We study foliations $\mathcal{F}$ on Hirzebruch surfaces $S_δ$ and prove that, similarly to those on the projective plane, any $\mathcal{F}$ can be represented by a bi-homogeneous polynomial affine $1$-form. In case $\mathcal{F}$ has isolated singularities, we show that, for $ δ=1 $, the singular scheme of $\mathcal{F}$ does determine the foliation, with some exceptions that we describe, as is the case of foliations in the projective plane. For $δ\neq 1$, we prove that the singular scheme of $\mathcal{F}$ does not determine the foliation. However we prove that, in most cases, two foliations $\mathcal{F}$ and $\mathcal{F}'$ given by sections $s$ and $s'$ have the same singular scheme if and only if $s'=Φ(s)$, for some global endomorphism $Φ$ of the tangent bundle of $S_δ$.
We prove that a generic holomorphic foliation on a weighted projective plane has no algebraic solutions when the degree is big enough. We also prove an analogous result for foliations on Hirzebruch surfaces.
We study properties of the Hirzebruch class of quotient singularities $\mathbb{C}^n/G$, where $G$ is a finite matrix group. The main result states that the Hirzebruch class coincides with the Molien series of $G$ under suitable substitution of variables. The Hirzebruch class of a crepant resolution can be described specializing the orbifold elliptic genus constructed by Borisov and Libgober. It is equal to the combination of Molien series of centralizers of elements of $G$. This is an incarnation of the McKay correspondence. The results are illustrated with several examples, in particular of 4…
We prove a new formula for the Hirzebruch-Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern-Milnor classes in the hypersurface case that was conjectured by S. Yokura and was proved by A. Parusinski and P. Pragacz. It also generalizes a formula of J. Seade and T. Suwa for the Chern-Milnor classes of complete intersections with isolated singularities.
We investigate the notion of the $p$-divisor for foliations on a smooth algebraic surface defined over a field of positive characteristic $p$ and we study some of their properties. We present a structure theorem for the $p$-divisor of foliations in the projective plane and the Hirzebruch surfaces where we show that, under certain conditions, such $p$-divisors are reduced.
In this paper, we classify del Pezzo foliations on projective manifolds of rank at least 3 and with log canonical singularities in the sense of McQuillan.
It is known that there is at least an invariant analytic curve passing through each of the components in the complement of nodal singularities, after the reduction of singularities of a germ of singular foliation in ${\mathbb C}^2,0$}. Here, we state and prove a generalization of this property to any ambient dimension.
The Hirzebruch $td_y(X)$ class of a complex manifold X is a formal combination of Chern characters of the sheaves of differential forms multiplied by the Todd class. The related $χ_y$-genus admits a generalization for singular complex algebraic varieties. The equivariant version of the Hirzebruch class can be developed as well. The general theory applied in the situation when a torus acts on a singular variety allows to apply powerful tools as the Localization Theorem of Atiyah and Segal for equivariant K-theory and Berline-Vergne formula for equivariant cohomology. We obtain a meaningful inva…
We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. For an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert-Riemenschneider type vanishing theorem for foliated surfaces with good log canonical foliation singularities.
We give a birational reduction of singularities for one dimensional foliations in ambient spaces of dimension three. To do this, we first prove the existence of a Local Uniformization in the sense of Zariski. The reduction of singularities is then obtained by a gluing procedure for Local Uniformization similar to Zariski's one, using an axiomatization of Zariski's work by Olivier Piltant.
We provide several applications of the minimal model program to the local and global study of co-rank one foliations on threefolds. Locally, we prove a singular variant of Malgrange's theorem, a classification of terminal foliation singularities and the existence of separatrices for log canonical singularities. Globally, we prove termination of flips, a connectedness theorem on lc centres, a non-vanshing theorem and some hyperbolicity properties of foliations.
Let $\mathcal F$ be a holomorphic one-dimensional foliation on $\mathbb{P}^n$ such that the components of its singular locus $Σ$ are curves $C_i$ and points $p_j$. We determine the number of $p_j$, counted with multiplicities, in terms of invariants of $\mathcal F$ and $C_i$, assuming that $\mathcal F$ is special along the $C_i$. Allowing just one nonzero dimensional component on $Σ$, we also prove results on when the foliation happens to be determined by its singular locus.
We give formulas for the degrees of the spaces of foliations in P2 with a dicritical singularity of prescribed order. Blowing up such singularity induces, generically, a foliation with all but finitely many leaves transversal to the exceptional line; we also find the degree of the locus defined by imposing a leaf of total contact with the exceptional line.
