Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGotzmann regularity for globally generated coherent sheaves
103
0
0.0
(
0
)
Authors
Roger Dellaca
Ask ChatGPT about the research
No Arabic abstract
In this paper, Gotzmanns Regularity Theorem is established for globally generated coherent sheaves on projective space. This is used to extend Gotzmanns explicit construction to the Quot scheme. The Gotzmann representation is applied to bound the second Chern class of a rank 2 globally generated coherent sheaf in terms of the first Chern class.
rate research
Read More
In algebraic geometry, one often encounters the following problem: given a scheme X, find a proper birational morphism from Y to X where the geometry of Y is nicer than that of X. One version of this problem, first studied by Faltings, requires Y to be Cohen-Macaulay; in this case Y is called a Macaulayfication of X. In another variant, one requires Y to satisfy the Serre condition S_r. In this paper, the authors introduce generalized Serre conditions--these are local cohomology conditions which include S_r and the Cohen-Macaulay condition as special cases. To any generalized Serre condition S_rho, there exists an associated perverse t-structure on the derived category of coherent sheaves on a suitable scheme X. Under appropriate hypotheses, the authors characterize those schemes for which a canonical finite S_rho-ification exists in terms of the intermediate extension functor for the associated perversity. Similar results, including a universal property, are obtained for a more general morphism extension problem called S_rho-extension.
We formulate a few conjectures on some hypothetical coherent sheaves on the stacks of arithmetic local Langlands parameters, including their roles played in the local-global compatibility in the Langlands program. We survey some known results as evidences of these conjectures.
Let $X$ be a compact Calabi-Yau 3-fold, and write $mathcal M,bar{mathcal M}$ for the moduli stacks of objects in coh$(X),D^b$coh$(X)$. There are natural line bundles $K_{mathcal M}tomathcal M$, $K_{bar{mathcal M}}tobar{mathcal M}$, analogues of canonical bundles. Orientation data on $mathcal M,bar{mathcal M}$ is an isomorphism class of square root line bundles $K_{mathcal M}^{1/2},K_{bar{mathcal M}}^{1/2}$, satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman arXiv:1006.270 in their theory of motivic Donaldson-Thomas invariants, and is important in categorifying Donaldson-Thomas theory using perverse sheaves. We show that natural orientation data can be constructed for all compact Calabi-Yau 3-folds, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi-Yau 3-folds $X$ with a spin smooth projective compactification $Xhookrightarrow Y$. This proves a long-standing conjecture in Donaldson-Thomas theory. These are special cases of a more general result. Let $X$ be a spin smooth projective 3-fold. Using the spin structure we construct line bundles $K_{mathcal M}tomathcal M$, $K_{bar{mathcal M}}tobar{mathcal M}$. We define spin structures on $mathcal M,bar{mathcal M}$ to be isomorphism classes of square roots $K_{mathcal M}^{1/2},K_{bar{mathcal M}}^{1/2}$. We prove that natural spin structures exist on $mathcal M,bar{mathcal M}$. They are equivalent to orientation data when $X$ is a Calabi-Yau 3-fold with the trivial spin structure. We prove this using our previous paper arXiv:1908.03524, which constructs spin structures (square roots of a certain complex line bundle $K_Ptomathcal B_P$) on differential-geometric moduli stacks $mathcal B_P$ of connections on a principal U$(m)$-bundle $Pto X$ over a compact spin 6-manifold $X$.
We introduce moduli spaces of stable perverse coherent systems on small crepant resolutions of Calabi-Yau 3-folds and consider their Donaldson-Thomas type counting invariants. The stability depends on the choice of a component (= a chamber) in the complement of finitely many lines (= walls) in the plane. We determine all walls and compute generating functions of invariants for all choices of chambers when the Calabi-Yau is the resolved conifold. For suitable choices of chambers, our invariants are specialized to Donaldson-Thomas, Pandharipande-Thomas and Szendroi invariants.
The aim of these notes is to generalize Laumons construction [18] of automorphic sheaves corresponding to local systems on a smooth, projective curve $C$ to the case of local systems with indecomposable unipotent ramification at a finite set of points. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article On the geometric Langlands conjecture by Frenkel, Gaitsgory and Vilonen [10] carry over to our situation. We show that our sheaves descend to the moduli space of parabolic bundles if the rank is $leq 3$ and that the general case can be deduced form a generalization of the vanishing conjecture of [10].
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
