A taxonomic revision of the endemic members of Varronia P. Br. (Cordiaceae) in the Galápagos Islands

The Galápagos Islands have long been an arena for biological diversity, scientific discovery, and conservation. Accurate identification and documentation of the flora of the Galápagos continue to aid conservation efforts. The purpose of this study was to conduct a taxonomic revision of the endemic members genus Varronia (Cordiaceae) on the Islands: V. revoluta (Hook. f.) Andersson, V. leucophlyctis (Hook. f.) Andersson, V. canescens Andersson, and V. scouleri (Hook. f.) Andersson. Taxonomic uncertainty among these species has resulted in difficult evaluation of their potential for conservation status by the International Union for the Conservation of Nature. The present taxonomic study concludes that there are four endemic species of Varronia: V. revoluta (Hook. f.) Andersson, V. leucophlyctis (Hook. f.) Andersson, V. canescens Andersson, and V. scouleri (Hook. f.) Andersson. Further, this study has resulted in a new dichotomous key for field identification, using hair types as the strongest characters for differentiation between species. A new distribution map also specifies on which islands each species has been found. Proper identification and distribution assessment adds valuable information for the evaluation of endemic Varronia populations on the Islands to determine the conservation status of each species

Residue currents of holomorphic morphisms

Given a generically surjective holomorphic vector bundle morphism $f\colon E\to Q$, $E$ and $Q$ Hermitian bundles, we construct a current $R^f$ with values in \Hom(Q,H), where $H$ is a certain derived bundle, and with support on the set $Z$ where $f$ is not surjective. The main property is that if $\phi$ is a holomorphic section of $Q$, and $R^f\phi=0$, then locally $f\psi=\phi$ has a holomorphic solution $\psi$. In the generic case also the converse holds. This gives a generalization of the corresponding theorem for a complete intersection, due to Dickenstein-Sessa and Passare. We also present results for polynomial mappings, related to M Noether's theorem and the effective Nullstellensatz. The construction of the current is based on a generalization of the Koszul complex. By means of this complex one can also obtain new global estimates of solutions to $f\psi=\phi$, and as an example we give new results related to the $H^p$-corona problem

Integral representation with weights II, division and interpolation

Let $f$ be a $r\times m$-matrix of holomorphic functions that is generically surjective. We provide explicit integral representation of holomorphic $\psi$ such that $\phi=f\psi$, provided that $\phi$ is holomorphic and annihilates a certain residue current with support on the set where $f$ is not surjective. We also consider formulas for interpolation. As applications we obtain generalizations of various results previously known for the case $r=1$

Stability estimates with a priori bound for the inverse local Radon transform

We consider the inverse problem for the $2$-dimensional weighted local Radon transform $R_m[f]$, where $f$ is supported in $y\geq x^2$ and $R_m[f](\xi,\eta)=\int f(x, \xi x + \eta) m(\xi, \eta, x)\,\text{d} x$ is defined near $(\xi,\eta)=(0,0)$. For weight functions satisfying a certain differential equation we give weak estimates of $f$ in terms of $R_m[f]$ for functions $f$ that satisfies an a priori bound.Comment: 34 page

Koppelman formulas on the A_1-singularity

In the present paper, we study the regularity of the Andersson-Samuelsson Koppelman integral operator on the $A_1$-singularity. Particularly, we prove $L^p$- and $C^0$-estimates. As applications, we obtain $L^p$-homotopy formulas for the $\bar{\partial}$-equation on the $A_1$-singularity, and we prove that the $\mathcal{A}$-forms introduced by Andersson-Samuelsson are continuous on the $A_1$-singularity.Comment: 23 pages. v3: Minor changes made for the final versio

On the duality theorem on an analytic variety

The duality theorem for Coleff-Herrera products on a complex manifold says that if $f = (f_1,\dots,f_p)$ defines a complete intersection, then the annihilator of the Coleff-Herrera product $\mu^f$ equals (locally) the ideal generated by $f$. This does not hold unrestrictedly on an analytic variety $Z$. We give necessary, and in many cases sufficient conditions for when the duality theorem holds. These conditions are related to how the zero set of $f$ intersects certain singularity subvarieties of the sheaf $\mathcal{O}_Z$.Comment: 21 pages, v2: Incorporate changes from the review proces