Related papers: Log-uniruled affine varieties without cylinder-lik…

Let X be a smooth, projective variety over the field of complex numbers. Here we focus on a conjecture attributed to Shigefumi Mori, which claims that X is uniruled if and only if the Kodaira dimension of X is negative.

We give conditions for a uniruled variety of dimension at least 2 to be non-solid. This study provides further evidence to a conjecture by Abban and Okada on the solidity of Fano 3-folds. To complement our results we write explicit…

We study pluricanonical systems on smooth projective varieties of positive Kodaira dimension, following the approach of Hacon-McKernan, Takayama and Tsuji succesfully used in the case of varieties of general type. We prove a uniformity…

We prove that any smooth rational projective surface over the field of complex numbers has an open covering consisting of 3 subsets isomorphic to affine planes.

Affine rotation surfaces are a generalization of the well-known surfaces of revolution. Affine rotation surfaces arise naturally within the framework of affine differential geometry, a field started by Blaschke in the first decades of the…

We prove that up to automorphisms a line admits a unique embedding into the regular part of of a simplicial toric variety of dimension n>=4 over an algebraically closed field of characteristic zero which is smooth in codimension 2.

We continue the program of classification of normal Q-acyclic surfaces defined over the field of complex numbers, so-called 'Q-homology planes'. Here we show that if a Q-homology plane has negative Kodaira dimension then its smooth locus is…

We give several new criteria for a quasi-projective variety to be affine. In particular, we prove that an algebraic manifold $Y$ with dimension $n$ is affine if and only if $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $\kappa(D,…

This paper continues the study of non-general type subvarieties begun in a joint paper with M.Schneider and A.Sommese (Int.L.Math. 10, 1999). We prove uniruledness of a projective manifold containing a submanifold not of general type whose…

We characterize embeddability of algebraic varieties into smooth toric varieties and prevarieties. Our embedding results hold also in an equivariant context and thus generalize a well known embedding theorem of Sumihiro on quasiprojective…

We prove that the affine cone over a general primitively polarised K3 surface of genus g is smoothable if and only if g\le 10 or g=12. We also give several examples of singularities with special behaviour, such as surfaces whose affine cone…

We give a general structure theorem for affine A 1-fibrations on smooth quasi-projective surfaces. As an application, we show that every smooth A 1-fibered affine surface non-isomorphic to the total space of a line bundle over a smooth…

Let F be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the DG-Lie algebra RHom(F,F) of derived endomorphisms of F is formal. The proof is based on the study of equivariant $L_{\infty}$ minimal models…

Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper arXiv:0805.0045v4 by Haines, Kottwitz, Reuman, and the first named author, about…

A smooth variety is called uniformly rational if every point admits a Zariski open neighborhood isomorphic to a Zariski open subset of the affine space. In this note we show that every smooth and rational affine variety endowed with an…

We exhibit a smooth complex rational affine surface with uncountably many nonisomorphic real forms.

We prove that the moduli spaces of K3 surfaces with non-symplectic involutions are unirational. As a by-product we describe configuration spaces of 4<d<9 points in the projective plane as arithmetic quotients of type IV.

An affine surface is said to be an affine Zoll surface if all affine geodesics close smoothly. It is said to be an affine almost Zoll surface if thru any point, every affine geodesic but one closes smoothly (the exceptional geodesic is said…

We prove sharp $L^2$ Fourier restriction inequalities for compact, smooth surfaces in $\mathbb{R}^3$ equipped with the affine surface measure or a power thereof. The results are valid for all smooth surfaces and the bounds are uniform for…

We prove that invariant subbundles of the Kontsevich-Zorich cocycle respect the Hodge structure. In particular, we establish a version of Deligne semisimplicity in this context. This implies that invariant subbundles must vary polynomially…