Related papers: Recent progress in the geometry of Q-acyclic surfa…
We survey some old and new results about acyclic (affine) complex surfaces, also called homology planes. We ask several questions and leave open directions for future research.
We survey some results on real rational surfaces focused on their topology and their birational geometry.
A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type. We show that if a Q-homology plane contains a non-quotient…
We study quartic surfaces that admit a group of projective automorphisms isomorphic to icosahedron group.
In this note we briefly review some recent results of the authors on the topological and geometrical properties of 3-cosymplectic manifolds.
This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent r{\^o}le. This selection is largely…
We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.
In this article we briefly survey some developments in gauged linear sigma models (GLSMs). Specifically, we give an overview of progress on constructions of GLSMs for various geometries, GLSM-based computations of quantum cohomology,…
A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We classify singular Q-homology planes which are C^1- or C*-ruled. We analyze their completions, the number of different rulings, the number of…
In this article we will represent some ideas and a lot of new theorems in Euclidean plane geometry.
In this survey paper, we give a complete list of known results on the first and the second homology groups of surface mapping class groups. Some known results on higher (co)homology are also mentioned.
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…
We show that a homology plane of general type has at worst a single cyclic quotient singular point. An example of such a surface with a singular point does exist. We also show that the automorphism group of a smooth contractible surface of…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
In recent years, several quantizations of real manifolds have been studied, in particular from the point of view of Connes' noncommutative geometry. Less is known for complex noncommutative spaces. In this paper, we review some recent…
Fix a prime number $p$. We report on some recent developments in algebraic geometry (broadly construed) over $p$-adically complete commutative rings. These developments include foundational advances within the subject as well as external…
Making a survey of recent constructions of universal cohomologies we suggest a new framework for a theory of motives in algebraic geometry.
The article surveys published and not yet published results about moduli spaces of algebraic surfaces.
The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably…
There are three complete plane geometries of constant curvature: spherical, Euclidean and hyperbolic geometry. We explain how a closed oriented surface can carry a geometry which locally looks like one of these. Focussing on the hyperbolic…