In this paper we extend to the singular setting the theory of Fano foliations developed in our previous paper. A Q-Fano foliation on a complex projective variety X is a foliation F whose anti-canonical class is an ample Q-Cartier divisor. In the spirit of Kobayashi-Ochiai Theorem, we prove that under some conditions the index i of a Q-Fano foliation is bounded by the rank r of F, and classify the cases in which i=r. Next we consider Q-Fano foliations F for which i=r-1. These are called del Pezzo foliations. We classify codimension 1 del Pezzo foliations on mildly singular varieties.
Let $ω$ be a differential $q$-form defining a foliation of codimension $q$ in a projective variety. In this article we study the singular locus of $ω$ in various settings. We relate a certain type of singularities, which we name \emph{persistent}, with the unfoldings of $ω$, generalizing previous work done on foliations of codimension $1$ in projective space. We also relate the absence of persistent singularities with the existence of a connection in the sheaf of $1$-forms defining the foliation. In the latter parts of the article we extend some of these results to toric varieties by making co…
We consider holomorphic foliations of dimension $k>1$ and codimension $\geq 1$ in the projective space $\mathbb{P}^n$, with a compact connected component of the Kupka set. We prove that, if the transversal type is linear with positive integers eigenvalues, then the foliation consist on the fibers of a rational fibration. As a corollary, if $\mathcal{F}$ is a foliation such that $dim(\mathcal{F})\geq cod(\mathcal{F})+2$ and has transversal type diagonal with different eigenvalues, then the Kupka component $K$ is a complete intersection and we get the same conclusion. The same conclusion holds i…
We present enumerative results for holomorphic foliations by curves on $\mathbb{P}^n$, $n\geq 3$, with singularities of positive dimension. Some of the results presented improve previous ones due to Corrêa--Fernández-Pérez--Nonato Costa--Vidal Martins and Nonato Costa. We also present an enumerative result bounding the number of isolated singularities in a projective subvariety invariant by a holomorphic foliation by curves on $\mathbb{P}^n$ with a singularity of positive dimension.
We introduce spectral Hirzebruch-Milnor classes for singular hypersurfaces. These can be identified with Steenbrink spectra in the isolated singularity case, and may be viewed as their global analogues in general. Their definition uses vanishing cycles of mixed Hodge modules and the Todd class transformation. These are compatible with the pushforward by proper morphisms, and the classes can be calculated by using resolutions of singularities. Formulas for Hirzebruch-Milnor classes of projective hypersurfaces in terms of these classes are given in the case where the multiplicity of a generic hy…
In this article, we give the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with klt singularities. Building on recent works of Spicer, Cascini - Spicer and Spicer - Svaldi, we then describe the birational geometry of rank two foliations with canonical singularities and canonical class of numerical dimension zero on complex projective threefolds.
A codimension one singular holomorphic foliation is Newton non-degenerate if it satisfies the classical conditions of Kouchnirenko and Oka, in terms of its Newton polyhedra system. In this paper we prove that a foliation is Newton non-degenerate if and only if it admits a logarithmic reduction of singularities of a combinatorial nature.
The reduction of singularities of codimension one foliations is known in the case of ambient dimension 2 (Seidenberg, A. (1968). Reduction of singularities of the differential equation Ady= Bdx. American Journal of Mathematics, 90(1), 248-269) and 3 (Cano, F. (2004). Reduction of the singularities of codimension one singular foliations in dimension three. Annals of mathematics, 160(3), 907-1011). However, in greater ambient dimension it is still an open problem. In this work we prove the existence of local uniformization for codimension one foliations in arbitrary ambient dimension, in the cas…
In this article, we describe the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with terminal singularities, extending a result of Loray, Pereira and Touzet to this context.
Formulae for indices of holomorphic foliations via reduction of singularities
Instituto de Matemática e Estatística, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学数学与统计研究所）；Pontificia Universidad Católica del Perú（秘鲁天主教 Pontificia 大学）；Universitat Autònoma de Barcelona（巴塞罗那自治大学）
In the main result of this paper we prove that a codimension one foliation of $\mathbb{P}^n$, which is locally a product near every point of some codimension two component of the singular set, has a Kupka component. In particular, we obtain a generalization of a known result of Calvo Andrade and Brunella about foliations with a Kupka component.
In this paper, we prove the boundedness of foliated surfaces $(X,\mathscr{F})$ which are minimal partial du Val resolutions of canonical models $(X_c,\mathscr{F}_c)$ of general type. For applications, we show the boundedness of non-cusp singularities on canonical models of foliated surfaces of general type and the effective generation on the complement of the cusp singularities.
