Moduli of Reflexive Sheaves on Smooth Projective 3-folds
Peter Vermeire
备注：
17 pages. Entire paper rewritten - discusses relationship between moduli spaces of reflexive rank 2 sheaves and the Hilbert scheme of curves; brief mention of Donaldson-Thomas invariants
We compute the expected dimension of the moduli space of torsion-free rank 2 sheaves at a point corresponding to a stable reflexive sheaf and give conditions for the existence of a perfect tangent-obstruction complex on a class of smooth projective threefolds; this class includes Fano and Calabi-Yau threefolds. We also explore both local and global relationships between moduli spaces of reflexive rank 2 sheaves and the Hilbert scheme of curves.
In this paper, we define $m$-tail reflexive sheaves as reflexive sheaves on projective spaces with the simplest possible cohomology. We prove that the rank of any $m$-tail reflexive sheaf $\mathcal{E}$ on $\mathcal{P}^n$ is greater or equal to $ nm-m$. We completely describe $m$-tail reflexive sheaves on $\mathcal{P}^n$ of minimal rank and we construct huge families of $m$-tail reflexive sheaves of higher rank.
Consider an ample and globally generated line bundle $L$ on a smooth projective variety $X$ of dimension $N\geq 2$ over $\mathbb{C}$. Let $D$ be a smooth divisor in the complete linear system of $L$. We construct reflexive sheaves on $X$ by an elementary transformation of a trivial bundle on $X$ along certain globally generated torsion-free sheaves on $D$. The dual reflexive sheaves are called the Lazarsfeld-Mukai reflexive sheaves. We prove the $μ_L$-(semi)stability of such reflexive sheaves under certain conditions.
We investigate the reflexive sheaves on $\PP^3$ spanned in codimension 2 with very low first Chern class $c_1$. We also give the sufficient and necessary conditions on numeric data of such sheaves for indecomposabiity. As a by-product we obtain that every reflexive sheaf on $\PP^3$ spanned in codimension 2 with $c_1=2$ is spanned.
It is well known that the Chern classes $c_i$ of a rank $n$ vector bundle on $\PP^N$, generated by global sections, are non-negative if $i\leq n$ and vanish otherwise. This paper deals with the following question: does the above result hold for the wider class of reflexive sheaves? We show that the Chern numbers $c_i$ with $i\geq 4$ can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for $i\leq 3$ we show positivity of the $c_i$ with weaker hypothesis. We obtain lower bounds for $c_1$, $c_2$ and $c_3$ for every reflexive sheaf $\FF$ which is generated by $H^0\FF$ on …
On a smooth projective threefold, we show that there are only two isomorphism types for the moduli of stable objects with respect to Bayer's standard polynomial Bridgeland stability - the moduli of Gieseker-stable sheaves and the moduli of PT-stable objects - under the following assumptions: no two of the stability vectors are collinear, and the degree and rank of the objects are relatively prime. We also describe a close relation between the intersection of the moduli spaces of PT-stable and dual-PT-stable objects, and the moduli of reflexive sheaves.
In this paper we give a complete algebro-geometric characterization of analytic tangent cones of admissible Hermitian-Yang-Mills connections over any reflexive sheaves.
We study the notion of algebraic tangent cones at singularities of reflexive sheaves. These correspond to extensions of reflexive sheaves across a negative divisor. We show the existence of optimal extensions in a constructive manner, and we prove the uniqueness in a suitable sense. The results here are an algebro-geometric counterpart of our previous study on singularities of Hermitian-Yang-Mills connections.
Given an isolated, quasi-homogeneous singularity $X$ we prove that there is a group isomorphism between the group of rank one reflexive sheaves on $X$ and the free abelian group generated by $\mathbb{C}^*$-divisors, modulo linear equivalence. When $\dim(X)=2$ we reduce the problem of finding matrix factorizations of arbitrary reflexive $\mathcal{O}_X$-modules to the same question on rank one reflexive sheaves. We then enumerate the matrix factorizations of all rank one reflexive sheaves. As a consequence, we prove a conjecture of Etingof and Ginzburg on point modules.
The purpose of this paper is to study the structure of reflexive sheaves over projective spaces through hyperplane sections. We give a criterion for a reflexive sheaf to split into a direct sum of line bundles. An application to the theory of free hyperplane arrangements is also given.
