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Curve-counting invariants and crepant resolutions of Calabi-Yau threefolds 2012
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Title | Curve-counting invariants and crepant resolutions of Calabi-Yau threefolds |
Creator |
Steinberg, David Christopher |
Publisher | University of British Columbia |
Date Created | 2012-07-19 |
Date Issued | 2012-07-19 |
Date | 2012 |
Description | The Donaldson-Thomas (DT) theory of a Calabi-Yau threefold X gives rise to subtle deformation invariants. They are considered to be the mathematical counterparts of BPS state counts in topological string theory compactified on X. Principles of physics indicate that the string theory of a singular Calabi-Yau threefold and that of its crepant resolution ought to be equivalent, so one might expect that the DT theory of a singular Calabi-Yau threefold ought to be equivalent to that of its crepant resolution. There is some difficulty in defining DT when X is singular, but Bryan, Cadman, and Young have (in some generality) defined DT theory in the case where X is the coarse moduli space of an orbifold χ. The crepant resolution conjecture of Bryan-Cadman-Young gives a formula determining the DT invariants of the orbifold in terms of the DT invariants of the crepant resolution. In this dissertation, we begin a program to prove the crepant resolution conjecture using Hall algebra techniques inspired by those of Bridgeland. |
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Thesis/Dissertation |
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Text |
Language | Eng |
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Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2012-07-19 |
DOI | 10.14288/1.0072891 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2012-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/42769 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/24/items/1.0072891/source |
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Curve-counting Invariants and Crepant Resolutions of Calabi-Yau Threefolds by David Christopher Steinberg A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2012 c© David Christopher Steinberg 2012 Abstract The Donaldson-Thomas (DT) theory of a Calabi-Yau threefold X gives rise to subtle deformation invariants. They are considered to be the mathemat- ical counterparts of BPS state counts in topological string theory compact- ified on X. Principles of physics (see [46], [50]) indicate that the string theory of a singular Calabi-Yau threefold and that of its crepant resolution ought to be equivalent, so one might expect that the DT theory of a singular Calabi-Yau threefold ought to be equivalent to that of its crepant resolution. There is some difficulty in defining DT when X is singular, but the authors of [14] have (in some generality) defined DT theory in the case where X is the coarse moduli space of an orbifold X . The crepant resolution conjec- ture of [14] gives a formula determining the DT invariants of the orbifold in terms of the DT invariants of the crepant resolution. In this dissertation, we begin a program to prove the crepant resolution conjecture using Hall algebra techniques inspired by those of Bridgeland [12]. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 pi-stable pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 The torsion pair and the stability condition . . . . . . . . . 29 5 The motivic Hall algebra . . . . . . . . . . . . . . . . . . . . . 37 6 Equations in the infinite-type Hall algebra and the fake proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7 Equations in the Laurent Hall algebra and the true proof 51 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 iii Acknowledgements I wish to thank my Ph. D. advisor, Jim Bryan, for his endless patience, en- thusiasm, and support in the face of adversity. I would also like thank Arend Bayer for help with lemma 3.16, Brian Conrad for help with lemma 2.16, Sándor Kovács, Kai Behrend, John Calabrese, Kalle Karu, Andrew Morri- son, and mathoverflow.com for their helpful conversations. Special thanks are due to Tom Bridgeland. On top of offering invaluable comments and suggestions on an early draft of this dissertation, his bril- liantly written papers [11] and [12] were the inspiration for most everything that follows. Finally, I would like to thank my parents, whose continued support has made all the difference. iv Dedicated to my friends. v Chapter 1 Introduction Donaldson-Thomas (DT) theory of a Calabi-Yau threefold X gives rise to subtle deformation invariants. They are considered to be the mathematical counterparts of BPS state counts in topological string theory compactified on X. Principles of physics (see [46], [50]) indicate that the string theory of a singular Calabi-Yau threefold and that of its crepant resolution ought to be equivalent, so one might expect that the DT theory of a singular Calabi- Yau threefold ought to be equivalent to that of its crepant resolution. There is some difficulty in defining DT when X is singular, but the authors of [14] have (in some generality) defined DT theory in the case where X is the coarse moduli space of an orbifold X . The crepant resolution conjecture of [14] gives a formula determining the DT invariants of the orbifold in terms of the DT invariants of the crepant resolution. In this dissertation, we begin a program to prove the crepant resolution conjecture using Hall algebra techniques inspired by those of Bridgeland [12]. Donaldson-Thomas theory Let Y be a smooth projective Calabi-Yau threefold. Let K(Y ) be the nu- merical K-theory of Y , i.e. the quotient of the K-group of coh(Y ) by the kernel of the Chern character map to cohomology. The Hilbert scheme of Y , Hilbα(Y ), parametrizes quotients OY → OZ , such that the class of OZ in K(Y ) is α. The group K(Y ) is filtered by the dimension of the support: F0K(Y ) ⊂ F1K(Y ) ⊂ F2K(Y ) ⊂ F3K(Y ) = K(Y ). 