Related papers: $\mathbb{A}^1$-curves on log smooth varieties

Let $X_0$ be an irreducible smooth projective curve defined over $\overline{\mathbb Q}$ and $\mathbb E$ a vector bundle on $X_0$. We give a criterion for connections on the base change ${\mathbb E}\otimes_{\overline{\mathbb Q}}{\mathbb C}…

Let $X$ be a connected smooth complex projective variety of dimension $n \geq 1$. Let $D$ be a simple normal crossing divisor on $X$. Let $G$ be a connected complex Lie group, and $E_G$ a holomorphic principal $G$-bundle on $X$. In this…

In this paper, we study the deformation theory of degenerate algebraic curves on singular varieties which appear as the degenerate limit of families of varieties. For this purpose, we systematically develop a new method to calculate the…

We show that pairs $(X,Y)$ of 1-spherical objects in $A_\infty$-categories, such that the morphism space ${\rm Hom}(X,Y)$ is concentrated in degree 0, can be described by certain noncommutative orders over (possibly stacky) curves. In fact,…

In order to understand the structure of the cohomologies involved in the study of projectively equivariant quantizations, we introduce a notion of affine representation of a Lie algebra.We show how it is related to linear representations…

We consider rationally connected complex projective manifolds M and show that their loop spaces--infinite dimensional complex manifolds--have properties similar to those of M. Furthermore, we give a finite dimensional application concerning…

Let $A$ be an abelian variety and $G$ a finite group of automorphisms of $A$ fixing the origin such that $A/G$ is smooth. The quotient $A/G$ can be seen as a fibration over an abelian variety whose fibers are isomorphic to a product of…

Following the work of Totaro and Pereira, we study sufficient conditions under which collections of pairwise-disjoint divisors on a variety over an algebraically closed field are contained in the fibers of a morphism to a curve. We prove…

We study relations among characteristic classes of smooth manifold bundles with highly-connected fibers. For bundles with fiber the connected sum of $g$ copies of a product of spheres $S^d \times S^d$ and an odd $d$, we find numerous…

We prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over non-archimedean fields. Our arguments rely on known structure theorems for the relevant Picard varieties, together with recent…

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…

This survey focuses on the geometric problem of log-surfaces, which are pairs consisting of a smooth projective surface and a reduced non-empty boundary divisor. In the first part, we focus on the geography problem for complex log-surfaces…

A theorem of Mumford states that, on complex surfaces, any normal isolated singularity whose link is diffeomorphic to a sphere is actually a smooth point. While this property fails in higher dimensions, McLean asks whether the contact…

We consider some conditions under which a smooth projective variety X is actually the projective space. We also extend to the case of positive characteristic some results in the theory of vector bundle adjunction. We use methods and…

Let $f\colon X \to \mathbb{A}^1_t$ be an affine flat morphism of finite type, and let $V = f^{-1}(0)$. Then, we obtain a morphism of log schemes $f\colon (X|V) \to (\mathbb{A}^1_t|0)$. In this article, we develop algorithmic tools to study…

In this article we study the cohomological and homological (due to Jannsen) Hodge conjecture for singular varieties. The motivation for studying singular varieties comes from the fact that any smooth projective variety X is birational to a…

Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures…

The first part of this note contains a review of basic properties of the variety of lines contained in an embedded projective variety and passing through a general point. In particular we provide a detailed proof that for varieties defined…

Let R^1(A,R) be the degree-one resonance variety over a field R of a hyperplane arrangement A. We give a geometric description of R^1(A,R) in terms of projective line complexes. The projective image of R^1(A,R) is a union of ruled…

We prove that for any closed surface of genus at least four, and any punctured surface of genus at least two, the space of ending laminations is connected. A theorem of E. Klarreich implies that this space is homeomorphic to the Gromov…