Fourier-Mukai transforms of slope stable torsion-free sheaves and stable 1-dimensional sheaves on Weierstrass elliptic threefolds
We focus on a class of Weierstraß elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically -trivial surface. In the first half of this article, we define the notion of limit tilt stability, which is closely related to Bayer’s polynomial stability. We 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 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.
In the second half of this article, we define a limit stability for complexes that vanish on the generic fiber of the fibration. We show that one-dimensional stable sheaves with positive twisted correspond to such limit stable complexes under a Fourier-Mukai transform. When the elliptic fibration has a numerically -trivial base, we show that these limit stable complexes are the stable objects with respect to a Bridgeland stability on a triangulated subcategory of the derived category of coherent sheaves on the threefold.
Key words and phrases:Weierstrass threefold, elliptic threefold, Fourier-Mukai transform, stability
2010 Mathematics Subject Classification:Primary 14J30; Secondary: 14J33, 14J60
Given the derived category of coherent sheaves on a smooth projective variety and an autoequivalence of , the natural and fundamental question of how acts on slope stability or Gieseker stability for sheaves on has been studied in a multitude of works. On threefolds, there is Friedman-Morgan-Witten’s spectral cover construction of stable sheaves on elliptic threefolds, which involves applying Fourier-Mukai transforms to stable sheaves supported on hypersurfaces (the ‘spectral covers’), i.e. rank-zero stable sheaves [FMW]. In Bridgeland-Maciocia’s work [BMef], where they established the existence of Fourier-Mukai transforms on Calabi-Yau fibrations of relative dimension at most two, they showed that any connected component of the moduli of rank-one torsion-free sheaves on an elliptic threefold is isomorphic to a moduli of higher-rank Gieseker stable sheaves via a Fourier-Mukai transform. In light of the connection between the Bridgeland stability manifold and mirror symmetry [SCK3, StabTC], it has also become important to understand the action of autoequivalences on Bridgeland stability conditions. More recently, the action of autoequivalences on specific moduli spaces of stable objects (e.g. stable pairs in the sense of Pandharipande-Thomas) has also proved relevant to understanding the structures of Pandharipande-Thomas (PT) invariants and Donaldson-Thomas (DT) invariants on elliptic threefolds, including their modularity [OS1, Dia15]. Since slope stable torsion-free sheaves are basic building blocks of other stable objects such as Bridgeland stable objects and PT stable pairs on the level of cohomology, we study the Fourier-Mukai transforms of slope stable torsion-free sheaves on elliptic fibrations in the first half of this article.
Throughout this article, we focus on a class of elliptic threefolds that are Weierstraß, where is -trivial, along with the Fourier-Mukai transform given by the relative Poincaré sheaf. Our first main result is Theorem LABEL:thm:paper14thm5-1analogue, which says that any slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 is taken by to a limit tilt stable object, while a limit tilt semistable object is taken by the quasi-inverse of to a slope semistable torsion-free sheaf up to modification in codimension 2. In particular, the aforementioned vanishing condition is always satisfied by a torsion-free reflexive sheaf (see Corollary LABEL:cor:thm1reflexiveexample). Since a torsion-free sheaf differs from its double dual - which is reflexive - only in codimension 2, we can also think of the first part of Theorem LABEL:thm:paper14thm5-1analogue as saying, that every slope stable torsion-free sheaf is taken to a limit stable object up to modification in codimension 2.
Limit tilt stability is constructed by varying the polarisation that appears in the definition of tilt stability on the threefold . Strictly speaking, limit tilt stability is ‘asymptotically’ a polynomial stability in the sense of Bayer [BayerPBSC]. Tilt stability on threefolds is a stability for 2-term complexes of coherent sheaves, and is an intermediary between slope stability and Bridgeland stability. A prevalent construction of Bridgeland stability conditions on threefolds hinges on a Bogomolov-Gieseker type inequality for tilt stable objects - this is a very active area of research [BMT1, toda2014note, Macri, Schmidt2, MacPir1, MacPir2, BMS, li2015stability, piyaratne2017stability, bernardara2016bridgeland, koseki2017stability, 2017arXiv170909351P, SMD].