1 In this dissertation, we will focus on curves, i.e., α ∈ F1K(Y ), with ch(α) = (0, 0, β, n), where β ∈ H4(Y,Z) is a curve class, and n ∈ H6(Y,Z) ∼= Z is the holomorphic Euler characteristic. In [43], an obstruction theory for this moduli space is constructed, which produces (by [5]) a virtual fundamental cycle. Donaldson-Thomas invariants are defined by integrating over the zero-dimensional virtual fundamental class: DTα(Y ) = ∫ [Hilbα(Y )]vir 1. Since the obstruction theory is symmetric, we may also express the in- variants as the Euler characteristic of Hilbα(Y ) weighted by Behrend’s mi- crolocal function [3]: DTα(Y ) = ∑ n∈Z nχ(ν−1(n)), where ν : Hilbα(Y )→ Z is Behrend’s function. Following [35], we assemble the invariants into a partition function DT(Y ) = ∑ α∈F1K(Y ) DTα(Y )qα. Remark 1.1. In [29], Donaldson-Thomas invariants are greatly generalized, from the case of structure sheaves of curves to that of arbitrary sheaves. The price of admission to this generality is the formidable machinery of Joyce [24, 25, 26, 27, 28]. An even more ambitious program of generalization is being lead by Kontsevich and Soibelman [31]. The Donaldson-Thomas crepant resolution conjecture We follow [14] in our treatment of the crepant resolution conjecture. An orbifold CY3 is defined to be a smooth, quasi-projective, Deligne- Mumford stack X over C of dimension three having generically trivial sta- 2 bilizers and trivial canonical bundle, KX ∼= OX . The definition implies that the local model for X at a point p is [C3/Gp] where Gp ⊂ SL(3,C) is the (finite) group of automorphisms of p. The orbifold CY3s that appear in this dissertation will all be projective and satisfy the hard Lefschetz condition ([16, definition 1.1]), which in this case is equivalent ([15, lemma 24]) to the condition that all Gp are finite subgroups of SO(3) ⊂ SU(3) or SU(2) ⊂ SU(3). Let X denote the coarse space of X . A crepant resolution of X is a resolution of singularities pi : Y → X such that pi∗KX ∼= KY . Lemma 1 and proposition 1 of [48] prove that R•pi∗OY = OX . (1.2) The results of [13] and [18] prove that one distinguished crepant resolution of X is Y = Hilb[Op](X ), (1.3) the Hilbert scheme parametrizing substacks in the class [Op] ∈ F0K(X ). The hard Lefschetz condition implies that the resolution is semi-small (i.e., that the fibres of pi are zero- or one-dimensional), and that the singular locus of X is one-dimensional; see [8, 15]. Furthermore, [13] and [18] prove that there is a Fourier-Mukai isomorphism Ψ : Db(Y )→ Db(X ) defined by E 7→ Rq∗p∗E where p : Z → Y, q : Z → X are the projections from the universal substack Z ⊂ X ×Y onto each factor. 3 This isomorphism descends to an isomorphism of K-theory also denoted Ψ : K(Y ) → K(X ). It does not respect the filtration by dimension. However, the hard Lefschetz condition implies that the image of F0K(X ) is contained in F1K(Y ), under the inverse Φ of Ψ. We call the image FexcK(Y ); its elements can be represented by formal differences of sheaves supported on the exceptional fibres of pi : Y → X. We define the multi-regular part of K-theory, Fmr(X ), to be the preimage of F1K(Y ) under Ψ. Its elements can be represented by formal differences of sheaves supported in dimension one where at the generic point of each curve in the support, the associated representation of the stabilizer group of that point is a multiple of the regular representation. The following filtrations are respected by Ψ: FexcK(Y ) ⊂ F1K(Y ) ⊂ K(Y ) F0K(X ) ⊂ FmRK(X ) ⊂ K(X ). Define the exceptional DT generating series of Y , the multi-regular gen- erating series, and degree zero generating series of X to be: DTexc(Y ) = ∑ α∈FexcK(Y ) DTα(Y )qα, DTmr(X ) = ∑ α∈FmrK(X ) DTα(X )qα DT0(X ) = ∑ α∈F0K(X ) DTα(X )qα We state the crepant resolution conjecture of [14, conjecture 1]: Conjecture 1.4. Let X be an orbifold CY3 satisfying the hard Lefschetz condition. Let Y be the Calabi-Yau resolution of X given by equation 1.3. Then using Ψ to identify the variables, we have an equality DTmr(X ) DT0(X ) = DT(Y ) DTexc(Y ) . This dissertation makes progress towards proving this conjecture. 4 pi-stable pairs Objects of the Hilbert scheme may be viewed as two-term complexes, OY γ→ G, where the cokernel of γ must be zero, and where G may be any sheaf ad- mitting such a map γ. The new invariants introduced in this dissertation, pi-stable pairs, are a modification of this idea. They have been constructed with a view towards proving the crepant resolution conjecture, and as such, they depend on a crepant resolution Y pi→ X as described in the previous sec- tion. The objects of our moduli space allow more variation in our cokernels, but less in the sheaf G. In particular, a two-term complex OY γ→ G is a pi-stable pair (c.f. definition 2.7) if: 1. R•pi∗ coker(γ) is a zero-dimensional sheaf on X, and 2. G admits only the zero map from any sheaf P with the property above, namely that R•pi∗P is a zero-dimensional sheaf. Remark 1.5. These pairs were inspired by, and are a generalization of, the