Related papers: Rank one foliations on toroidal varieties

Let $(X,D)$ be an open log del Pezzo surface of rank one, that is, $X$ is a normal projective surface of Picard rank one, the boundary $D$ is a reduced nonzero divisor on $X$, and the anti-log canonical divisor $-(K_X+D)$ is ample. We show…

We prove that, under mild restrictions, the space of codimension-one foliations of degree one on a smooth projective complete intersection has two irreducible components of logarithmic type. We also prove that the same conclusion holds for…

A singular foliation $\mathcal F$ gives a partition of a manifold $M$ into leaves whose dimension may vary. Associated to a singular foliation are two complexes, that of the diffeological differential forms on the leaf space $M / \mathcal…

Let (X,D) be a dlt pair, where X is a normal projective variety. Let S denote the support of the rounddown of D, and K the canonical divisor of X. We show that any smooth family of canonically polarized varieties over X\S is isotrivial if…

We prove that the characteristic foliation $F$ on a non-singular divisor $D$ in an irreducible projective hyperkaehler manifold $X$ cannot be algebraic, unless the leaves of $F$ are rational curves or $X$ is a surface. More generally, we…

Given a variety defined over a field of characteristic zero and an algebraically integrable foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on the non-empty open set of stable points,…

Let $\pi\colon\mathcal{X}\to B$ be a family over a smooth connected analytic variety $B$, not necessarily compact, whose general fiber $X$ is smooth of dimension $n$, with irregularity $\geq n+1$ and such that the image of the canonical map…

Let $X$ be any variety in characteristic zero. Let $V \subset X$ be an open subset that has toroidal singularities. We show the existence of a canonical desingularization of $X$ except for V. It is a morphism $f: Y \to X$ , which does not…

We give a classification of pairs (F, f) where F is a holomorphic foliation on a projective surface and f is a non-invertible dominant rational map preserving F. We prove that both the map and the foliation are integrable in a suitable…

We interpret Galois covers in terms of particular monoidal functors, extending the correspondence between torsors and fiber functors. As applications we characterize tame $G$-covers between normal varieties for finite and \'etale group…

Given a smooth, projective variety $X$ and an effective divisor $D\,\subseteq\, X$, it is well-known that the (topological) obstruction to the deformation of the fundamental class of $D$ as a Hodge class, lies in $H^2(\mathcal{O}_X)$. In…

We classify the irreducible components of the space of foliations on Fano 3-folds with rank one Picard group. As a corollary we obtain a classification of holomorphic Poisson structures on the same class of 3-folds.

We use recent results by Bainbridge-Chen-Gendron-Grushevsky-Moeller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in…

Deformation of morphisms along leaves of foliations define the tangential foliation on the corresponding space of morphisms. We prove that codimension one fo-liations having a tangential foliation with at least one non-algebraic leaf are…

We prove that if $f:X \rightarrow A$ is a morphism from a smooth projective variety $X$ to an abelian variety $A$ over a number field $K$, and $G$ is a subgroup of automorphisms of $X$ satisfying certain properties, and if a prime $p$…

Given a normal complete variety $Y$ over an algebraically closed field $\mathbb K$, distinct effective Weil divisors $D_1,... D_n$ of $Y$ and positive integers $d_1,... d_n$, we spell out the conditions for the existence of an abelian cover…

Let $\mathcal{F}$ be a transversely orientable codimension one minimal foliation without vanishing cycles of a manifold $M$. We show that if the fundamental group of each leaf of $\mathcal{F}$ has polynomial growth of degree $k$ for some…

Let R\_n be the ring of Laurent polynomials in n variables over a field k of characteristic zero and let K\_n be its fraction field.Given a linear algebraic k-group $G$, we show that a K\_n-torsor under G which is unramified with respect to…

Given a category $\mathcal{E}$, we establish sufficient conditions on a faithful isofibration $\mathcal{E}\rightarrow\operatorname{Mon}(\mathcal{V})$ valued in the category of monoids internal to a monoidal additive category $\mathcal{V}$…

The canonical ring $S_D = \bigoplus_{d \geq 0} H^0(X, \lfloor dD \rfloor)$ of a divisor D on a curve X is a natural object of study; when D is a Q-divisor, it has connections to projective embeddings of stacky curves and rings of modular…