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Register a new userFoliations with isolated singularities on Hirzebruch surfaces
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We study foliations $mathcal{F}$ on Hirzebruch surfaces $S_delta$ 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 $ delta=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 $delta eq 1$, 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 $s=Phi(s)$, for some global endomorphism $Phi $ of the tangent bundle of $S_delta$.
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Let $M$ be a smooth manifold and let $F$ be a codimension one, $C^infty$ foliation on $M$, with isolated singularities of Morse type. The study and classification of pairs $(M,F)$ is a challenging (and difficult) problem. In this setting, a classical result due to Reeb cite{Reeb} states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper cite{Ku-Ee} classify manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices). In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, $Sing(F)$ of the foliation $F$, we consider {em{weakly stable}} components, that we define as those components admitting a neighborhood where all leaves are compact. If $Sing(F)$ admits only weakly stable components, given by smoothly embedded curves diffeomorphic to $S^1$, we are able to extend Haefligers theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result.
By the fundamental work of Griffiths one knows that, under suitable assumption, homological and algebraic equivalence do not coincide for a general hypersurface section of a smooth projective variety $Y$. In the present paper we prove the same result in case $Y$ has isolated singularities.
The decomposition of a two dimensional complex germ with non-isolated singularity into semi-algebraic sets is given. This decomposition consists of four classes: Riemannian cones defined over a Seifert fibered manifold, a topological cone over thickened tori endowed with Cheeger-Nagase metric, a topological cone over mapping torus endowed with Hsiang-Pati metric and a topological cone over the tubular neighbourhoods of the links singularities. In this decomposition there exist semi-algebraic sets that are metrically conical over the manifolds constituting the link. The germ is reconstituted up to bi-Lipschitz equivalence to a model describing its geometric behavior.
Let $Y$ be a complex projective variety of dimension $n$ with isolated singularities, $pi:Xto Y$ a resolution of singularities, $G:=pi^{-1}{rm{Sing}}(Y)$ the exceptional locus. From Decomposition Theorem one knows that the map $H^{k-1}(G)to H^k(Y,Ybackslash {rm{Sing}}(Y))$ vanishes for $k>n$. Assuming this vanishing, we give a short proof of Decomposition Theorem for $pi$. A consequence is a short proof of the Decomposition Theorem for $pi$ in all cases where one can prove the vanishing directly. This happens when either $Y$ is a normal surface, or when $pi$ is the blowing-up of $Y$ along ${rm{Sing}}(Y)$ with smooth and connected fibres, or when $pi$ admits a natural Gysin morphism. We prove that this last condition is equivalent to say that the map $H^{k-1}(G)to H^k(Y,Ybackslash {rm{Sing}}(Y))$ vanishes for any $k$, and that the pull-back $pi^*_k:H^k(Y)to H^k(X)$ is injective. This provides a relationship between Decomposition Theorem and Bivariant Theory.
Let $E$ be the essential part of the exceptional locus of a good resolution of an isolated, log canonical singularity of index one. We describe the dimension of the dual complex of $E$ in terms of the Hodge type of $H^{n-1}(E, O_E)$, which is one of the main results of the paper [1] of Fujino. Our proof uses only an elementary classical method, while Fujinos argument depends on the recent development in minimal model theory.
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