stable pairs of Pandharipande and Thomas [39]. In fact, when X = Y and pi is the identity map, the above definition reduces to their definition of stable pairs. Below, we prove that there is a finite-type algebraic space, pi-Hilbα parametrizing these objects with [G] = α ∈ K(Y ). We may then define invariants pi-PTα(Y ) = ∑ n∈Z nχ(ν−1(n)), where ν : pi-Hilbα → Z is Behrend’s microlocal function. Note that if pi : Y → X is the identity, then pi-PTα(Y ) = PTα(Y ), the usual Pandharipande- Thomas invariants of Y . As with Donaldson-Thomas theory, we collect the 5 invariants into a generating series, pi-PT(Y ) = ∑ α∈F1K(Y ) pi-PTα(Y )qα. Main result The following theorem rests the work of Bridgeland [12] and Joyce–Song [29], and we therefore require our Calabi-Yau threefold Y to satisfy H1(Y,OY ) = 0. Theorem 1.6. LetX be a projective Calabi-Yau threefold that is the coarse space of an CY3 orbifold X that satisfies the hard Lefschetz condition. Let pi : Y → X be the resolution given by equation 1.3. Then the generating series for the pi-stable pair invariants and the DT invariants are related by the equation pi-PT(Y ) = DT(Y ) DTexc(Y ) . The aim of this dissertation is to prove this theorem. We summarize the chapters below. In chapter 2, we describe a torsion pair (Ppi,Qpi) that is crucial to our definition of pi-stable pairs. We explain the similarities between pi-stable pairs and PT stable pairs and objects of the Hilbert scheme. The chapter ends by establishing results about the moduli space of pi-stable pairs. In chapter 3, we recall the concept of a stability condition in the sense of Joyce. We then define the stability condition that we will use through out. The rest of the chapter is dedicated to proving that we may apply Joyce’s powerful machinery. In chapter 4, we introduce the Harder-Narasimhan filtration for our sta- bility condition, which will be our main tool to prove the relationship be- tween the stability condition and the torsion pair from chapter 2. In chapter 5, we introduce the motivic Hall algebra. In chapter 6, we introduce the infinite-type Hall algebra as a purely pedagogical tool. It helps us to give the essence of the idea of many re- 6 sults, without having to concern ourselves with convergence issues, which are handled in the next chapter. In chapter 7, we introduce the Laurent Hall algebra, address the conver- gence issues alluded to in the previous chapter, and prove theorem 1.6. Remark 1.7. To prove the crepant resolution conjecture, we need to prove that pi-PT(Y ) = PT(X ) and then use ([2]) Bayer’s proof of the PT/DT correspondence on X , PT(X ) = DTmR(X ) DT0(X ) . The hope is that the Fourier-Mukai isomorphism Ψ takes pi-stable pairs (as an object in Db(Y )) to a PT pair on X . In his recent article [17], John Calabrese proves a relationship between the DT invariants of a Calabi-Yau threefold and its flop. This problem is similar in many respects to the crepant resolution conjecture studied in this thesis, and Calabrese uses many similar techniques. He constructs a torsion pair and new counting invariants which he relates to invariants on the flop via equations in the Hall algebra and the integration map. While this is very similar to our approach in outline, the actual torsion pair and counting invariants that Calabrese considers (even when adapted to the orbifold setting) are quite different from ours. It would be very interesting to find the precise relationship between the two approaches. 7 Chapter 2 pi-stable pairs In this section, we define pi-stable pairs, and prove some basic results. Categorical constructions Let A be an abelian category. Here we recall the notion of torsion pairs. Definition 2.1. Let (P,Q) be a pair of full subcategories of A. We say (P,Q) is a torsion pair if the following conditions hold. • Hom(T, F ) = 0 for any T ∈ P and F ∈ Q. • Any object E ∈ A fits into a unique exact sequence, 0→ T → E → F → 0, (2.2) with T ∈ P and F ∈ Q. We borrow the following lemma from Toda [45]. Lemma 2.3. Suppose that A is a noetherian abelian category. (i) Let P ⊂ A be a full subcategory which is closed under extensions and quotients in A. Then for Q = {E ∈ A : Hom(P, E) = 0}, the pair (P,Q) is a torsion pair on A. (ii) Let Q ⊂ A be a full subcategory which is closed under extensions and subobjects in A. Then for P = {E ∈ A : Hom(E,Q) = 0}, the pair (P,Q) is a torsion pair on A. Proof: We only show (i), as the proof of (ii) is similar. Take E ∈ A with E /∈ Q. Then there is T ∈ P and a non-zero morphism T → E. Since P is 8 closed under quotients, we may assume that T → E is a monomorphism in A. Take an exact sequence in A, 0→ T → E → F → 0. (2.4) By the noetherian property of A and the assumption that P is closed under extensions, we may assume that there is no T ( T ′ ⊂ E with T ′ ∈ T . Then we have F ∈ Q and (2.4) gives the desired sequence. Example 2.5. Let P = {0-dimensional sheaves on Y }, and let Q = {E ∈ coh(Y ) : Hom(P,E) = 0 for all P ∈ P}. Lemma 2.3 easily proves that the pair (P,Q) is a torsion pair. Let C = coh≤1(Y ) denote the full subcategory of coherent sheaves on Y whose support is of dimension no more than one. We make the following definitions: Ppi = {P ∈ C|R•pi∗P is a zero-dimensional sheaf on X}, and Qpi = {F ∈ C| for all P ∈ Ppi,Hom(P, F ) = 0} = P⊥pi . Lemma 2.6. The pair (Ppi,Qpi) is a torsion pair in C. Proof: By lemma 2.3, it suffices to prove that Ppi is closed under extensions and quotients. Let P ′, P ′′ ∈ Ppi, and consider the short exact sequence 0→ P ′ → P → P ′′ → 0. We are to show that such a P must live in Ppi. Consider now the long exact sequence, 9 0→ pi∗P ′ → pi∗Ppi → pi∗P ′′ → R1pi∗P ′ → R1pi∗Ppi → R1pi∗P ′′ → 0. Since P ′, P ′′ ∈ Ppi, we know that R1pi∗P ′ = 0 and R1pi∗P ′′ = 0, so R1pi∗P = 0. We also know that pi∗P ′ and pi∗P ′′ are zero-dimensional sheaves, and so it is clear then that pi∗P must be so as well. This proves that Ppi is closed under extensions. Let P ∈ Ppi, and consider the quotient P → B → 0. Denote the kernel of this map by K. As before, we get a long exact sequence, 0→ pi∗K → pi∗P → pi∗B → R1pi∗K → R1pi∗P → R1pi∗B → 0. Since P ∈ Ppi, R1pi∗P = 0, and so R1pi∗B = 0. It remains to show that pi∗B is zero-dimensional. We know that pi∗K is zero dimensional, since it is a subsheaf of (the zero-dimensional sheaf) pi∗P . The support of R1pi∗K is contained in the singular locus. Suppose dim supp(R1pi∗K) = 1. Then, K must have been supported in dimension two, however this contradicts the fact that K ∈ coh≤1(Y ). Hence R1pi∗K is zero dimensional. Now, pi∗B is the extension of zero- dimensional sheaves, so it too is zero-dimensional. This completes the proof that B ∈ Ppi, and that (Ppi,Qpi) is a torsion pair. Definition 2.7. A map γ : OY → G is a pi-stable pair if G ∈ Qpi and coker(γ) ∈ Ppi. Remark 2.8. Our notion of pi-stable pair is a generalization of the stable pairs of Pandharipande and Thomas [39]. In the trivial case when X = Y and pi = the identity, we have that (Ppi,Qpi) = (P,Q) of example 2.5, and the pi-stable pairs are exactly PT stable pairs. Definition 2.9. Two pi-stable pairs γ1 : OY → G1 and γ2 : OY → G2 are isomorphic if there exists a isomorphism of sheaves θ : G1 → G2 making the 10 following diagram commute: OY γ1 // γ2 !!C CC CC CC C G1 θ G2 Remark 2.10. In [12], the tilt of A with respect to the torsion pair (P,Q) of example 2.5 is denoted A#, and lemma 2.3 of [12] proves that A#- epimorphisms of the form OY → F are precisely stable pairs. The abelian category generated by OY and C has a tilt whose epimorphisms of the form OY → G are precisely pi-stable pairs. This is analogous to the tilt used in [12]. However, since it is not strictly necessary for any of our arguments, we will not present a proof here. We associate to every pi-stable pair γ : OY → G a short exact sequence 0→ OC → G→ P → 0, where P = coker(γ) ∈ Ppi and OC = OY / ker(γ). Proposition 2.11. Let G be a non-zero sheaf. If OY → G is a pi-stable pair, then G is not supported exclusively on exceptional curves, and R1pi∗G = 0. Proof: Let 0→ OC → G→ P → 0 be the associated short exact sequence. We will first show that R1pi∗OC = 0. Consider the short exact sequence 0→ IC → OY → OC → 0. Pushing forward yields 0→ pi∗IC → pi∗OY → pi∗OC → R1pi∗IC → R1pi∗OY → R1pi∗OC → 0 which is exact since the dimension of the fibres of Y → X is at most one. Thus the vanishing of R1pi∗OY by equation 1.2 implies that of R1pi∗OC . 11 Now consider the following long exact sequence. 0→ pi∗OC → pi∗G→ pi∗P → R1pi∗OC → R1pi∗G→ R1pi∗P → 0. From above, we know R1pi∗OC = 0. As well, P ∈ Ppi implies R1pi∗P = 0. Thus R1pi∗G = 0. Now, if C consists of only exceptional curves, then pi∗OC is zero-dimensional. This implies that pi∗G is the extension of zero-dimensional sheaves, and therefore zero-dimensional. This means that G ∈ Ppi. By definition of pi- stable pair, G ∈ Qpi. By definition of Qpi the only map from an object of Ppi to an object of Qpi is the zero map, hence the identity of G is the zero map, and G is the zero object. Let us introduce some terminology and results taken from [12] (modified for our purposes, since we are only interested in sheaves supported in di- mension no more than one). LetM denote the stack of objects of coh≤1(Y ). It is an algebraic stack, locally of finite type over C. Let M(O) denote the stack of framed sheaves, that is, the stack whose objects over a scheme S are pairs (E, γ) where E is a S-flat coherent sheaf on S × Y , of relative dimension no more than one, together with a map OS×Y γ→ E. Given a morphism of schemes f : T → S, and an object (F, δ) over T , a morphism in M(O) lying over f is an isomorphism θ : f∗(E)→ F such that the following diagram commutes f∗(OS×Y ) can f∗(γ) // f∗(E) θ OT×Y δ // F. (2.12) The symbol “can” denotes the canonical isomorphism of pullbacks. 12 There is a natural map M(O) q→M (2.13) sending a sheaf with a section to the underlying sheaf. The following lemmas are 2.4 and 2.5 of [12]. Lemma 2.14. The stack M(O) is algebraic and the morphism q is repre- sentable and of finite type. Lemma 2.15. There is a stratification of M by locally-closed substacks Mr ⊂M such that objects F of Mr(C) are coherent sheaves satisfying dimCH0(Y, F ) = r. Furthermore, the pullback of the morphism q to Mr is a Zariski fibration with fibre Cr. Both of these are proven in [12]. Lemma 2.16. The moduli space of pi-stable pairs with a fixed Hilbert poly- nomial is an algebraic space of finite type. Proof: Let pi-Hilb(β,n) denote the stack of pi-stable pairs on Y whose sheaf G has Chern character (0, 0, β, n). Notice that pi-Hilb(β,n) ⊂M(O). Lemma 2.14 states that M(O) is an algebraic stack. Using the forth-coming description of Ppi and Qpi in terms of stability, and using theorem 4.8 of [26], we know that Ppi and Qpi are algebraic sub- stacks ofM. We use this fact to proven that pi-Hilb is an algebraic substack of M. 13 Consider the following cartesian diagram, Qpi(O) // M(O) q Qpi //M. Since Qpi and M(O) are algebraic stacks, their fibre product, Qpi(O) is an algebraic