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Register a new userOn the topology of a resolution of isolated singularities, II
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Let $Y$ be a complex projective variety of dimension $n$ with isolated singularities, $pi:Xto Y$ a resolution of singularities, $G:=pi^{-1}left(rm{Sing}(Y)right)$ the exceptional locus. From the Decomposition Theorem one knows that the map $H^{k-1}(G)to H^k(Y,Ybackslash {rm{Sing}}(Y))$ vanishes for $k>n$. It is also known that, conversely, assuming this vanishing one can prove the Decomposition Theorem for $pi$ in few pages. The purpose of the present paper is to exhibit a direct proof of the vanishing. As a consequence, it follows a complete and short proof of the Decomposition Theorem for $pi$, involving only ordinary cohomology.
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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.
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 $X$ be a complex, irreducible, quasi-projective variety, and $pi:widetilde Xto X$ a resolution of singularities of $X$. Assume that the singular locus ${text{Sing}}(X)$ of $X$ is smooth, that the induced map $pi^{-1}({text{Sing}}(X))to {text{Sing}}(X)$ is a smooth fibration admitting a cohomology extension of the fiber, and that $pi^{-1}({text{Sing}}(X))$ has a negative normal bundle in $widetilde X$. We present a very short and explicit proof of the Decomposition Theorem for $pi$, providing a way to compute the intersection cohomology of $X$ by means of the cohomology of $widetilde X$ and of $pi^{-1}({text{Sing}}(X))$. Our result applies to special Schubert varieties with two strata, even if $pi$ is non-small. And to certain hypersurfaces of $mathbb P^5$ with one-dimensional singular locus.
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 $S_delta$.
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.
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