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Register a new userHodge filtration on local cohomology, Du Bois complex, and local cohomological dimension
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We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions, and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing, and reflexive differentials associated to Z.
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In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,m)$ be a local ring, we prove that if $R_{red}$ is Du Bois, then $H_m^i(R)to H_m^i(R_{red})$ is surjective for every $i$. We find many applications of this result. For example we answer a question of Kovacs and the second author on the Cohen-Macaulay property of Du Bois singularities. We obtain results on the injectivity of $Ext$ that provide substantial partial answers of questions of Eisenbud-Mustata-Stillman in characteristic $0$, and these results can be viewed as generalizations of the Kodaira vanishing theorem for Cohen-Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen-Macaulayness of the defining ideal of Du Bois singularities, which are characteristic $0$ analog of results of Singh-Walther and answer some of their questions. We extend results of Hochster-Roberts on the relation between Koszul cohomology and local cohomology for $F$-injective and Du Bois singularities, see Hochster-Roberts. We also prove that singularities of dense $F$-injective type deform.
We consider a series of four subexceptional representations coming from the third line of the Freudenthal-Tits magic square; using Bourbaki notation, these are fundamental representations $(G,X)$ corresponding to $(C_3, omega_3),, (A_5, omega_3), , (D_6, omega_5)$ and $(E_7, omega_6)$. In each of these four cases, the group $G=Gtimes mathbb{C}^*$ acts on $X$ with five orbits, and many invariants display a uniform behavior, e.g. dimension of orbits, their defining ideals and the character of their coordinate rings as $G$-modules. In this paper, we determine some more subtle invariants and analyze their uniformity within the series. We describe the category of $G$-equivariant coherent $mathcal{D}_X$-modules as the category of representations of a quiver with relations. We construct explicitly the simple $G$-equivariant $mathcal{D}_X$-modules and compute the characters of their underlying $G$-structures. We determine the local cohomology groups with supports given by orbit closures, determining their precise $mathcal{D}_X$-module structure. As a consequence, we calculate the intersection cohomology groups and Lyubeznik numbers of the orbit closures. While our results for the cases $(A_5, omega_3), , (D_6, omega_5)$ and $(E_7, omega_6)$ are still completely uniform, the case $(C_3, omega_3)$ displays a surprisingly different behavior. We give two explanations for this phenomenon: one topological, as the middle orbit of $(C_3, omega_3)$ is not simply-connected; one geometric, as the closure of the orbit is not Gorenstein.
If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow that the morphism is an isomorphism? This problem is called the local isomorphism problem. In this paper, we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem. This approach leads to a partial solution of the local isomorphism problem, which is shown to have a negative answer in general and a positive one in several situations of geometric interest.
The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $xin (X,Delta)$ satisfies the ACC if the coefficients of $Delta$ belong to a DCC set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of $delta$-plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities.
v2: We improved a little bit according to the referees wishes. v1: On $X$ projective smooth over a field $k$, Pink and Roessler conjecture that the dimension of the Hodge cohomology of an invertible $n$-torsion sheaf $L$ is the same as the one of its $a$-th power $L^a$ if $a$ is prime to $n$, under the assumptions that $X$ lifts to $W_2(k)$ and $dim Xle p$, if $k$ has characteristic $p>0$. They show this if $k$ has characteristic 0 and if $n$ is prime to $p$ in characteristic $p>0$. We show the conjecture in characteristic $p>0$ if $n=p$ assuming in addition that $X$ is ordinary (in the sense of Bloch-Kato).
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