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-Forstneri\v{c} 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 1930's, 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 article we present 3 classes of properties: 1. density property, 2. flexibility 3. Oka-Forstneri\v{c}. 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.Comment: thanks added, minor correction

Sharp version of the Goldberg-Sachs theorem

We reexamine from first principles the classical Goldberg-Sachs theorem from General Relativity. We cast it into the form valid for complex metrics, as well as real metrics of any signature. We obtain the sharpest conditions on the derivatives of the curvature that are sufficient for the implication (integrability of a field of alpha planes)$\Rightarrow$(algebraic degeneracy of the Weyl tensor). With every integrable field of alpha planes we associate a natural connection, in terms of which these conditions have a very simple form.Comment: In this version we made a minor change in Remark 5.5 and simplified Section 6, starting at Theorem 6.

Fujita's Conjecture and Frobenius amplitude

We prove a version of Fujita's Conjecture in arbitrary characteristic, generalizing results of K.E. Smith. Our methods use the Frobenius morphism, but avoid tight closure theory. We also obtain versions of Fujita's Conjecture for coherent sheaves with certain ampleness properties.Comment: 8 pages. Erratum added to replace Lemma 2.

Horizontal Ga-actions on affine T-varieties of complexity one

We classify the $\mathbb{G}_{a}$-actions on normal affine varieties defined over any field that are horizontal with respect to a torus action of complexity one. This generalizes previous results that were available for perfect ground fields (cf. [Flenner-Zaidenberg2005, Liendo2010, L-Liendo2016]).Comment: 14 pages. To appear in Le Matematich