stack, too. There is a map of stacks of stack c : Qpi(O) → M which takes γ : OY → G to the cokernel coker(γ). Since pi-Hilb fits into the following cartesian diagram, pi-Hilb // Ppi Qpi(O) c //M, we conclude that it too is an algebraic stack. Now we prove pi-Hilb(β,n) is of finite type. In lemma 3.16, we prove that the sheaves of Chern character (0, 0, β, n) underlying pi-stable pairs form a bounded family. In particular, this means that there exists a coherent sheaf E such that every sheaf underlying a pi-stable pair G of Chern character (0, 0, β, n) is a quotient of E. Now if we choose a polarization O(1) of Y , we have a finite type scheme Quot(E,P ) where P is the Hilbert polynomial P (t) = β · O(1)t + n. There is a morphism Quot(E,P ) →M that takes a quotient E → G and sends it to G. This map is of finite type. Consider the diagram: T f // f ′ Quot(E,P ) q′ M(O) q //M where T is the pullback of q′ by q. The image of the map q′ contains all sheaves underlying pi-stable pairs with Hilbert polynomial P . This means that the image of T in M(O) contains all 2-term complexes [OY → G] where G is a sheaf underlying a pi-stable pair with Hilbert polynomial P . 14 In particular, the image of T contains all pi-stable pairs. Now since q is of finite type, so must f be of finite type. The scheme Quot(E,P ) is of finite type over C, so then we may conclude that T is finite type over C. Thus the image of T under f ′ is of finite type. The stack of all pi-stable pairs is in this image, hence of finite type. It remains to prove that this stack is in fact an algebraic space. It suffices to prove that all closed points of the stack pi-Hilb(β,n) have trivial automorphism. Let OY → G be a pi-stable pair. We will show that it has only the trivial automorphism. Consider the associated short exact sequence, 0→ OC → G→ P → 0. An automorphism of this pi-stable pair leads to a diagram of the form, 0 // OC id γ // G g h // P // g 0 0 // OC γ // G h // P // 0. We will show that g is the identity map. Consider the following diagram, obtained by subtracting the identity from the diagram above, 0 // OC zero γ // G g− id h // P // g−id 0 0 // OC // G h // P // 0. (2.17) Since the left-most vertical arrow is zero, a diagram chase proves that the dotted morphism exists and commutes with the diagram. However, P ∈ Ppi and G ∈ Qpi, so this map must be zero. As a consequence, g − id = 0. Homological algebra then implies that g− id must be the zero map, forcing g = id. This proves that closed points of the stack of pi-stable pairs have no au- tomorphisms, and that pi-stable pairs form an algebraic space. 15 The Behrend function identity We state and prove a variation of [12, theorem 3.1] of Bridgeland. Lemma 2.18. Let γ : OY → G be a pi-stable pair. Then there is an equality of Behrend’s microlocal functions νM(O)(γ) = (−1)χ(G)νM(G). Proof: The case when G is a stable pair is taken care of by theorem 3.1 of [12]. Thus, we may assume that the cokernel P of γ : OY → G has one-dimensional support. Let OC ⊂ G be the image of γ. It is the structure sheaf of a subscheme C ⊂ Y of dimension 1. There is a line bundle L on Y such that H i(Y,G⊗ L) = 0 (2.19) for all i > 0, and there is a divisor H ∈ |L| such that H meets C at finitely many points, none of which are in the support of coker(γ). This claim is verified in lemma 2.22. From here, the proof is identical to Bridgeland’s, but we include a portion of it to illustrate his ideas. There is a short exact sequence 0→ OY s→ L→ OH(H)→ 0 where s is the section of L corresponding to the divisor H. Tensoring it with G, and using the above assumptions yields a diagram of sheaves OY γ δ !!B BB BB BB B 0 // G α // F β // K // 0 (2.20) where F = G ⊗ L. The support of the sheaf K is zero-dimensional, and 16 disjoint from the support of coker(γ). In particular, HomY (K,F ) = 0. (2.21) Consider two points of the stack M(O) corresponding to the maps γ : OY → G and δ : OY → F. The statement of lemma 2.18 holds for the map δ because lemma 2.15 to- gether with equation 2.19 implies that q :M(O)→M is smooth of relative dimension χ(F ) = H0(Y, F ) over an open neighbour- hood of the point F ∈ M(C). On the other hand, tensoring sheaves with L defines an automorphism of M, so the microlocal function of M at the points corresponding to G and F are equal. To prove the lemma, it suffices to show that (−1)χ(G) · νM(O)(γ) = (−1)χ(F ) · νM(O)(δ). Consider the stack W whose S-valued points are diagrams of S-flat sheaves on S × Y of the form OS×Y γS δS ##F FF FF FF F 0 // GS αS // FS βS // KS // 0. There are two morphisms p :W →M(O), q :W →M(O), taking such a diagram to the maps γS and δS respectively. By passing to an open substack of W , we may assume that equation 2.21 holds for all C-valued points of W . It follows that p and q induce injective maps on 17 stabilizer groups of C-valued points, and hence are representable. Recall that Behrend’s microlocal function satisfies the property that when f : T → S is a smooth morphism of relative dimension d, there is an identity [3, proposition 1.5] νT = (−1)df∗(νS). Using this identity, it will be enough to show that at the point w ∈ W (C) corresponding to the diagram 2.20, the morphisms p and q are smooth of relative dimension χ(K) and 0, respectively. For the proof of these facts, see the [12, pages 11–13]. Lemma 2.22. Given a pi-stable pair γ : OY → G, we may choose a very ample divisor H on X such that its pull-back is equal to its proper transform (we denote both by H̃), it satisfies