On the other hand, the moduli spaces of stable sheaves with 1-dimensional supports on threefolds (also referred to as 1-dimensional sheaves in the literature), as well as the action of autoequivalences of on these moduli spaces, play an important role in connecting different types of counting invariants on Calabi-Yau threefolds such as PT stable pair invariants and Donaldson-Thomas (DT) invariants for ideal sheaves. They also arise naturally in describing the internal symmetries within these invariants [Dia15, OS1, toda2012stability, Toda2]. Motivated by this, in the second half of this article, we study Fourier-Mukai transforms of 1-dimensional stable sheaves with positive twisted on elliptic threefolds. In Theorem LABEL:thm:1dimEequivalence, we show that the Fourier-Mukai transform induces an equivalence of categories, between the category of 1-dimensional stable sheaves with positive twisted , and the category of stable objects with respect to a stability denoted as -stability. As in the case of limit tilt stability, -stability is ‘asymptotically’ a polynomial stability.
Intuitively, one can think of -stability as a limit of ‘Bridgeland stability conditions for singular surfaces’. In fact, if one is willing to restrict to a triangulated subcategory of , then when the base of the elliptic fibration is numerically -trivial, Corollary LABEL:cor:1dimstableshZwpDstableobj states that the Fourier-Mukai transform induces an equivalence of categories, between the category of 1-dimensional stable sheaves with positive twisted , and the category of stable objects with respect to a Bridgeland stability condition on .
Theorem LABEL:thm:1dimEequivalence and Corollary LABEL:cor:1dimstableshZwpDstableobj give alternatives to Diaconescu’s description of Fourier-Mukai transforms of 1-dimensional stable sheaves in [Dia15, Theorem 1.1]. In Diaconescu’s work, he considers a Weierstraß elliptic threefold where is a Fano surface, and shows that the Fourier-Mukai transform induces an isomorphism between the moduli of 1-dimensional semistable sheaves with positive , and the moduli of adiabatically semistable 2-dimensional sheaves that vanish on the generic fiber of . The moduli of adiabatically semistable sheaves is both open and closed in the moduli of Gieseker semistable sheaves, and constitute the entire moduli of Gieseker semistable sheaves when admits a K3 fibration and the 2-dimensional sheaves are supported on the K3 fibers. It will be interesting to describe the precise relations among the stabilities for 1-dimensional sheaves considered in this article, adiabatic stability in Diaconescu’s work, and Gieseker stability.
n joint work with Zhang [LZ2], the author considered the product threefold where is a smooth elliptic curve and is a K3 surface of Picard rank 1. The second projection is an elliptic fibration where all the fibers are copies of , and hence are smooth, while the Fourier-Mukai transform given by the Poincaré line bundle can be lifted to a Fourier-Mukai transform . In the predecessor [Lo14] to this article, the author proved a coarser version of Theorem LABEL:thm:paper14thm5-1analogue for the product threefold . In the present article, more care is needed in dealing with technical issues such as a section of the fibration not being nef, or that there are nontrivial contributions from the base of the fibration towards cohomological Fourier-Mukai transforms. Therefore, even though the results in the preceding article [Lo14] were less general, because of the simpler notation in the product case, the exposition there provides a more illuminating picture of the key ideas in this article.
n Section 2, we lay out the assumptions we place on our elliptic threefold , and review preliminary concepts that will be used throughout the paper. In Section 3, we explain how a matrix notation for Chern characters makes it easier to understand the cohomological Fourier-Mukai transform ; we also solve a numerical equivalence that is necessary for comparing slope stability with tilt stability up to the autoequivalence . In Section 4, we give the construction of limit tilt stability. Section 5 includes computations of phases of various objects with respect to limit tilt stability. In Section LABEL:sec:slopestabvslimittilstab, we prove the main theorem in the first half of the paper, Theorem LABEL:thm:paper14thm5-1analogue, which is a comparison theorem between slope stability and limit tilt stability via the Fourier-Mukai transform . We end the first half of the paper with Section LABEL:sec:HNpropertylimittiltstab, in which we verify the Harder-Narasimhan property of limit tilt stability.
The second half of the paper begins with Section LABEL:sec:limitstab2dimcplxdefn, in which we restrict our attention to the triangulated subcategory of that consists of complexes where every cohomology sheaf vanishes on the generic fiber of . Via an approach similar to the construction of limit tilt stability, we define -stability on the triangulated category . In Section LABEL:sec:1dimstablimitstab2dimcplxcomparision, we study the Fourier-Mukai transforms of 1-dimensional stable sheaves with positive twisted , and prove the first main result of the second half of the paper, Theorem LABEL:thm:1dimEequivalence. This theorem states that the Fourier-Mukai transform identifies the category of such 1-dimensional sheaves with the category of -stable objects. The Harder-Narasimhan property of -stability is shown in Section LABEL:sec:ZlpDstabHNpropertyproof. Finally, in Section 11, we restrict further to a triangulated subcategory of and construct a Bridgeland stability condition on which we call -stability. We end the article with Corollary LABEL:cor:1dimstableshZwpDstableobj, which states that when the base of the elliptic fibration is numerically -trivial, the Fourier-Mukai transform identifies the category of 1-dimensional stable sheaves having positive twisted with the category of -stable objects.
