We give formulas for the extension groups between tautological sheaves and more general between tautological objects twisted by a determinant line bundle on the Hilbert scheme of points on a smooth quasi-projective surface. We do this using L. Scala's results about the image of tautological sheaves under the Bridgeland-King-Reid equivalence. We also compute the Yoneda products in terms of these formulas.
Results on stability of tautological sheaves on Hilbert schemes of points are extended to higher dimensions and transferred to abelian surfaces and to the restriction of tautological sheaves to generalised Kummer varieties. This provides a big class of new examples of stable sheaves on higher dimensional irreducible symplectic manifolds. Some aspects of deformations of tautological sheaves are studied.
We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on surfaces within the framework of Nakajima's oscillator algebra. This leads to an identification of the cohomology ring of Hilbert schemes of the affine plane with a ring of differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilb^n up to n=7, and give a conjecture for the generating series.
We prove stability of rank two tautological bundles on the Hilbert square of a surface (under a mild positivity condition) and compute their Chern classes.
We explore the structure of the cohomology ring of the moduli space of stable 1-dimensional sheaves on $\mathbb{P}^2$ of any degree. We obtain a minimal set of tautological generators, which implies an optimal generation result for both the cohomology and the Chow ring of the moduli space. Our approach is through a geometric study of tautological relations.
We propose a conjecture on the generating series of Chern numbers of tautological bundles on Hilbert schemes of points on curves and establish the rank 1 and rank -1 case of this conjecture. Thus we compute explicitly the generating series of integrals of the Segre classes of tautological bundles of line bundles on curves, which has a similar structure as Lehn's conjecture for surfaces.
We develop a formula for tautological integrals over geometric subsets of the Hilbert scheme of points on complex manifolds. As an illustration of the theory, we derive a new iterated residue formula for the number of nodal curves in sufficiently ample linear systems.
We develop a new method to study intersection theory of the main component of the Hilbert scheme of points on complex manifolds. The main result is an iterated residue formula for tautological integrals. We formulate a Chern-Segre-type positivity conjecture for these integrals, and we introduce higher dimensional Segre and Chern integrals.
We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves. In particular, we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface.
We take a new look at the curvilinear Hilbert scheme of points on a smooth projective variety $X$ as a projective completion of the non-reductive quotient of holomorphic map germs from the complex line into $X$ by polynomial reparametrisations. Using an algebraic model of this quotient coming from global singularity theory we develop an iterated residue formula for tautological integrals over curvilinear Hilbert schemes.
We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each K-trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the K3 case, we extend recent constructions and results of Bini, Boissière and Flamini from the Hilbert scheme of 2 and 3 points to an arbitrary number of points. Among the K-trivial surfaces, the case of Enriques surfaces is the most involved. Our techniques apply to other smooth projective surfaces, including blowups of K3s and minimal surfaces of general type,…
The purpose of this paper is to explore the geometry and establish the slope stability of tautological vector bundles on Hilbert schemes of points on smooth surfaces. By establishing stability in general we complete a series of results of Schlickewei and Wandel who proved the slope stability of these vector bundles for Hilbert schemes of 2 points or 3 points on K3 or abelian surfaces with Picard group restrictions. In exploring the geometry we show that every sufficiently positive semistable vector bundle on a smooth curve arises as the restriction of a tautological vector bundle on the Hilber…
Let (X,H) be a polarized smooth projective surface satisfying H^1(X,O_X)=0 and let F be either a rank one torsion-free sheaf or a rank two μH-stable vector bundle on X. Assume that c_1(F)/=0. In this article it is shown that the rank two, respectively rank four tautological sheaf F^{[2]} associated with F on the Hilbert square X^{[2]} is μ-stable with respect to a certain polarization.