In this note, we discuss the concept of pseudoeffective vector bundle and also introduce pseudoeffective torsion-free sheaves over compact Kähler manifolds. We show that a pseudoeffective reflexive sheaf over a compact Kähler manifold with vanishing first Chern class is in fact a numerically flat vector bundle. A proof is obtained through a natural construction of positive currents representing the Segre classes of pseudoeffective vector bundles.
We show that codimension one distributions with at most isolated singularities on certain smooth projective threefolds with Picard rank one have stable tangent sheaves. The ideas in the proof of this fact are then applied to the characterization of certain irreducible components of the moduli space of stable rank 2 reflexive sheaves on $\mathbb{P}^3$, and to the construction of stable rank 2 reflexive sheaves with prescribed Chern classes on general threefolds. We also prove that if $\mathscr{G}$ is a subfoliation of a codimension one distribution $\mathscr{F}$ with isolated singularities, the…
We study moduli spaces $\mathcal{N}$ of rank 2 stable reflexive sheaves on $\mathbb{P}^3$. Fixing Chern classes $c_1$, $c_2$, and summing over $c_3$, we consider the generating function $\mathsf{Z}^{\mathrm{refl}}(q)$ of Euler characteristics of such moduli spaces. The action of the torus $T$ on $\mathbb{P}^3$ lifts to $\mathcal{N}$ and we classify all sheaves in $\mathcal{N}^T$. This leads to an explicit expression for $\mathsf{Z}^{\mathrm{refl}}(q)$. Since $c_3$ is bounded below and above, $\mathsf{Z}^{\mathrm{refl}}(q)$ is a polynomial. We find a simple formula for its leading term when $c_…
For (X,L) a polarized toric variety and G a torus of automorphisms of (X,L), denote by Y the GIT quotient X/G. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on Y to the category of torus equivariant reflexive sheaves on X. We show, under a genericity assumption on G, that slope stability is preserved by these functors if and only if the pair ((X, L), G) satisfies a combinatorial criterion. As an application, when (X,L) is a polarized toric orbifold of dimension n, we relate stable equivariant reflexive sheaves on (X, L) to stable equivar…
We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface $V$ with isolated singularities and the Torelli properties of $V$ (in the sense of Dolgachev-Kapranov). We show in particular that hypersurfaces with a small Tjurina numbers are Torelli in this sense. When $V$ is a plane curve, or more interestingly, a surface in $P^3$, we discuss the stability of the reflexive sheaf of logarithmic vector fields along $V$. A new lower bound for the minimal degree of a syzygy associated to a 1-dimensional complete intersection is also g…
Let \sF be a coherent rank 2 sheaf on a scheme Y \subset \proj{n} of dimension at least two. In this paper we study the relationship between the functor which deforms a pair (\sF,σ), σ\in H^0(\sF), and the functor which deforms the corresponding pair (X,ξ) given as in the Serre correspondence. We prove that the scheme structure of e.g. the moduli scheme M_Y(P) of stable sheaves on a threefold Y at (\sF), and the scheme structure at (X) of the Hilbert scheme of curves on Y are closely related. Using this relationship we get criteria for the dimension and smoothness of M_Y(P) at (\sF), without a…
We construct a moduli space of slope-semistable pure sheaves, building upon previous work of Le Potier and Jun Li on torsion-free sheaves over smooth surfaces. In particular, our construction provides a compactification of the Simpson moduli space of slope-stable reflexive sheaves. We also prove an effective restriction theorem for slope-(semi)stable pure sheaves following an approach due to Langer.
We define reflexive sheaves on a singular quadric Q that generalize the spinor bundles on smooth quadrics, using matrix factorizations of the equation of Q. We study the first properties of these spinor sheaves, give a Horrocks-type criterion, and show that they are semi-stable, and indeed stable in some cases.
In this paper, we study almost nef regular foliations. We give a structure theorem of a smooth projective variety $X$ with an almost nef regular foliation $\mathcal{F}$: $X$ admits a smooth morphism $f: X \rightarrow Y$ with rationally connected fibers such that $\mathcal{F}$ is a pullback of a numerically flat regular foliation on $Y$. Moreover, $f$ is characterized as a relative MRC fibration of an algebraic part of $\mathcal{F}$. As a corollary, an almost nef tangent bundle of a rationally connected variety is generically ample. For the proof, we generalize Fujita's decomposition theorem. A…
Based on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive differentials. First, we generalise the extension theorem to holomorphic forms on locally algebraic complex spaces. We investigate the (non-)existence of reflexive pluri-differentials on singular rationally connected varieties, using a semistability analysis with respect to movable curve classes. The necessary foundational material concerning this stability notion is dev…
We give a complete classification of semistable rank two sheaves on three-dimensional projective space with maximal third Chern character. This implies an explicit description of their moduli spaces. As an open subset they contain rank two reflexive sheaves with maximal number of singularities. These spaces are irreducible, and apart from a single special case, they are also smooth. This extends a result by Okonek and Spindler to all missing cases and gives a new proof of their result. The key technical ingredient is variation of stability in the derived category.
