Related papers: Codimension one foliations with numerically trivia…
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…
We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields…
We consider resolution of singularities for $1$-foliations on varieties of dimension at most three in positive characteristic. We prove that such singularities can be completely resolved if we allow tame regular Deligne--Mumford stacks as…
In this paper, we classify the configurations of the singular points which appear on the quotients of the projective plane by the $1$-foliations of degree $-1$ in characteristic $2$.
We study the invariant theory of singular foliations of the projective plane. Our first main result is that a foliation of degree m>1 is not stable only if it has singularities in dimension 1 or contains an isolated singular point with…
We show that an everywhere regular foliation $\mathcal F$ with compact canonically polarized leaves on a quasi-projective manifold $X$ has isotrivial family of leaves when the orbifold base of this family is special. By a recent work of…
We classify singular foliations admitting a given leaf and a given transverse singular foliation.
We present examples of foliations with infinite dimensional basic symplectic and com- plex cohomologies, along with a general sufficient condition for such phenomena. This puts re- strictions on possible generalizations of several…
We study local, global and local-to-global properties of threefolds with certain singularities. We prove criteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincar\'e duality.…
We classify irreducible polar foliations of codimension $q$ on quaternionic projective spaces $\mathbb H P^n$, for all $(n,q)\neq(7,1)$. We prove that all irreducible polar foliations of any codimension (resp. of codimension one) on…
Androulidakis and Skandalis showed how to associate a holonomy groupoid, a smooth convolution algebra and a C*-algebra to any singular foliation. In this note, we consider the singular foliations of a one-dimensional manifold given by…
Let $X$ be a compact K\"ahler manifold and $\alpha$ be a class in the Dolbeault cohomology class of bidegree $(1, 1)$ on $X$. When the numerical dimension of $\alpha$ is one and $\alpha$ admits at least two smooth semi-positive…
Following T. Suwa, we study unfoldings of algebraic foliations and their relationship with families of foliations, making focus on those unfoldings related to trivial families. The results obtained in the study of unfoldings are then…
We study analytic deformations and unfoldings of holomorphic foliations in complex projective plane $\mathbb{C}P(2)$. Let $\{\mathcal{F}_t\}_{t \in \mathbb{D}_{\epsilon}}$ be topological trivial (in $\mathbb{C}^2$) analytic deformation of a…
The Kodaira dimension of Shimura varieties has been studied by many people. Kondo and Gritsenko-Hulek-Sankaran studied the singularities of orthogonal Shimura varieties related to the moduli spaces of polarized K3 surfaces. They proved that…
Campana introduced the class of special varieties as the varieties admitting no Bogomolov sheaves i.e. rank one coherent subsheaves of maximal Kodaira dimension in some exterior power of the cotangent bundle. Campana raised the question if…
The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincar\'e complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more…
The main purpose of this paper is to provide a structure theorem for codimension one singular transversely projective foliationson projective manifolds. To reach our goal, we firstly extend Corlette-Simpson's classification of rank two…
We study del Pezzo fibrations of degree 1 with terminal singularities. A connection between singularities on del Pezzo surfaces of degree 1 and Kodaira's classification of elliptic singular fibers will be studied in this paper. By this…
The paper deals with a modified Godbillon-Vey class defined by Losik for codimension-one foliations. This characteristic class takes values in the cohomology of the second order frame bundle over the leaf space of the foliation. The…