We study intersection theory on the relative Hilbert scheme of a family of nodal-or-smooth curves, over a base of arbitrary dimension. We introduce an additive group called 'discriminant module', generated by diagonal loci, node scrolls, and twists thereof, and determine the action of the discriminant or big diagonal divisor on this group by intersection. We show that this suffices to determine arbitrary polynomials in Chern classes, in particular Chern numbers, for the tautological vector bundles on the Hilbert schemes, which are closely related to enumerative geometry. The latest version inc…
We first give an alternative proof, based on a simple geometric argument, of a result of Marian, Oprea and Pandharipande on top Segre classes of the tautological bundles on Hilbert schemes of $K3$ surfaces equipped with a line bundle. We then turn to the blow-up of $K3$ surface at one point and establish vanishing results for the corresponding top Segre classes in a certain range. This determines, at least theoretically, all top Segre classes of tautological bundles for any pair $(Σ,H),\,H\in {\rm Pic}\,Σ$.
We compute the tautological ring for a Hilbert modular variety at an unramified prime. The method generalises that of van der Geer from the Siegel case.
We give a description of the image of tensor products of tautological bundles on Hilbert schemes of points on surfaces under the Bridgeland-King-Reid-Haiman equivalence. Using this, some new formulas for cohomological invariants of these bundles are obtained. In particular, we give formulas for the Euler characteristic of arbitrary tensor products on the Hilbert scheme of two points and of triple tensor products in general.
We show that tautological integrals on Hilbert schemes of points can be written in terms of universal polynomials in Chern numbers. The results hold in all dimensions, though they strengthen known results even for surfaces by allowing integrals over arbitrary "geometric" subsets (and their Chern-Schwartz-MacPherson classes). We apply this to enumerative questions, proving a generalised Göttsche Conjecture for all singularity types and in all dimensions. So if L is a sufficiently ample line bundle on a smooth variety X, in a general subsystem P^d of |L| of appropriate dimension the number of hy…
Given a vector bundle $E$ on a smooth projective curve or surface $X$ carrying the structure of a $V$-twisted Hitchin pair for some vector bundle $V$, we observe that the associated tautological bundle $E^{[n]}$ on the punctual Hilbert scheme of points $X^{[n]}$ has an induced structure of a $((V^\vee)^{[n]})^\vee$-twisted Hitchin pair, where $(V^\vee)^{[n]}$ is a vector bundle on $X^{[n]}$ constructed using the dual $V^\vee$ of $V$. In particular, a Higgs bundle on $X$ induces a logarithmic Higgs bundle on the Hilbert scheme $X^{[n]}$. We then show that the known results on stability of tauto…
Let S be a smooth projective surfaces and S^[n] the Hilbert scheme of zero-dimensional subschemes of S of length n. We proof that the class of S^[n] in the complex cobordism ring depends only on the class of the surface itself. Moreover, we compute the cohomology and holomorphic Euler characterisitcs of certain tautological sheaves on S^[n] and prove results on the general structure of certain integrals over polynomials in Chern classes of tautological sheaves.
For $X$ a smooth quasi-projective variety and $X^{[n]}$ its associated Hilbert scheme of $n$ points, we study two canonical Fourier--Mukai transforms $D(X)\to D(X^{[n]})$, the one along the structure sheaf and the one along the ideal sheaf of the universal family. For $\dim X\ge 2$, we prove that both functors admit a left-inverse. This means in particular that both functors are faithful and injective on isomorphism classes of objects. Using another method, we also show in the case of an elliptic curve that the Fourier--Mukai transform along the structure sheaf of the universal family is faith…
Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S. When s=1, the top Segre classes of the tautological bundles are given by a recently proven formula conjectured in 1999 by M. Lehn. We calculate here the Segre classes of tautological bundles for all ranks s over all K-trivial surfaces. Furthermore, in rank s=2, the Segre integrals are determined for all surfaces, thus establishing a full analogue of Lehn's formula. We also give conjectural formulas for certain series of Verlinde Euler cha…
Recently, Oh and Thomas constructed algebraic virtual cycles for moduli spaces of sheaves on Calabi-Yau 4-folds. The purpose of this paper is to provide a virtual pullback formula between these Oh-Thomas virtual cycles. We find a natural compatibility condition between 3-term symmetric obstruction theories that induces a virtual pullback formula. There are two types of applications. Firstly, we introduce a Lefschetz principle in Donaldson-Thomas theory, which relates the tautological DT4 invariants of a Calabi-Yau 4-fold with the DT3 invariants of its divisor. As corollaries, we prove the Cao-…
Given a quiver, we consider its cohomological Hall algebra (CoHA) as well as CoHA modules built of cohomology groups of non-commutative Hilbert schemes. We investigate cell decompositions of non-commutative Hilbert schemes and the corresponding (non-canonical) bases of CoHA modules. We construct canonical bases of CoHA modules, which consist of products of Chern classes of tautological vector bundles over non-commutative Hilbert schemes. This result generalizes classical results for Grassmannians and (partial) flag varieties.