Let $F$ be a torsionfree semistable coherent sheaf on a polarized normal projective variety. We prove that $F$ has a unique maximal locally free subsheaf $V$ such that $F/V$ is torsionfree and $V$ admits a filtration of subbundles for which each successive quotient is stable whose slope is $μ(F)$. We also prove that $F$ has a unique maximal reflexive subsheaf $W$ such that $F/W$ is torsionfree and $W$ admits a filtration of subsheaves for which each successive quotient is a stable reflexive sheaf whose slope is $μ(F)$. We show that these canonical subsheaves behave well with respect to the pul…
We describe new irreducible components of the Gieseker-Maruyama moduli scheme $\mathcal{M}(3)$ of semistable rank 2 coherent sheaves with Chern classes $c_1=0,\ c_2=3,\ c_3=0$ on $\mathbb{P}^3$, general points of which correspond to sheaves whose singular loci contain components of dimensions both 0 and 1. These sheaves are produced by elementary transformations of stable reflexive rank 2 sheaves with $c_1=0,\ c_2=2,\ c_3=2$ or 4 along a disjoint union of a projective line and a collection of points in $\mathbb{P}^3$. The constructed families of sheaves provide first examples of irreducible co…
We construct a new infinite series of irreducible components of the Gieseker-Maruyama moduli scheme $\mathcal{M}(k), ~ k \geq 3$ of coherent semistable rank 2 sheaves with Chern classes $c_1=0,~ c_2=k,~ c_3=0$ on $\mathbb{P}^3$ whose general points are sheaves with singularities of mixed dimension. These sheaves are constructed by elementary transformations of stable and properly $μ$-semistable reflexive sheaves along disjoint union of collections of points and smooth irreducible curves which are rational or complete intersection curves. As a special member of this series we obtain a new compo…
We identify Le Potier's moduli spaces of limit stable pairs $(F,s)$, where $F$ is a 2-dimensional sheaf on a nonsingular projective 4-fold $X$ and $s \in H^0(F)$, with the moduli spaces of polynomial stable 2-term complexes in derived category. These stable pairs are 2-dimensional analogs of Pandharipande-Thomas' stable pairs defined for 3-folds. We establish categorical correspondences involving these stable pairs, ideal sheaves of 2-dimensional subschemes of $X$, and 1-dimensional sheaves on $X$. Under some conditions on the Chern character, these lead to Hall algebra correspondences. The ge…
We prove a Bogomolov-Gieseker type inequality for the third Chern characters of stable sheaves on Calabi-Yau 3-folds and a large class of Fano 3-folds with given rank and first and second Chern classes. The proof uses the spreading-out technique, vanishings from the tilt-stability conditions, and Langer's estimation theorem of the global sections of torsion free sheaves. In particular, the result implies that the conjectural sufficient conditions on the Chern numbers for the existence of stable sheaves on a Calabi-Yau 3-fold by Douglas-Reinbacher-Yau needs to be modified.
Reflexive polytopes in n dimensions have attracted much attention both in mathematics and theoretical physics due to their connection to Fano n-folds and mirror symmetry. This work focuses on the 18 regular reflexive polytopes corresponding to smooth Fano 3-folds. For the first time, we show that all 18 regular reflexive polytopes have corresponding 2d (0,2) gauge theories realized by brane brick models. These 2d gauge theories can be considered as the worldvolume theories of D1-branes probing the toric Calabi-Yau 4-singularities whose toric diagrams are given by the associated regular reflexi…
A $h$-instanton sheaf on a closed subscheme $X$ of some projective space endowed with an ample and globally generated line bundle $\mathcal{O}_X(h)$ is a coherent sheaf whose cohomology table has a certain prescribed shape. In this paper we deal with $h$-instanton sheaves relating them to Ulrich sheaves. Moreover, we study $h$-instanton sheaves on smooth curves and surfaces, cyclic $n$-folds, Fano $3$-folds and scrolls over arbitrary smooth curves. We also deal with a family of monads associated to $h$-instanton bundles on varieties satisfying some mild extra technical conditions.
