Related papers: Local vanishing for toric varieties

We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional…

We study the local cohomology modules H^i_B(R) for a reduced monomial ideal B in a polynomial ring R=k[X_1,...,X_n]. We consider a grading on R which is coarser than the Z^n-grading such that each component of H^i_B(R) is finite dimensional…

In this paper, we prove that: For any given finitely many distinct points $P_1,...,P_r$ and a closed subvariety $S$ of codimension $\geq 2$ in a complete toric variety over a uncountable (characteristic 0) algebraically closed field, there…

We revisit generic vanishing results for perverse sheaves with any field coefficients on a complex semi-abelian variety, and indicate several topological applications. In particular, we obtain finiteness properties for the integral…

For an equivariant log pair $(X, D)$ where $X$ is a normal toric variety and $D$ a reduced Weil divisor, we study slope-stability of the logarithmic tangent sheaf $\mathcal{T}_{X}(- \log D)$. We give a complete description of divisors $D$…

Let $\omega$ be a differential $q$-form defining a foliation of codimension $q$ in a projective variety. In this article we study the singular locus of $\omega$ in various settings. We relate a certain type of singularities, which we name…

We study the local invariants that a meromorphic $k$-differential on a Riemann surface of genus $g\geq0$ can have. These local invariants are the orders of zeros and poles, and the $k$-residues at the poles. We show that for a given pattern…

Let $\K$ be a finite field and $X$ be a complete simplicial toric variety over $\K$. We give an algorithm relying on elimination theory for finding generators of the vanishing ideal of a subgroup $Y_Q$ parameterized by a matrix $Q$ which…

Let $(\mathrm{Spec }R, \frak m)$ be a rational double point defined over an algebraically closed field $k$ of characteristic $p\geq 0$. We evaluate further the dimensions of the local cohomology groups which were treated by Wahl in 1975 as…

Kaneyama and Klyachko have shown that any torus equivariant vector bundle of rank $r$ over $\mathbb{CP}^n$ splits if $r < n$. In particular, any such bundle is not slope stable. In contrast, we provide explicit examples of stable…

We say an excellent local domain $(S,n)$ satisfies the vanishing conditions for maps of Tor, if for every $A\to R\to S$ with $A$ regular and $A\to R$ module-finite torsion-free extension, and every $A$-module $M$, the map $Tor^A_i(M, R)\to…

We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo $p^2$ - we expect that such varieties, after a finite \'etale cover, admit a toric fibration over…

We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X. We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of…

Let $\mathcal E$ be a torus-linearised reflexive sheaf over a smooth projective toric variety. Generalising a theorem of Perlman and Smith, we prove an explicit sufficient condition for $\mathcal E$ to be acyclic via Weil decorations.

Fix a variety X with a transitive (left) action by an algebraic group G. Let E and F be coherent sheaves on X. We prove that, for elements g in a dense open subset of G, the sheaf Tor_i^X(E, g F) vanishes for all i > 0. When E and F are…

We introduce the class of weakly log canonical singularities, a natural generalization of semi-log canonical singularities. Toric varieties (associated to toric face rings, possibly non-normal or reducible) which have weakly (semi-) log…

In a seminal work of A. N. Varchenko, the behavior at infinity of oscillatory integrals with real analytic phase is precisely investigated by using the theory of toric varieties based on the geometry of the Newton polyhedron of the phase.…

We extend the dimension and strong linearity results of generic vanishing theory to bundles of holomorphic forms and rank one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin associated to irregular…

Let $X$ be any variety in characteristic zero. Let $V \subset X$ be an open subset that has toroidal singularities. We show the existence of a canonical desingularization of $X$ except for V. It is a morphism $f: Y \to X$ , which does not…

We provide a new proof of the following result: Let $X$ be a variety of finite type over an algebraically closed field $k$ of characteristic 0, let $Z\subset X$ be a proper closed subset. There exists a modification $f:X_1 \rar X$, such…