We study general symmetric powers $S^k L^{[n]}$ of a tautological bundle $L^{[n]}$ on the Hilbert scheme $X^{[n]}$ of $n$ points over a smooth quasi-projective surface $X$, associated to a line bundle $L$ on $X$. Let $V_L$ be the $\mathfrak{S}_n$-vector bundle on $X^n$ defined as the exterior direct sum $L \boxplus \cdots \boxplus L$. We prove that the Bridgeland-King-Reid transform $\mathbfΦ(S^k L^{[n]})$ of symmetric powers $S^k L^{[n]}$ is quasi isomorphic to the last term of a finite decreasing filtration on the natural vector bundle $S^k V_L$, defined by kernels of operators $D^l_L$, whic…
Derived categories of Quot schemes on smooth curves and tautological bundles
The Abdus Salam International Centre for Theoretical Physics（阿卜杜斯·萨拉姆国际理论物理中心）；Department of Mathematics, Northeastern University（东北大学数学系）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Simion Stoilow Institute of Mathematics (IMAR)（西奥多·斯托伊洛瓦数学研究所）
We find certain relations between flag Hilbert schemes of points on plane curves and moduli spaces of one-dimensional plane sheaves. We show that some of these moduli spaces are unirational.
We study the Simpson moduli space of semi-stable sheaves on the complex projective plane that have dimension 1, multiplicity 6 and Euler characteristic 2. We describe concretely these sheaves as cokernels of morphisms of locally free sheaves and we stratify the moduli space according to the types of sheaves that occur.
We investigate the moduli space of sheaves supported on space curves of degree 4 and having Euler characteristic 1. We give an elementary proof of the fact that this moduli space consists of three irreducible components.
We classify all Gieseker semi-stable sheaves on the complex projective plane that have dimension 1, multiplicity 6 and Euler characteristic 3. We show that their moduli space is birational to the blow-up at a special point of a certain moduli space of semi-stable Kronecker modules.
We compute the Hodge numbers of the moduli space of semi-stable sheaves on the complex projective plane supported on quintic curves and having Euler characteristic 3. For this purpose we study the fixed-point set for a certain torus action on the moduli space.
We prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert schemes of points of a K3 surface X. We derive relations among the Segre classes via equivariant localization of the virtual fundamental classes of Quot schemes on X. The resulting recursions are then solved explicitly. The formula proves the K-trivial case of a conjecture of M. Lehn from 1999. The relations determining the Segre classes fit into a much wider theory. By localizing the virtual classes of certain relative Quot schemes on surfaces, we obtain new systems of relations among…
We show how the "finite Quot scheme method" applied to Le Potier's strange duality on del Pezzo surfaces leads to conjectures (valid for all smooth complex projective surfaces) relating two sets of universal power series on Hilbert schemes of points on surfaces: those for top chern classes of tautological sheaves, and those for Euler characteristics of line bundles. We have verified these predictions computationally for low order. We then give an analysis of these conjectures in small ranks. We also give a combinatorial proof of a formula predicted by our conjectures: the top chern class of th…
We compute some cohomology spaces for the symmetric power of the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme of a complex smooth projective surface.
We compute the space of global sections for the symmetric power of the tautological bundle on the punctual Hilbert scheme of a complex smooth projective surface.
