Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA resolution of singularities of Drinfeld compactification with an Iwahori structure
80
0
0.0
(
0
)
Authors
Ruotao Yang
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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.
We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl-Oort stratification on the former, the Kottwitz-Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case $g$ is even.
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.
Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 - the symplectically primitive but complex imprimitive groups - and 10 exceptional groups up to dimension 10, for which it is still open. In this paper, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. We furthermore prove non-existence of a symplectic resolution for one exceptional group, leaving 39+9=48 open cases in total. We do not expect any of the remaining cases to admit a symplectic resolution.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