We provide a tool how one can view a polynomial on the affine plane of bidegree $(a,b)$ - by which we mean that its Newton polygon lies in the triangle spanned by $(a,0)$, $(0,b)$ and the origin - as a curve in a Hirzebruch surface having nice geometric properties. As an application, we study maximal $A_k$-singularities of curves of bidegree $(3,b)$ and find the answer for $b\leq 12$.
This article is dedicated to the study of singular codimension $1$ foliations $\mathcal{F}$ on a simplicial complete toric variety $X$ and their pullbacks by dominant rational maps $φ:\mathbb{P}^n\dashrightarrow X$. First, we describe the singularities of $\mathcal{F}$ and $φ^*\mathcal{F}$ for a generic pair $(φ,\mathcal{F})$. Then we show that the first order deformations of $φ^*\mathcal{F}$ arising from first order unfoldings are the families of the form $φ_\varepsilon^*\mathcal{F}$, where $φ_\varepsilon$ is a perturbation of $φ$. We also prove that the deformations of the form $φ^*\mathcal{…
In this paper we address Fano foliations on complex projective varieties. These are foliations whose anti-canonical class is ample. We focus our attention on a special class of Fano foliations, namely del Pezzo foliations on complex projective manifolds. We show that these foliations are algebraically integrable, with one exceptional case when the ambient space is a projective space. We also provide a classification of del Pezzo foliations with mild singularities.
We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria assuring that a rational map between projective manifolds of the same dimension defines a Galois covering. Then, these criteria are used to give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities. We also characterize Galois foliations on $\mathbb P^2$ admitting continuous symmetries, obtaining a com…
Foliations with small singular set in arbitrary characteristic
Universidade Federal Fluminense（弗鲁米嫩塞联邦大学）；Institut Montpelliérain Alexander Grothendieck, Université de Montpellier（蒙彼利埃亚历山大·格罗滕迪克研究所，蒙彼利埃大学）；Univ Rennes, CNRS, IRMAR, UMR 6625（雷恩大学，法国国家科学研究中心，IRMAR）
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 multiplicity at least (m^2-1)/(2m+1). Our second main result is the construction of an invariant map from the space of foliations of degree m to that of curves of degree m^2+m-2. We describe this map explicitly in case m=2.
When $p$ is inert in the quadratic imaginary field $E$ and $m<n$, unitary Shimura varieties of signature $(n,m)$ and a hyperspecial level subgroup at $p$, carry a natural foliation of height 1 and rank $m^2$ in the tangent bundle of their special fiber $S$. We study this foliation and show that it acquires singularities at deep Ekedahl-Oort strata, but these singularities are resolved if we pass to a natural smooth moduli problem $S^\sharp$, a successive blow-up of $S$. Over the ($μ$-)ordinary locus we relate the foliation to Moonen's generalized Serre-Tate coordinates. We study the quotient o…
We look at natural foliations on the Painlevé VI moduli space of regular connections of rank 2 on $\pp ^1 -{t_1,t_2,t_3,t_4}$. These foliations are fibrations, and are interpreted in terms of the nonabelian Hodge filtration, giving a proof of the nonabelian Hodge foliation conjecture in this case. Two basic kinds of fibrations arise: from apparent singularities, and from quasiparabolic bundles. We show that these are transverse. Okamoto's additional symmetry, which may be seen as Katz's middle convolution, exchanges the quasiparabolic and apparent-singularity foliations.
We show that if $(X_t,\mathcal{F}_t)_t$ is a family of foliations with reduced singularities on a smooth family of surfaces, then invariance of plurigenera holds for sufficiently large $m$. On the other hand, we provide examples on which the result fails, for small values of $m$.
This article studies codimension one foliations on projective man-ifolds having a compact leaf (free of singularities). It explores the interplay between Ueda theory (order of flatness of the normal bundle) and the holo-nomy representation (dynamics of the foliation in the transverse direction). We address in particular the following problems: existence of foliation having as a leaf a given hypersurface with topologically torsion normal bundle, global structure of foliations having a compact leaf whose holonomy is abelian (resp. solvable), and factorization results.
A foliation is of toric type when it has a combinatorial reduction of singularities. We show that every toric type foliation on (C3, 0), without saddle-nodes, has invariant surface. We extend the argument of Cano-Cerveau, done for the nondicritical case, to the compact dicritical components of the exceptional divisor. These components are projective toric surfaces and the isolated invariant branches of the induced foliation extend to global curves. We build the invariant surface as a germ along the singular locus and those global invariant curves. The result of Ortiz-Rosales-Voronin, about the…
