Related papers: Naive $\mathbb A^1$-homotopies on ruled surfaces

Sheaves on non-reduced curves can appear in moduli space of 1-dimensional semistable sheaves over a surface, and moduli space of Higgs bundles as well. We estimate the dimension of the stack $\mathbf{M}_{X}(nC,\chi)$ of pure sheaves…

In this paper we begin to study curves on a weighted projective plane with one trivial weight, ${\mathbb P}(1,m,n)$, by determining the genus of curves of Fermat type. These are curves defined by a ``homogeneous'' polynomial analagous to…

We classify (semi)stable sheaves on a rational curve with one node. The results are based on the classification of indecomposable torsion-free sheaves due to Drozd and Greuel "Tame and wild projective curves and classification of vector…

We provide an explicit formula for localizing $A^1$-homotopy invariants of topological Fukaya categories of marked surfaces. Following a proposal of Kontsevich, this differential $\mathbb Z$-graded category is defined as global sections of…

A smooth Hermitian surface $X$ is a projective surface isomorphic to the Fermat surface of degree $q+1$ in positive characteristic. We study incidence relations of the rational curves of degree $q+1$ contained in $X$, and show that such…

In this note we prove that the $\mathbb{A}^1$-connected component sheaf $a_{Nis}(\pi_0^{\mathbb{A}^1}(\mathcal{X}))$ of an $H$-group $\mathcal{X}$ is $\mathbb{A}^1$-invariant.

We prove necessary and sufficient conditions for a smooth surface in a 4-manifold X to be pseudoholomorphic with respect to some almost complex structure on X. This provides a systematic approach to the construction of pseudoholomorphic…

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli…

Let G be a connected, compact, semisimple Lie group. It is known that for a compact closed orientable surface $\Sigma$ of genus $l >1$, the order of the group $H^2(\Sigma,\pi_1(G))$ is equal to the number of connected components of the…

We confirm the quasi-projective case of Saito's conjecture, namely that the cohomological characteristic classes defined by Abbes and Saito can be computed in terms of the characteristic cycles. We construct a cohomological characteristic…

Let X be a projective complex K3 surface. Beauville and Voisin singled out a 0-cycle c_X on X of degree 1: it is represented by any point lying on a rational curve in X. Huybrechts proved that the second Chern class of a rigid simple…

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…

We study the cohomology of a general stable sheaf on an abelian surface. We say that a moduli space satisfies weak Brill-Noether if the general sheaf has at most one non-zero cohomology group. Let $(X,H)$ be a polarized abelian surface and…

For a smooth subvariety $X\subset\Bbb P^N$, consider (analogously to projective normality) the vanishing condition $H^1(\Bbb P^N,\Cal I^2_X(k))=0$, $k\ge3$. This condition is shown to be satisfied for all sufficiently large embeddings of a…

In this paper, we extend the structure theorem for smooth projective varieties with nef tangent bundle to projective klt varieties whose tangent sheaf is either positively curved or almost nef. Specifically, we show that such a variety $X$,…

We study foliations by curves on the three-dimensional projective space with no isolated singularities, which is equivalent to assuming that the conormal sheaf is locally free. We provide a classification of the topological and algebraic…

The goal of this paper is to extend the work of Voevodsky and Morel on the homotopy $t$-structure on the category of motivic complexes to the context of motives for logarithmic schemes. To do so, we prove an analogue of Morel's connectivity…

Let $X$ be a smooth projective horospherical variety of Picard number one. We show that a uniruled projective manifold of Picard number one is biholomorphic to $X$ if its variety of minimal rational tangents at a general point is…

We prove a well known conjecture of Nikolai Ivanov which states that if $X$ is a surface of genus $\geq 3$ (with any number of punctures and boundary components), $\rm{Mod}(X)$ is the mapping class group of $X$, and $K < \rm{Mod}(X)$ is a…

We define a new version of $\mathbb A^1$-homology, called cellular $\mathbb A^1$-homology, for smooth schemes over a field that admit an increasing filtration by open subschemes with cohomologically trivial closed strata. We provide several…