Coisotropic Subvarieties Research Articles

Constant cycle and co-isotropic subvarieties in a Mukai system

Combining theorems of Voisin and Marian, Shen, Yin and Zhao, we compute the dimensions of the orbits under rational equivalence in the Mukai system of rank two and genus two. We produce several examples of algebraically coisotropic and constant cycle subvarieties.

The Voisin map via families of extensions

Let Y be a cubic fourfold not containing any plane, F(Y) be the variety of lines in Y, Z(Y) be the Lehn-Lehn-Sorger-van Straten hyperkahler eightfold constructed in Lehn et al. (J fur die Reine und Angewandte Math (Crelles J) 2017(731):87–128, 2017). In Voisin (Remarks and questions on Coisotropic subvarieties and 0-cycles of hyper-Kahler varieties. Springer International Publishing, Berlin, 2016), Voisin defined a degree six rational map $$v: F(Y)\times F(Y) \dashrightarrow Z(Y),$$ relating the two hyperkahler varieties F(Y) and Z(Y). In this note, we reinterpret this map v using moduli spaces of Bridgeland stable objects in a triangulated category associated with Y, called a Kuznetsov component of Y. We prove that the Voisin map v can be resolved by blowing up the incident locus of intersecting lines in $$F(Y)\times F(Y)$$ endowed with the reduced scheme structure. As a consequence of our approach, we also show that the above-mentioned blowup is a relative Quot scheme over Z(Y) parameterizing quotients in a heart of the Kuznetsov component of Y.

Isotropic and coisotropic subvarieties of Grassmannians

We generalize the notion of coisotropic hypersurfaces to subvarieties of Grassmannians having arbitrary codimension. To every projective variety X, Gel'fand, Kapranov and Zelevinsky associate a series of coisotropic hypersurfaces in different Grassmannians. These include the Chow form and the Hurwitz form of X. Gel'fand, Kapranov and Zelevinsky characterized coisotropic hypersurfaces by a rank one condition on conormal spaces, which we use as the starting point for our generalization. We also study the dual notion of isotropic varieties by imposing rank one conditions on tangent spaces instead of conormal spaces.

Derived categories of surfaces, O’Grady’s filtration, and zero-cycles on holomorphic symplectic varieties

Moduli spaces of stable objects in the derived category of a $K3$ surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on holomorphic symplectic varieties which arise as moduli spaces. First, we show that the second Chern class of any object in the derived category lies in a suitable piece of O’Grady’s filtration on the $\text{CH}_{0}$-group of the $K3$ surface. This solves a conjecture of O’Grady and improves on previous results of Huybrechts, O’Grady, and Voisin. Second, we propose a candidate for the Beauville–Voisin filtration on the $\text{CH}_{0}$-group of the moduli space of stable objects. We discuss its connection with Voisin’s recent proposal via constant cycle subvarieties, and prove a conjecture of hers on the existence of special algebraically coisotropic subvarieties for the moduli space.

The generalized Franchetta conjecture for some hyper-Kähler varieties

The generalized Franchetta conjecture for hyper-Kähler varieties predicts that an algebraic cycle on the universal family of certain polarized hyper-Kähler varieties is fiberwise rationally equivalent to zero if and only if it vanishes in cohomology fiberwise. We establish Franchetta-type results for certain low (Hilbert) powers of low degree K3 surfaces, for the Beauville–Donagi family of Fano varieties of lines on cubic fourfolds and its relative square, and for 0-cycles and codimension-2 cycles for the Lehn–Lehn–Sorger–van Straten family of hyper-Kähler eightfolds. We also draw many consequences in the direction of the Beauville–Voisin conjecture as well as Voisin's refinement involving coisotropic subvarieties. In the appendix, we establish a new relation among tautological cycles on the square of the Fano variety of lines of a smooth cubic fourfold and provide some applications.