supp(coker γ) ∩ H̃ = ∅, H̃ ∩ suppG is 0-dimensional, and H1(Y,G(H̃)) = 0. Proof: First we collect a little notation. Let E be the exceptional locus of Y , let E′ be the image of E in X. Define a subset Z to be Z = {p ∈ X : G|pi−1(p) is one dimensional} ⊂ E′. Notice that Z is a finite collection of points. Since coker γ ∈ Ppi, we have that pi(supp(coker γ)) is zero dimensional. Moreover, pi(suppG) is one dimensional. Thus we may choose an ample divisor H on X so that H ∩ pi(supp(coker γ)) = ∅ and H ∩ pi(suppG) is zero dimensional and does not contain any of the points in Z. It follows that H̃ ∩ supp(coker γ) is empty, and H̃ ∩ supp(G) is zero dimensional. Moreover, by Serre vanishing, we may assume that H is 18 sufficiently ample on X so that H1(X, (pi∗G)(H)) = 0. (2.23) We now show that H1(Y,G(H̃)) = 0. By proposition 2.11, we know that R1pi∗G = 0, since OY → G is a pi-stable pair. The sequence 0→ G→ G(H̃)→ G(H̃)| eH → 0 gives . . .→ R1pi∗G→ R1pi∗G(H̃)→ R1pi∗G(H̃)| eH → 0. However, we know R1pi∗G = 0 and G(H̃)| eH is supported on points so R1pi∗G(H̃)| eH = 0, so R1pi∗G(H̃) = 0. Now by the Leray spectral sequence, H1(Y,G(H̃)) = H1(X,pi∗(G(H̃)) = H1(X,pi∗(G⊗ pi∗OX(H))) = H1(X, (pi∗(G))(H)) = 0, where the last equality comes from equation 2.23. 19 Chapter 3 Stability conditions In this section, we define a stability condition on C = coh≤1 Y . We follow Joyce’s treatment of stability conditions as found in section 4 of [26], though not in as great generality. Let N1(Y ) denote the abelian group of cycles of dimension one modulo numerical equivalence. We begin by quoting lemmas 2.1 and 2.2 of [12]. Lemma 3.1. An element β ∈ N1(Y ) has only finitely many decompositions of the form β = β1 + β2 with βi effective. Lemma 3.2. The Chern character map induces an isomorphism ch = (ch2, ch3) : F1K(Y )→ N1(Y )⊕ Z. Define ∆ = {[E] ∈ F1K(Y ) : E ∈ C} to be the positive or effective cone of F1K(Y ). Definition 3.3. A stability condition on C is a triple (T, τ,≤) where (T,≤) is a set T with a total ordering ≤, and τ is a map ∆ τ→ T from the effective cone to T , satisfying the following condition: whenever α+β = γ in ∆, then τ(α) < τ(γ) < τ(β), or τ(β) < τ(γ) < τ(α), or τ(α) = τ(γ) = τ(β). 20 A triple (T, τ,≤) is called a weak stability condition if it satisfies the weaker condition that whenever α + β = γ in ∆, τ(α) ≤ τ(γ) ≤ τ(β) or τ(β) ≤ τ(γ) ≤ τ(α). Definition 3.4. A non-zero sheaf G is 1. τ -semistable if for all S ⊂ G, such that S 6∼= 0, we have that τ(S) ≤ τ(G/S); 2. τ -stable if for all S ⊂ G, such that S 6∼= 0, we have that τ(S) < τ(G/S); 3. τ -unstable if it is not τ -semistable. Lemma 3.5. Let F and G be τ -semistable sheaves, and let F f→ G be a map of sheaves. Then either τ(F ) ≤ τ(G) or f = 0. Proof: Consider the inclusion map ι : im(f) → G. Since G is semistable, we know either im(f) = 0 or τ(im(f)) ≤ τ(G). Consider now the core- striction of f , cor(f) : F → im(f). Since F is semistable, we know that either im(f) = 0 or τ(F ) ≤ τ(im(f)). This implies either im(f) = 0 or τ(F ) ≤ τ(G). Now we define a stability condition on C. Definition 3.6. Let E be the exceptional divisor of pi. Choose an ample divisor H on X, let H̃ denote the total transform of H in Y . Let L be a very ample line bundle on Y . Given a sheaf G in C, define the pi-slope of G to be µpi(G) = (χ(G) β · H̃ , χ(G) β · L ) ∈ (−∞,+∞]× (−∞,+∞], where (−∞,+∞]× (−∞,+∞] is ordered lexicographically, and β = βG the homology class associated to the support of G. Theorem 3.7. The map µpi : ∆→ (−∞,+∞]× (−∞,+∞] 21 defines a weak permissible stability condition. Unpacking the definitions 4.1 and 4.7 of [26], we see that a weak permis- sible stability condition is one that satisfies three conditions: 1. (weak seesaw property) for any short exact sequence 0→ A→ G→ B → 0, either µpi(A) ≤ µpi(G) ≤ µpi(B) or µpi(A) ≥ µpi(G) ≥ µpi(B), 2. µpi is dominated by a permissible stability condition, and 3. the family of all µpi-stable sheaves of a fixed Chern class is bounded. We will prove theorem 3.7 by proving the above three properties in lemma 3.8, corollary 3.10, and lemma 3.16. Lemma 3.8. The function µpi satisfies the seesaw property. Proof: Let 0 → A → G → B → 0 be a short exact sequence of sheaves, and suppose µpi(A) ≤ µpi(G). Less concisely, we are supposing ( χ(A) βA · H̃ , χ(A) βA · L ) ≤ ( χ(G) βG · H̃ , χ(G) βG · L ) , from which we are to deduce that µpi(G) ≤ µpi(B). Before we start a case- by-case analysis, notice that χ(G) = χ(A) + χ(B) and βG = βA + βB. case 1: χ(A) βA· eH < χ(G)βG· eH and no denominator is zero. Then this follows from the observation a b < a+ c b+ d ⇒ a+ c b+ d < c d , provided b, d > 0. In particular, we assume χ(A) βA · H̃ ≤ χ(G) βG · H̃ . 