Let $X$ a smooth quasi-projective algebraic surface, $L$ a line bundle on $X$. Let $X^{[n]}$ the Hilbert scheme of $n$ points on $X$ and $L^{[n]}$ the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. We explicitely compute the image $\bkrh(L^{[n]})$ of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence $\bkrh : \B{D}^b(X^{[n]}) \ra \B{D}^b_{\perm_n}(X^n)$ in terms of a complex $\comp{\mc{C}}_L$ of $\perm_n$-equivariant sheaves in $\B{D}^b_{\perm_n}(X^n)$. We give, moreover, a characterization of the image $\bkrh(L^{[n]} \tens ... \ten…
Let $C$ be a hyperelliptic curve embedded in its Jacobian $J$ via an Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme component of $\textrm{Hilb}_J$ containing the Abel-Jacobi curve as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism $\mathcal M_g\rightarrow \mathcal A_g$ along the hyperelliptic locus. As an application, we determine the scheme structure of the moduli space of Picard sheaves (introduced by Mukai) on a hyperelliptic Jacobian.
The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let $X$ be a non-singular irreducible complex surface and let $E$ be a vector bundle of rank $n$ on $X$. We use the $m$-elementary transformation of $E$ at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;c_1,c_2)$ to $Hilb_{\, \ma…
We compute the cohomology spaces for the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme H of subschemes of length n of a smooth projective surface X. We show that for L and A invertible vector bundles on X, and w the canonical bundle of X, if $w^{-1}\otimes L$, $w^{-1}\otimes A$ and A are ample vector bundles, then the higher cohomology spaces on H of the tautological bundle associated to L tensor the determinant bundle associated to A vanish, and the space of global sections is computed in terms of $H^0(A)$ and $H^0(L\otimes A)$. This result is motivated by t…
The convolution powers of a perverse sheaf on an abelian variety define an interesting family of branched local systems whose geometry is still poorly understood. We show that the generating series for their generic rank is a rational function of a very simple shape and that a similar result holds for the symmetric convolution powers. We also give formulae for other Schur functors in terms of characteristic classes on the dual abelian variety, and as an example we discuss the case of Prym-Tjurin varieties.
Hecke operators on symplectic surfaces and $χ$-independence
University of Edinburgh（爱丁堡大学）；Université de Picardie Jules Verne（约尔·凡尔纳皮卡第大学）；Kyoto University（京都大学）；Université Paris-Saclay（巴黎萨克雷大学）；Simion Stoilow Institute of Mathematics（西米翁·斯托伊洛瓦数学研究所）；Université Paris-Cité（巴黎城市大学）
As a continuation of the work of Freiermuth and Trautmann, we study the geometry of the moduli space of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $4m+1$. The moduli space has three irreducible components whose generic elements are, respectively, sheaves supported on rational quartic curves, on elliptic quartic curves, or on planar quartic curves. The main idea of the proof is to relate the moduli space with the Hilbert scheme of curves by wall crossing. We present all stable sheaves contained in the intersections of the three irreducible components. We also classify stable sheav…
We prove de Cataldo-Hausel-Migliorini's P=W conjecture in arbitrary rank for parabolic Higgs bundles labeled by the affine Dynkin diagrams $\tilde{A}_0$, $\tilde{D}_4$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$. Our proof relies on the study of the tautological classes on the Hilbert scheme of points on an elliptic surface with respect to the perverse filtration.
In this paper, we study the higher codimensional cycle structure of the Hilbert scheme of three points in the projective plane. In particular, we compute all Chern (and Segre) classes of all tautological bundles on it and compute the nef (effective) cones of cycles in codimensions 2 and 3 (dimensions 2 and 3).
In this paper, we give three bases for the cohomology groups of the Hilbert scheme of two points on projective space. Then, we use these bases to compute all effective and nef cones of higher codimensional cycles on the Hilbert scheme. Next, we compute the class in one of these bases of the Chern classes of tautological bundles coming from line bundles. Finally, we provide an application of these results to the degrees of secant varieties of complete intersections.
We investigate the geometry of the Simpson moduli space M of stable sheaves on P_3 with Hilbert polynomial H(m)=3m+1 and describe explicitly the two smooth, rational components, their 11-dimensional smooth, transversal intersection and the relation between M and the Hilbert scheme Hilb_H(P_3) of cubic space curves. We also compute the Betti numbers of the Chow groups of the moduli space M.
We define formal vector bundles with marked sections on Hilbert modular schemes and we show how to use them to construct modular sheaves with an integrable meromorphic connection and a filtration which, in degree 0, gives to us a $p$-adic interpolation of the usual Hodge filtration. We relate it with the sheaf of overconvergent Hilbert modular forms.
