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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.
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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.
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.
This paper considers a tiny modification of Justin Campbells construction of the Kontsevich compactification in cite{[Camp]}. We construct a resolution of singularities of Drinfeld compactification with an Iwahori structure and use it to prove the universally local acyclity of !-extension D-module on the Drinfeld compactification with an Iwahori structure.
A $k$-differential on a Riemann surface is a section of the $k$-th power of the canonical line bundle. Loci of $k$-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification of the moduli space of $k$-differentials. In this paper we give a complete description for the compactification of the strata of $k$-differentials in terms of pointed stable $k$-differentials, for all $k$. The upshot is a global $k$-residue condition that can also be reformulated in terms of admissible covers of stable curves. Moreover, we study properties of $k$-differentials regarding their deformations, residues, and flat geometric structure.
Let $p$ be a prime. Let $ninmathbb N-{0}$. Let $mathcal C$ be an $F^n$-crystal over a locally noetherian $mathbb F_p$-scheme $S$. Let $(a,b)inmathbb N^2$. We show that the reduced locally closed subscheme of $S$ whose points are exactly those $xin S$ such that $(a,b)$ is a break point of the Newton polygon of the fiber $mathcal C_x$ of $mathcal C$ at $x$ is pure in $S$, i.e., it is an affine $S$-scheme. This result refines and reobtains previous results of de Jong--Oort, Vasiu, and Yang. As an application, we show that for all $min mathbb N$ the reduced locally closed subscheme of $S$ whose points are exactly those $xin S$ for which the $p$-rank of $mathcal C_x$ is $m$ is pure in $S$; the case $n=1$ was previously obtained by Deligne (unpublished) and the general case $nge 1$ refines and reobtains a result of Zink.
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