Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka–Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930s, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview chapter we present three classes of properties: (1) density property, (2) flexibility, and (3) Oka–Forstnerič. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.

The algebraic density property for affine toric varieties

In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Andersén–Lempert theory. We show that an affine toric variety X satisfies this algebraic density property relative to a closed T-invariant subvariety Y if and only if X∖Y≠TX∖Y≠T. For toric surfaces we are able to classify those which possess a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella's famous classification of complete algebraic vector fields in the affine plane.

IL28B alleles associated with poor hepatitis C virus (HCV) clearance protect against inflammation and fibrosis in patients infected with non-1 HCV genotypes

Genetic polymorphisms near IL28B are associated with spontaneous and treatment-induced clearance of hepatitis C virus (HCV), two processes that require the appropriate activation of the host immune responses. Intrahepatic inflammation is believed to mirror such activation, but its relationship with IL28B polymorphisms has yet to be fully appreciated. We analyzed the association of IL28B polymorphisms with histological and follow-up features in 2335 chronically HCV-infected Caucasian patients. Assessable phenotypes before any antiviral treatment included necroinflammatory activity (n = 1,098), fibrosis (n = 1,527), fibrosis progression rate (n = 1,312), and hepatocellular carcinoma development (n = 1,915). Associations of alleles with the phenotypes were evaluated by univariate analysis and multivariate logistic regression, accounting for all relevant covariates. The rare G allele at IL28B marker rs8099917-previously shown to be at risk of treatment failure-was associated with lower activity (P = 0.04), lower fibrosis (P = 0.02) with a trend toward lower fibrosis progression rate (P = 0.06). When stratified according to HCV genotype, most significant associations were observed in patients infected with non-1 genotypes (P = 0.003 for activity, P = 0.001 for fibrosis, and P = 0.02 for fibrosis progression rate), where the odds ratio of having necroinflammation or rapid fibrosis progression for patients with IL28B genotypes TG or GG versus TT were 0.48 (95% confidence intervals 0.30-0.78) and 0.56 (0.35-0.92), respectively. IL28B polymorphisms were not predictive of the development of hepatocellular carcinoma.

Genome-wide association study identifies variants associated with progression of liver fibrosis from HCV infection

Polymorphisms in IL28B were shown to affect clearance of hepatitis C virus (HCV) infection in genome-wide association (GWA) studies. Only a fraction of patients with chronic HCV infection develop liver fibrosis, a process that might also be affected by genetic factors. We performed a 2-stage GWA study of liver fibrosis progression related to HCV infection.