22 Rewriting the second term yields χ(A) βA · H̃ ≤ χ(A) + χ(B) (βA + βB) · H̃ . The observation above then proves that χ(A) + χ(B) (βA + βB) · H̃ ≤ χ(B) βB · H̃ , as desired. case 2: χ(A) βA· eH = χ(G)βG· eH , χ(A)βA·L ≤ χ(G)βG·L and no denominator is zero. We are given that χ(A) βA· eH = χ(G)βG· eH , so χ(A)(βG · H̃) = (βA · H̃)χ(G). Writing everything in terms of A and B, χ(A)((βA + βB) · H̃) = βA · H̃(χ(A) + χ(B)), which implies χ(A)(βB · H̃) = βA · H̃χB. Since we assume that all denominators are non-zero, we have χ(A) βA · H̃ = χ(B) βB · H̃ . So we must show that χ(G) βG · L < χ(B) βB · L. This follows from the same observation made in case 1. case 3: βA · H̃ = 0. Then +∞ = χ(A) βA· eH ≤ χ(A)+χ(B)(βA+βB)· eH . This implies that χ(A)+χ(B) (βA+βB)· eH = +∞, so (βA + βB) · H̃ = 0, and hence βB · H̃ = 0. This 23 reduces us to Gieseker stability on Y , which we know satisfies the seesaw property. case 4: βG · H̃ = 0. We know β · pi∗H = pi∗(β · pi∗H) = pi∗(β) ·H ≥ 0, hence β · H̃ ≥ 0 for any effective curve class β, so we must have βA · H̃ = 0 and βB · H̃ = 0. This lands us back in the case of Gieseker stability on Y , and lemma 3.8 is proven. Let us now recall from Joyce the following facts and definitions. A stability condition τ on C is permissible if: • C is τ -artinian, i.e. there exists no infinite descending chain · · ·A2 ⊂ A1 ⊂ A in C such that Ai 6= Ai+1, and τ(Ai+1) ≥ τ(Ai/Ai+1) for all i; and • the substack of τ -semistable objects of a fixed Chern character of the stack parametrizing objects of C is a constructible substack of M. Thankfully, the second condition amounts to showing that the family of µpi- semistable sheaves of a fixed Chern character is bounded. See the proof of theorem 4.30 from [26] for details. Joyce proves that these properties are inherited by domination. Recall that a stability condition τ̃ is said to dominate τ if for any A,B in C with τ(A) ≤ τ(B) then τ̃(A) ≤ τ̃(B). Define the following stability condition: δ(G) = −dim suppG ∈ Z. In [26], it is proven that the abelian category coh(Y ) is δ-artinian. To prove that C = coh≤1(Y ) is µpi-artinian, we will show that µpi is dominated by δ. Lemma 3.9. The stability condition µpi is dominated by the stability con- dition δ. 24 Proof: Let µpi(A) ≤ µpi(B). We need to show that this implies that δ(A) ≤ δ(B). Expanding, we have that ( χ(A) βA · H̃ , χ(A) βA · L ) ≤ ( χ(B) βB · H̃ , χ(B) βB · L ) . We proceed with a case-by-case analysis. case 1: the denominators are non-zero. Since we have restricted our attention to sheaves supported in dimension ≤ 1, it follows that neither A nor B is 0-dimensional. Thus, they are both one dimensional, and δ(A) = δ(B). In particular, δ(A) ≤ δ(B). case 2: βB · H̃ = 0 and βA · H̃ 6= 0. Then dim suppA ≥ 1 ≥ dim suppB. So δ(A) ≤ δ(B). case 3: Both βB · H̃ = 0 and βA · H̃ = 0. Then µpi(A) ≤ µpi(B) amounts to regular Geiseker stability, which is dominated by δ as demonstrated by Joyce [26, §4.4]. Corollary 3.10. The category C is µpi-artinian. Proof: As mentioned above, this follows from the previous lemma since it is inherited by domination. To finish the proof that µpi is a permissible weak stability condition, it remains only to prove the family of all µpi-semistable sheaves of a fixed Chern class is bounded. This is proven in the next section. Boundedness Let us recall the basic results concerning boundedness (cf. [23]). Definition 3.11. A subcategory U of coh(Y ) is bounded if there exists a scheme S of finite type and a sheaf U on X × S such that for every object Ui of U , there exists a closed point si ∈ S such that Ui ∼= U |X×{si}. 25 Notice that this definition still makes sense if we have a set of isomor- phism classes of sheaves instead of a category. Definition 3.12. Let Y be a scheme, let O(1) be an ample line bundle, and let m be an integer. A sheaf F on Y is m-regular if, for all i > 0, H i(Y, F (m− i)) = 0. A proof for the following may be found in [30], as well as in [37]. Lemma 3.13. If F is m-regular, then the following statements are true: 1. F is m′-regular for all m′ ≥ m. 2. F (m) is globally generated. 3. For all n ≥ 0, the natural mapH0(Y, F (m))⊗H0(Y,O(n))→ H0(Y, F (n+ m)) is surjective. Definition 3.14. The Mumford-Castelnuovo regularity of a sheaf F is the number reg(F ) = inf{m ∈ Z : F is m-regular }. Lemma 3.15. Let U be a category of sheaves on Y . The following state- ments are equivalent. 1. U is bounded. 2. The set of Hilbert polynomials of objects Ui of U is finite, and there is an integer N such that for all objects Ui of U , regUi < N . 3. The set of Hilbert polynomials of objects Ui of U is finite, and there exists a sheaf F such that each object of U is isomorphic to a quotient of F . Proof of this lemma may be found in [20]. Now we prove the boundedness result we require. Lemma 3.16. The family of pi-stable pairs OY γ→ G with a fixed Chern class is bounded. 26 Proof: To each such pi-stable pair there is an associated a short exact sequence, 0→ OC → G→ P → 0, where OC is the image of the map γ, and P is the cokernel. We will show that the family of possibilities for OC and the family of possible P s are both bounded families. Once this is established, it is clear that the family of sheaves underlying a pi-stable pair is a bounded family of sheaves. First we will consider the family of possibilities for OC . To show that this family is bounded, we will show: 1. the Hilbert polynomials of this family take on only a finite number of values; and 2. there exists a single sheaf that surjects onto each member of this family. The second requirement is trivially satisfied, since each member of this fam- ily is the structure sheaf of a subscheme of Y , and hence, admits a surjective maps fromOY . It remains to find upper and lower bounds for the coefficients of the Hilbert polynomial of a general element from this family. In contrast to PT -theory, the support of OC is not equal to the support of G, since P is not necessarily zero-dimensional. However, we still have βG = βC+βP , where all β are effective. We know that there are only finitely many decompositions of βG into the sum of two effective curve classes. This forces an upper and lower bound on the linear coefficient of the Hilbert polynomial of OC . It remains to find upper and lower bounds for the Euler characteristic of OC (the constant coefficient