CATEGORIES, ONE-CYCLES ON CUBIC FOURFOLDS, AND THE BEAUVILLE–VOISIN FILTRATION

We explore the connection between$K3$categories and 0-cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov’s noncommutative$K3$category associated to a nonsingular cubic 4-fold.By introducing a filtration on the$\text{CH}_{1}$-group of a cubic 4-fold$Y$, we conjecture a sheaf/cycle correspondence for the associated$K3$category ${\mathcal{A}}_{Y}$. This is a noncommutative analog of O’Grady’s conjecture concerning derived categories of$K3$surfaces. We study instances of our conjecture involving rational curves in cubic 4-folds, and verify the conjecture for sheaves supported on low degree rational curves.Our method provides systematic constructions of (a) the Beauville–Voisin filtration on the$\text{CH}_{0}$-group and (b) algebraically coisotropic subvarieties of a holomorphic symplectic variety which is a moduli space of stable objects in${\mathcal{A}}_{Y}$.

Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds

Rational curves on Hilbert schemes of points on K 3 K3 surfaces and generalised Kummer manifolds are constructed by using Brill–Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up to isometry, as dual divisors to such rational curves. The locus covered by the rational curves is then described, thus exhibiting algebraically coisotropic subvarieties. This provides strong evidence for a conjecture by Voisin concerning the Chow ring of irreducible holomorphic symplectic manifolds. Some general results concerning the birational geometry of irreducible holomorphic symplectic manifolds are also proved, such as a non-projective contractibility criterion for wall divisors.

On the Chow group of zero-cycles of a generalized Kummer variety

For a generalized Kummer variety X of dimension 2n, we will construct for each 0≤i≤n some co-isotropic subvarieties in X foliated by i-dimensional constant cycle subvarieties. These subvarieties serve to prove that the rational orbit filtration introduced by Voisin on the Chow group of zero-cycles of a generalized Kummer variety coincides with the induced Beauville decomposition from the Chow ring of abelian varieties. As a consequence, the rational orbit filtration is opposite to the conjectural Bloch–Beilinson filtration for generalized Kummer varieties.

CALIBRATIONS IN HYPER-KÄHLER GEOMETRY

We describe a family of calibrations arising naturally on a hyper-Kähler manifold M. These calibrations calibrate the holomorphic Lagrangian, holomorphic isotropic and holomorphic coisotropic subvarieties. When M is an HKT (hyper-Kähler with torsion) manifold with holonomy SL (n, ℍ), we construct another family of calibrations Φi, which calibrates holomorphic Lagrangian and holomorphic coisotropic subvarieties. The calibrations Φi are (generally speaking) not parallel with respect to any torsion-free connection on M.

A Deodhar-type stratification on the double flag variety

We describe a stratification on the double flag variety $G/B^+\times G/B^-$ of a complex semisimple algebraic group $G$ analogous to the Deodhar stratification on the flag variety $G/B^+$, which is a refinement of the stratification into orbits both for $B^+\times B^-$ and for the diagonal action of $G$, just as Deodhar's stratification refines the orbits of $B^+$ and $B^-$. We give a coordinate system on each stratum, and show that all strata are coisotropic subvarieties. Also, we discuss possible connections to the positive and cluster geometry of $G/B^+\times G/B^-$, which would generalize results of Fomin and Zelevinsky on double Bruhat cells and Marsh and Rietsch on double Schubert cells.

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)

Gerstenhaber-Batalin-Vilkoviski structures on coisotropic intersections

Let $Y,Z$ be a pair of smooth coisotropic subvarieties in a smooth algebraic Poisson variety $X$. We show that any data of first order deformation of the structure sheaf $\oo_X$ to a sheaf of noncommutative algebras and of the sheaves $\oo_Y$ and $\oo_Z$ to sheaves of right and left modules over the deformed algebra, respectively, gives rise to a Batalin-Vilkoviski algebra structure on the Tor-sheaf ${\scr{T}\!}or^{\oo_X}_\idot(\oo_Y,\oo_Z)$. The induced Gerstenhaber bracket on the Tor-sheaf turns out to be canonically defined; it is independent of the choices of deformations involved. There are similar results for Ext-sheaves as well. Our construction is motivated by, and is closely related to, a result of Behrend-Fantechi \cite{BF}, who studied intersections of Lagrangian submanifolds in a symplectic manifold.