Acknowledgements. The author thanks Wanmin Liu for sharing his knowledge on elliptic fibrations through many helpful discussions, Ziyu Zhang for generously sharing his insight on cohomological Fourier-Mukai transforms on elliptic fibrations, and Ching-Jui Lai for answering the author’s questions on elliptic fibrations. The author would also like to thank Antony Maciocia for a suggestion raised during the author’s talk at Edinburgh in May 2017, which led to an improvement in the proofs of HN properties in this article. In addition, he thanks Conan Leung, Jun Li and Arend Bayer for conversations that provided the impetus for completing this work. He thanks the Institute for Basic Science in Pohang, South Korea, for their support throughout his visit in June-July 2017, during which a substantial portion of this work was completed. Finally, he would like to thank Emanuel Diaconescu for valuable discussions during a visit to Rutgers in May 2016, which gave inspiration to the second half of this article.
hroughout this article, we will write to denote a Weierstraß threefold in the sense of [FMNT, Section 6.2] where and are both smooth projective varieties. In particular, this means that is an elliptic threefold that is a Weierstraß fibration. By being an elliptic threefold, we mean that is a threefold and is a surface, while is a flat morphism whose fibers are Gorenstein curves of arithmetic genus 1; by also being Weierstraß, we mean that all the fibers of are geometrically integral and there exists a section such that the image does not intersect any singular point of the singular fibers. The smoothness of implies that the generic fiber of is a smooth elliptic curve. The existence of a section ensures that the singular fibers of can only be nodal or cuspidal curves.
We will assume the following on our elliptic fibration from Section 4 through the end of the article:
is -trivial, i.e. .
The cohomology ring over has a decomposition
There exists an ample class on satisfying:
There exists such that, for all , the divisor is ample on .
for some , i.e. the canonical divisor of is numerically equivalent to some real (possibly zero) multiple of .
We do not expect assumption (1) to be essential. Assumption (2) is needed for computing intersection numbers; in principle, however, it can be replaced by any explicit description of the structure of the cohomology ring of . Assumption (3) is the most crucial among the three, as it is needed for comparing stability ‘before’ and stability ‘after’ applying the Fourier-Mukai transform.
We do not impose the vanishing as in the case of [FMNT, Section 6.2.6] or [MV, Section 5]. In particular, we allow the base to be a K3 surface.
Example. Conditions (1) and (2) are assumed for the Calabi-Yau threefolds considered in [FMNT, Section 6.6.3], where only four possibilities of are considered: must be a Fano (i.e. del Pezzo) surface, a Hirzebruch surface, an Enriques surface, or the blowup of a Hirzebruch surface. Condition (3a) is satisfied for all these four possibilities of (see [FMNT, Section 6.6.3]), while condition (3b) is satisfied when is Fano (in which case can be chosen to be and ) or Enriques (in which case and ).
Example. Conditions (1), (2) and (3) are all satisfied for the elliptic threefolds considered by Diaconescu in [Dia15]. In the article [Dia15], it is assumed that (1) holds and that the base of the fibration is Fano, and so condition (3b) holds if we choose and . A decomposition of the form (2) is proved for the torsion-free part of the cohomology ring over in [Dia15, Lemma 2.1]; by considering the cohomology ring over , the torsion elements vanish and so condition (2) holds. By [Dia15, Corollary 2.2(i)], if we choose then condition (3a) holds for .
Example. Conditions (1), (2) and (3) are all satisfied when is the product of a smooth elliptic curve and a K3 surface , and the fibration is the second projection (i.e. is a trivial fibration) as in the case of the author’s previous work [Lo14, LZ2]. In this case, condition (1) clearly holds, condition (2) follows from [LZ2, Section 4.2], condition (3a) follows from [LZ2, Lemma 4.7] by choosing , while condition (3b) holds by taking since .
e collect here some notions and notations that will be used throughout the article.
or any divisor on a smooth projective threefold and any , the twisted Chern character is defined as
We write where
We refer to as the ‘-field’ involved in the twisting of the Chern character. In this article, there should be no risk of confusion as to whether ‘’ refers to a -field or the base of our elliptic threefold as in 2.1, as this will always be clear from the context.
uppose is an abelian category and is the heart of a t-structure on . For any object , we will write to denote the -th cohomology object of with respect to the t-structure with heart . When , i.e. when the aforementioned t-structure is the standard t-structure on , we will write instead of .
