Perverse Coherent Sheaves Research Articles

Cohomological Hall algebras and perverse coherent sheaves on toric Calabi–Yau $3$-folds

Cohomological Hall algebras and perverse coherent sheaves on toric Calabi–Yau $3$-folds

Perverse Coherent Sheaves on Blow-ups at Codimension 2 Loci

Abstract Let $f \colon X \to Y$ be the blow-up of a smooth projective variety $Y$ along its codimension two smooth closed subvariety. In this paper, we show that the moduli space of stable sheaves on $X$ and $Y$ are connected by a sequence of flip-like diagrams. The result is a higher dimensional generalization of the result of Nakajima and Yoshioka, which is the case of $\dim Y=2$. As an application of our general result, we study the birational geometry of the Hilbert scheme of two points.

Perverse coherent sheaves and Fourier–Mukai transforms on surfaces, II

We study perverse coherent sheaves on the resolution of rational double points. As examples, we consider rational double points on 2 -dimensional moduli spaces of stable sheaves on K3 and elliptic surfaces. Then we show that perverse coherent sheaves appear in the theory of Fourier–Mukai transforms. As an application, we generalize the Fourier–Mukai duality for K3 surfaces to our situation.

Parity sheaves on the affine Grassmannian and the Mirković–Vilonen conjecture

We prove the Mirkovic–Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no p-torsion, as long as p is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when p is a bad prime.) The main idea is to convert this topological question into an algebraic question about perverse-coherent sheaves on the dual nilpotent cone using the Juteau–Mautner–Williamson theory of parity sheaves.

Generalized Serre conditions and perverse coherent sheaves

In algebraic geometry, one often encounters the following problem: given a scheme X, find a proper birational morphism Y→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→X is called a Macaulayfication of X. In another variant, one requires Y to satisfy the Serre condition Sr. In this paper, the authors introduce generalized Serre conditions—these are local cohomology conditions which include Sr and the Cohen–Macaulay condition as special cases. To any generalized Serre condition Sρ, 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 Sρ-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 Sρ-extension.

Operations on -structures and perverse coherent sheaves

We introduce the notions of consistent pairs and consistent chains of -structures and prove that two consistent chains of -structures generate a distributive lattice. The technique developed is then applied to the pairs of chains obtained from the standard -structure on the derived category of coherent sheaves and the dual -structure by means of the shift functor. This yields a family of -structures whose hearts are known as perverse coherent sheaves.

Perverse coherent sheaves and Fourier–Mukai transforms on surfaces, I

Perverse coherent sheaves and Fourier–Mukai transforms on surfaces, I

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

AbstractWe extend most of the 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 with irregular varieties. Our main tools are Saito’s mixed Hodge modules, the Fourier–Mukai transform for $\mathscr{D}$-modules on abelian varieties introduced by Laumon and Rothstein, and Simpson’s harmonic theory for flat bundles. In the process, we also discover two natural categories of perverse coherent sheaves.

The ADHM variety and perverse coherent sheaves

We study the full set of solutions to the ADHM equation as an affine algebraic set, the ADHM variety. We determine a filtration of the ADHM variety into subvarieties according to the dimension of the stabilizing subspace. We compute dimension, and analyze singularity and reducibility of all of these varieties. We also establish a connection between arbitrary solutions of the ADHM equation and coherent perverse sheaves on P 2 in the sense of Kashiwara.

Counting Invariant of Perverse Coherent Sheaves and its Wall-crossing

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.

ESPACES DE MODULES DE SYSTÈMES COHÉRENTS

Etant donne un polynome > 0 a coefficients rationnels, nous definissons la notion de systeme coherent -semi-stable sur une variete algebrique projective lisse. Sur p#2(c), nous montrons l'irreductibilite et la normalite des espaces de modules s# de systemes coherents -semi-stables a = (,f(l)), ou f est un faisceau algebrique coherent de rang 2 sur le plan projectif p#2(c), de classes de chern (0,n), l un entier 1, et h#0(f(l)) un sous-espace vectoriel de rang 1. Quand n'est pas valeur critique, nous definissons ici sur chaque variete s# les fibres determinants, et l'integrale correspondante. La description du saut des espaces de modules de systemes coherents -semi-stables au passage d'une valeur critique, est analogue a celle des flips de thaddeus, avec la difficulte supplementaire due au fait que les espaces de modules ne sont pas forcement lisses. En etudiant le saut de ces integrales au passage d'une valeur critique, nous obtenons des relations entre les nombres de donaldson sur p#2(c) et certains nombres d'intersection sur le schema de hilbert hilb#n#+#l#2 (p#2) (3 n 11)