Galois descent of semi-affinoid spaces

We study the Galois descent of semi-affinoid non-archimedean analytic spaces. These are the non-archimedean analytic spaces which admit an affine special formal scheme as model over a complete discrete valuation ring, such as for example open or closed polydiscs or polyannuli. Using Weil restrictions and Galois fixed loci for semi-affinoid spaces and their formal models, we describe a formal model of a $K$-analytic space $X$, provided that $X\otimes_KL$ is semi-affinoid for some finite tamely ramified extension $L$ of $K$. As an application, we study the forms of analytic annuli that are trivialized by a wide class of Galois extensions that includes totally tamely ramified extensions. In order to do so, we first establish a Weierstrass preparation result for analytic functions on annuli, and use it to linearize finite order automorphisms of annuli. Finally, we explain how from these results one can deduce a non-archimedean analytic proof of the existence of resolutions of singularities of surfaces in characteristic zero.Comment: Exposition improved and minor modifications. 37 pages. To appear in Math.

Inner geometry of complex surfaces: a valuative approach

Given a complex analytic germ $(X, 0)$ in $(\mathbb C^n, 0)$, the standard Hermitian metric of $\mathbb C^n$ induces a natural arc-length metric on $(X, 0)$, called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity $(X,0)$ by means of an infinite family of numerical analytic invariants, called inner rates. Our main result is a formula for the Laplacian of the inner rate function on a space of valuations, the non-archimedean link of $(X,0)$. We deduce in particular that the global data consisting of the topology of $(X,0)$, together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of $(X,0)$, completely determine all the inner rates on $(X,0)$, and hence the local metric structure of the germ. Several other applications of our formula are discussed in the paper.Comment: Proposition 5.3 strengthened, exposition improved, some typos corrected, references updated. 42 pages and 10 figures. To appear in Geometry & Topolog

Management of acute pancreatitis: current knowledge and future perspectives

In recent years, a number of articles have been published on the treatment of acute pancreatitis in experimental models and most of them concerned animals with mild disease. However, it is difficult to translate these results into clinical practice. For example, infliximab, a monoclonal TNF antibody, was experimentally tested in rats and it was found to significantly reduce the pathologic score and serum amylase activity and also to alleviate alveolar edema and acute respiratory distress syndrome; however, no studies are available in clinical human acute pancreatitis. Another substance, such as interleukin 10, was efficacious in decreasing the severity and mortality of lethal pancreatitis in rats, but seems to have no effect on human severe acute pancreatitis. Thus, the main problem in acute pancreatitis, especially in the severe form of the disease, is the difficulty of planning clinical studies capable of giving reliable statistically significant answers regarding the benefits of the various proposed therapeutic agents previously tested in experimental settings. According to the pathophysiology of acute pancreatitis, the efficacy of the drugs already available, such as gabexate mesilate, lexipafant and somatostatin should be re-evaluated and should be probably administered in a different manner. Of course, also in this case, we need adequate studies to test this hypothesis

On Lipschitz Normally Embedded complex surface germs

We undertake a systematic study of Lipschitz Normally Embedded normal complex surface germs. We prove in particular that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In the appendix, we give a new example of a Lipschitz Normally Embedded surface singularity.Comment: v3: Section 8 has been removed and will be posted separately in expanded and vastly improved form; the title had been edited accordingly. 31 pages, 2 figure