Given a smooth projective variety , the dimension of an object will be denoted by , and refers to the dimension of its support, i.e.
That is, for a coherent sheaf , we have .
torsion pair in an abelian category is a pair of full subcategories such that
for all .
Every object fits in an -short exact sequence
for some .
The decomposition of in (ii) is canonical [HRS, Chapter 1], and we will occasionally refer to it as the -decomposition of in .
Whenever we have a torsion pair in an abelian category , we will refer to (resp. ) as the torsion class (resp. torsion-free class) of the torsion pair. The extension closure in
is the heart of a t-structure on , and hence an abelian subcategory of . We call the tilt of at the torsion pair . More specifically, the category is the heart of the t-structure on where
A subcategory of will be called a torsion class (resp. torsion-free class) if it is the torsion class (resp. torsion-free class) in some torsion pair in . By a lemma of Polishchuk [Pol, Lemma 1.1.3], if is a noetherian abelian category, then every subcategory that is closed under extension and quotient in is a torsion class in . The reader may refer to [HRS] for the basic properties of torsion pairs and tilting.
For any subcategory of an abelian category , we will set
when is clear from the context. Note that whenever is noetherian and is closed under extension and quotient in , the pair gives a torsion pair in .
torsion -tuple in an abelian category as defined in [Pol2, Section 2.2] is a collection of full subcategories of such that
for any where .
Every object of admits a filtration in
where for each .
The same notion also appeared in Toda’s work [Toda2, Definition 3.5]. Note that, given a torsion -tuple in as above, the pair is a torsion pair in for any .
or any Weierstraß elliptic fibration in the sense of [FMNT, Section 6.2] where is smooth (which implies that is Cohen-Macaulay [FMNT, Proposition C.1]), there is a pair of relative Fourier-Mukai transforms whose kernels are both sheaves on , satisfying
In particular, the kernel of is the relative Poincaré sheaf for the fibration , which is a universal sheaf for the moduli problem that parametrises degree-zero, rank-one torsion-free sheaves on the fibers of . An object is said to be -WIT if is a coherent sheaf sitting at degree . In this case, we write to denote the coherent sheaf satisfying up to isomorphism. The notion of -WIT can similarly be defined. The identities \maketag@@@(2.2.1\@@italiccorr) imply that, if a coherent sheaf on is -WIT for , then is -WIT. For , we will define the category
and similarly for . Due to the symmetry between and , the properties held by also hold for . The reader may refer to [FMNT, Section 6.2] for more background on the functors .
e fix our notation for various full subcategories of that will be used throughout this paper. For any integers , we set
where is the category of coherent sheaves supported in dimension 0 on the fiber , for the closed point . We will refer to coherent sheaves that are supported on a finite number of fibers of as fiber sheaves; then is precisely the category of fiber sheaves on .
or any integer , the categories as well as are all torsion classes in . From 2.2.3, each of these torsion classes determines a tilt of , and hence determines a t-structure on . The behaviours of these t-structures under the Fourier-Mukai transforms were studied by Zhang and the author in [Lo7, Lo11, LZ2]. In particular, we have the torsion pairs and in .
uppose is a smooth projective threefold with a fixed ample divisor and a fixed divisor . For any coherent sheaf on , we define
A coherent sheaf on is said to be -stable or slope stable (resp. -semistable or slope semistable) if, for every short exact sequence in of the form
where , we have . The slope function has the Harder-Narasimhan (HN) property, i.e. every coherent sheaf on has a filtration by coherent sheaves
where each is -semistable and .
For any coherent sheaf with , we have
Hence -stability is equivalent to -stability for coherent sheaves.
The HN property of the slope function on implies that the subcategories of
form a torsion pair in and give rise to the tilt of
For any object , we set
An object is said to be -stable or tilt stable (resp. -semistable or tilt semistable) if, for every short exact sequence in
where , we have . Tilt stability is a key notion in a now-standard construction of Bridgeland stability conditions on threefolds [BMT1].
When , we often drop the subscript in and above; for instance, we simply write instead of .
e will write
to denote the strict upper-half complex plane together with the negative real axis, and write
When is a smooth projective threefold and is an ample divisor on , we define the reduced central charge where
Then for any nonzero , we have [BMT1, Remark 3.3.1]. We can also define the phase of a nonzero object in by setting if , and requiring
if . An object is then said to be -stable (resp. -semistable) if, for every short exact sequence \maketag@@@(2.2.2\@@italiccorr) in , we have . Note that -stability is equivalent to -stability.
