | f 2 | 2 = ε{lunate} 2 } is the residue integral I f1 , f 2 φ (ε{lunate} 1 , ε{lunate} 2 ). It is in general discontinuous at the origin. Let χ 1 and χ 2 be smooth functions on [0, ∞] such that χ j (0) = 0 and χ j (∞) = 1. We prove that the regularized residue integral defined as the integral of over(∂, ̄) χ 1 ∧ over(∂, ̄) χ 2 ∧ φ / (f 1 f 2 ), where χ j = χ j (| f j | 2 / ε{lunate} j ), is H\uf6lder continuous on the closed first quarter and that the value at zero is the Coleff-Herrera residue current acting on φ. In fact, we prove that if φ is a test form of bidegree (n, n - 1) then the integral of χ 1 over(∂, ̄) χ 2 ∧ φ / (f 1 f 2 ) is H\uf6lder continuous and tends to the over(∂, ̄)-potential [(1 / f 1 ) ∧ over(∂, ̄) (1 / f 2 )] of the Coleff-Herrera current, acting on φ. More generally, let f 1 and f 2 be sections of some vector bundles and assume that f 1 ⊕ f 2 defines a complete intersection. There are associated principal value currents U f and U g and residue currents R f and R g . The residue currents equal the Coleff-Herrera residue currents locally. One can give meaning to formal expressions such as e.g. U f ∧ R g in such a way that formal Leibnitz rules hold. Our results generalize to products of these currents as well. \ua9 2006 Elsevier Inc. All rights reserved

Holomorphic forms, the ∂ˉ\bar{\partial}-equation, and duality on a reduced complex space

We study two natural notions of holomorphic forms on a reducedpure $n$-dimensional complex space $X$: sections of the sheaves $\Omega_X^{\bullet}$ of germs ofholomorphic forms on $X_{reg}$ that have a holomorphic extension to some ambient complex manifold, and sections of the sheaves $\omega_X^{\bullet}$ introduced by Barlet. We show that $\Omega_X^p$ and $\omega_X^{n-p}$ are Serre dual to each otherin a certain sense. We also provide explicit, intrinsic and semi-global Koppelman formulas for the $\bar{\partial}$-equation on $X$ and introduce fine sheaves $\mathscr{A}_X^{p,q}$and $\mathscr{B}_X^{p,q}$ of $(p,q)$-currents on $X$, that are smooth on $X_{reg}$,such that $(\mathscr{A}_X^{p,\bullet},\bar{\partial})$ is a resolution of \Om_X^pand, if $\Omega_X^{n-p}$ is Cohen-Macaulay, $(\mathscr{B}_X^{p,\bullet},\bar{\partial})$is a resolution of $\omega_X^{p}$

Holomorphic forms, the ∂ˉ\bar{\partial}-equation, and duality on a reduced complex space

We study two natural notions of holomorphic forms on a reducedpure $n$-dimensional complex space $X$: sections of the sheaves $\Omega_X^{\bullet}$ of germs ofholomorphic forms on $X_{reg}$ that have a holomorphic extension to some ambient complex manifold, and sections of the sheaves $\omega_X^{\bullet}$ introduced by Barlet. We show that $\Omega_X^p$ and $\omega_X^{n-p}$ are Serre dual to each otherin a certain sense. We also provide explicit, intrinsic and semi-global Koppelman formulas for the $\bar{\partial}$-equation on $X$ and introduce fine sheaves $\mathscr{A}_X^{p,q}$and $\mathscr{B}_X^{p,q}$ of $(p,q)$-currents on $X$, that are smooth on $X_{reg}$,such that $(\mathscr{A}_X^{p,\bullet},\bar{\partial})$ is a resolution of \Om_X^pand, if $\Omega_X^{n-p}$ is Cohen-Macaulay, $(\mathscr{B}_X^{p,\bullet},\bar{\partial})$is a resolution of $\omega_X^{p}$