In this paper we study the cohomology of the de Rham complex of sheaves of reflexive differential forms on a normal complex space. First, we prove that the complex is exact in degree one under suitable conditions on the underlying topological space, but that exactness in general depends on the complex structure. Second, we show exactness in high degrees for holomorphically contractible spaces under mild assumptions on the nature of singularities, e.g. klt singularities. Subsequently, the exactness of the de Rham complex of reflexive differential forms is related to the Lipman-Zariski conjectur…
A 10-dimensional symplectic moduli space of torsion sheaves on the cubic 4-fold is constructed. It parametrizes the stable rank 2 vector bundles on the hypeplane sections of the cubic 4-fold which are obtained by Serre's construction from normal elliptic quintics. The natural projection to the dual projective 5-space parametrizing the hyperplane sections is a Lagrangian fibration. The symplectic structure is closely related (and conjecturally, is equal) to the quasi-symplectic one, induced by the Yoneda pairing on the moduli space.
On the product elliptic threefold $X = C \times S$ where $C$ is an elliptic curve and $S$ is a K3 surface of Picard rank 1, we define a notion of limit tilt stability, which satisfies the Harder-Narasimhan property. We show that under the Fourier-Mukai transform $Φ$ on $D^b(X)$ induced by the classical Fourier-Mukai transform on $D^b(C)$, a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is taken to a limit tilt stable object. We also show that a limit tilt semistable object on $X$ is taken by $Φ$ to a slope semistable sheaf, up to mod…
We focus on a class of Weierstrass elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically $K$-trivial surface. We define the notion of limit tilt stability, and show that the Fourier-Mukai transform of a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is a limit tilt stable object. We also show that the inverse Fourier-Mukai transform of a limit tilt semistable object of nonzero fiber degree is a slope semistable torsion-free sheaf, up to modification in codimension 2.
We compute the Donaldson-Thomas invariants for two types of Calabi-Yau 3-folds. These invariants are associated to the moduli spaces of rank-2 Gieseker semistable sheaves. None of the sheaves are locally free, and their double duals are locally free stable sheaves investigated earlier by Donaldson and Thomas, Li and Qin respectively. We show that these Gieseker moduli spaces are isomorphic to some Quot-schemes. We prove a formula for Behrend's functions when torus actions present with positive dimensional fixed point sets, and use it to obtain the generating series of the relevant Donaldson-Th…
It is shown that there are globally defined Lagrangian distributions on the stable loci of derived Quot-stacks of coherent sheaves on Calabi--Yau four-folds. Dividing by these distributions produces perfectly obstructed smooth stacks with globally defined $-1$-shifted potentials, whose derived critical loci give back the stable loci of smooth stacks of sheaves in global Darboux form.
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 $K_{\mathcal M}\to\mathcal M$, $K_{\bar{\mathcal M}}\to\bar{\mathcal M}$, 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 Donaldso…
We give an explicit proof of a Bogomolov-type inequality for $c_3$ of reflexive sheaves on $\mathbb{P}^3$. Then, using resolutions of rank-two reflexive sheaves on $\mathbb{P}^3$, we prove that some strata of the moduli of rank-two complexes that are both PT-stable and dual-PT-stable are quotient stacks. Using monads, we apply the same techniques to $\mathbb{P}^2$ and show that some strata of the moduli of Bridgeland-stable complexes are quotient stacks.
We study the conormal sheaves and singular schemes of 1-dimensional foliations on smooth projective varieties $X$ of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is $μ$-stable whenever the tangent bundle $TX$ is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on $\mathbb{P}^3$ and on a smooth quadric hypersurface $Q_3\subset\mathbb{P}^4$. Finally, we give a classification of local complete intersection foliations, that is, foliations with locally…
We construct a surprisingly large class of new Calabi-Yau 3-folds $X$ with small Picard numbers and propose a construction of their mirrors $X^*$ using smoothings of toric hypersurfaces with conifold singularities. These new examples are related to the previously known ones via conifold transitions. Our results generalize the mirror construction for Calabi-Yau complete intersections in Grassmannians and flag manifolds via toric degenerations. There exist exactly 198849 reflexive 4-polytopes whose 2-faces are only triangles or parallelograms of minimal volume. Every such polytope gives rise to …
We determine the structure of certain moduli spaces of ideal sheaves by generalizing an earlier result of the first author. As applications, we compute the (virtual) Hodge polynomials of these moduli space, and calculate the Donaldson-Thomas invariants of certain 3-folds with trivial canonical classes.