In the second half of this article, for any divisor on , we will also make use of the full (twisted) central charge given by
which is used in the construction of Bridgeland stability conditions on select smooth projective threefolds; modifications of this central charge have been used on other threefolds.
uppose is an abelian category. We call a function on a slope-like function if is defined by
where are a pair of group homomorphisms satisfying: (i) for any ; (ii) if satisfies , then . The additive group in the definition of a slope-like function can be replaced by any discrete additive subgroup of . Whenever is a noetherian abelian category, every slope-like function possesses the Harder-Narasimhan property [LZ2, Section 3.2]; we will then say an object is -stable (resp. -semistable) if, for every short exact sequence in where , we have .
3. Cohomological Fourier-Mukai transforms
In this section, we will assume conditions (1), (2) and (3a) in 2.1. We will see how (3b) arises as a necessary condition for comparing slope stability and tilt stability under a Fourier-Mukai transform, and explain what is meant by taking a ‘limit’ when we speak of limit tilt stability. The author thanks Ziyu Zhang for his insight that the cohomological Fourier-Mukai transforms become easier to understand when the Chern characters are twisted before applying the transform.
ake any object . By assumption \maketag@@@(2.1.1\@@italiccorr), we can write
for some , , where denotes the class of a fiber of . Then
where from the formula [FMNT, (6.26)]. Rewriting in terms of , we obtain
e introduce a notation for Chern characters that makes the cohomological Fourier-Mukai transform formula as easy to understand as in the product case of [Lo14]. The assumption \maketag@@@(2.1.1\@@italiccorr) on the cohomology ring of says that it is the direct sum of the following six vector spaces:
We can therefore think of the Chern character in \maketag@@@(3.1.1\@@italiccorr) as a matrix
Let us write to denote the -field . In the above matrix notation, twisting by this -field gives
Conceptually, this makes the cohomological Fourier-Mukai transform very similar to the case of the product threefold (see [Lo14, 4.1]): In the product case, the cohomological FMT simply swaps the two rows of in matrix notation, and changes the signs of the lower row (ignoring and in the expressions). The situation for Weierstraß threefolds here is similar: up to adding the terms and , the cohomological FMT also swaps the two rows of and changes the signs of the lower row. Indeed, if , then the above cohomological FMT formula reduces to that of the product case in [Lo14].
rom the Nakai-Moishezon Criterion [KM, Theorem 1.42], the section is a -ample divisor on the Weierstraß threefold . Then, for any ample divisor on , there exists some such that is an ample divisor on for all [KM, Proposition 1.45]. We will always use polarisations of this form on .
n the elliptic threefold , in order for us to compare slope stability for a coherent sheaf with respect to a polarisation , and limit tilt stability for with respect to another polarisation , we need to find a way to deform the polarisation so that, under an appropriate limit in , there exists a positive constant and an asymptotic equivalence
for any . By 3.3, we can write the polarisations on as
where and are ample classes on . With as in \maketag@@@(3.1.1\@@italiccorr), we will also write
to simplify some of the calculations to come. Then
Later in the article, we will let , in which case we need the following numerical equivalence for the ‘coefficients’ of and to hold in in order to have a solution to the numerical equivalence \maketag@@@(3.4.1\@@italiccorr):
y adjunction, we have
[FMNT, (6.6), Section 6.2.6]. In order for there to be a solution to \maketag@@@(3.4.4\@@italiccorr), the 1-cycles
must be real scalar multiples of each other under numerical equivalence. Two ways through which this can happen are:
are not -multiples of each other under numerical equivalence as cycles on , as is the case for ; however, and are -multiples of each other up to numerical equivalence;
are -scalar multiples of one another up to numerical equivalence as cycles on , such as when is a Fano, Enriques or K3 surface.
In case (a), we can solve \maketag@@@(3.4.4\@@italiccorr) by setting
and then taking to be suitable -multiples of each other up to numerical equivalence. Since we would like to let as mentioned in 3.4, this solution would force as well. This amounts to taking limits in both and , and corresponds to the approach taken by Yoshioka [YosAS, YosPII] on elliptic surfaces and the author’s joint work with Zhang in [LZ2]. However, we would like to find a solution to \maketag@@@(3.4.4\@@italiccorr) for an arbitrary polarisation , and so will not pursue this approach in this article.
In case (b), let us write
for some ample class on and some . We allow the possibility of being zero, as is the case when is an Enriques or a K3 surface. Then we write
Since the fibration is a local complete intersection [FMNT, Section 6.2.1], the numerical equivalence \maketag@@@(3.5.2\@@italiccorr) on pulls back to the numerical equivalence
on [Fulton, Example 19.2.3]. It follows that