of the Hilbert polynomial). The Leray spectral sequence proves that χ(P ) = χ(R•pi∗P ) ≥ 0, the inequality following from the fact that R•pi∗P is zero dimensional. Now, χ(G) = χ(P ) + χ(OC), and χ(P ) ≥ 0 implies χ(OC) ≤ χ(G). This gives us an upper bound, since the Chern character, and hence, the Euler character- istic, of G is fixed. For the lower bound, let αG = (βG, nG) be the Chern character of G, and let αC = (βC , nC) be the Chern character of C. In general, if Hilb(β,n) is non-empty (say one of its points represents a curve J), then dimHilb(β,n+k) ≥ 3k since we get a 3k-dimensional space 27 of curves coming from the curve J with k “wandering points.” This line of reasoning tells us that dimHilb(βC ,nG) ≥ 3(nG − nC). Rearranging this yields nC ≥ nG − 13 dimHilb(βC , nG). This gives us a lower bound for nC , which completes the proof that the corresponding family is bounded. Now to show that the family of cokernels is bounded, we will show 1. there is a common upper-bound to the index of regularity; and 2. the Hilbert polynomials of this family take on only a finite number of values Using the Leray spectral sequence again, we note that for all P ∈ Ppi, H1(Y, P ) = H1(X,pi∗P ) = 0, thus, all P ∈ Ppi are 1-regular. To show this family is bounded, it remains to find upper and lower bounds for the coefficients of the Hilbert polynomial of a general object. As above, there are only a finite number of options for the support curve of P . This yields upper and lower bounds on the linear coefficient of the Hilbert polynomial. We know that χ(G) = χ(P ) + χ(OC). Since χ(OC) is bounded, and χ(G) is fixed, so too must χ(P ) be bounded. This completes the proof that the sheaves underlying a pi-stable pair of fixed K-class forms a bounded family of sheaves. This also completes the proof that µpi is a permissible weak stability condition. 28 Chapter 4 The torsion pair and the stability condition In this section, we show that Ppi may be conveniently expressed in terms of the stability condition, and similarly for Qpi. First we give a rapid intro- duction to the modern Harder-Narasimhan property, a generalization of the Harder-Narasimhan filtration of [22]. Definition 4.1. A weak stability condition (T, τ,≤) on C is said to have the Harder-Narasimhan property if for every sheaf G, there exists a unique filtration of G 0 = HNτ (G)0 ⊂ HNτ (G)1 ⊂ · · · ⊂ HNτ (G)N−1 ⊂ HNτ (G)N = G (where the inclusions are strict) such that the quotients Qi = HNτ (G)i/HNτ (G)i−1 are τ -semistable and τ(Qi) > τ(Qi+1) for all i > 0. When it is clear from context, most of the notation will be suppressed, and we will denote the Harder-Narasimhan filtration of G with respect to τ by 0 ⊂ G1 ⊂ G2 ⊂ · · · ⊂ GN−1 ⊂ GN = G. The Gi are called the filtered objects of the Harder-Narasimhan filtration, and the Qi are called the quotient objects. We borrow the following definition and theorem from Joyce [26]. In [24, §9], Joyce proves that the category of coherent sheaves satisfies assumptions 29 3.7 of [26]. This is enough for us to conclude that the assumptions are also true of C the category of coherent sheaves supported in dimension one or less. Theorem 4.2 ([26], Theorem 4.4). Let (T, τ,≤) be a weak stability con- dition on an abelian category A. If A is Noetherian and τ -artinian, then (T, τ,≤) has the Harder-Narasimhan property. Corollary 4.3. The weak stability condition µpi on C has the Harder- Narasimhan property. Proof: The category C is Noetherian because it is a subcategory of the cat- egory of coherent sheaves, which is Noetherian. Corollary 3.10 proves that C is µpi-artinian. When we refer to the Harder-Narasimhan filtration in what follows, we will always be referring to the filtration with respect to the stability condition µpi. We present some notation before we state and prove the main result of this section. Recall that our slope function µpi takes values in the lexico- graphically ordered set (−∞,+∞]×(−∞,+∞]. To avoid awkwardly writing the ordered pairs (+∞,+∞) and (+∞, 0) through-out, let us denote ∞ := (+∞,+∞), and ∞ 2 := (+∞, 0). Given an interval I ⊂ (−∞,+∞] × (−∞,+∞], we define SS(I) ⊂ C to be the full subcategory of zero objects together with those one-dimensional sheaves whose Harder-Narasimhan quotients have µpi-value in the interval I. If a, b ∈ (−∞,+∞]× (−∞,+∞] such that a < b, then we denote the closed interval between a and b by a ≤ ≤ b, and similarly for open, half-open, etc. intervals. 30 Lemma 4.4. Ppi = SS( ≥ ∞2 ), and Qpi = SS( < ∞2 ). Proof: First we will show that Ppi ⊂ SS( ≥ ∞2 ). Case 1: let P ∈ Ppi be semi-stable. We will show that P ∈ SS( ≥ ∞2 ). Since P is semi-stable, it suffices to show that µpi(P ) ≥ ∞2 . Now P ∈ Ppi implies that χ(P ) ≥ 0. By ampleness of L, we know βP · L ≥ 0. Hence µpi(P ) ≥ (+∞, 0). Case 2: Let P ∈ Ppi be general, let the following be its Harder-Narasimhan (HN) filtration, 0 = P0 ⊂ P1 ⊂ · · · ⊂ PN−1 ⊂ PN = P, and let Qi = PiPi−1 be the ith quotient; we must show that µpi(Qi) ≥ ∞2 . Since PN = P ∈ Ppi and Ppi is closed under quotients, it follows that QN ∈ Ppi. By definition of the HN filtration, QN is semi-stable, hence µpi(QN ) ≥ ∞2 and QN ∈ SS( ≥ ∞2 ) by the previous case. Another defining property is that µpi(Q1) > µpi(Q2) > · · · > µpi(QN ). Hence, µpi(Qi) ≥ ∞2 for all i, in other words, P ∈ SS( ≥ ∞2 ). Now we will show that SS( ≥ ∞2 ) ⊂ Ppi. Case 1: Let G ∈ SS( ≥ ∞2 ) be semi-stable. In this case, G ∈ SS( ≥ ∞ 2 ) implies that µpi(G) ≥ ∞2 . Since (Ppi,Qpi) is a torsion pair, for every sheaf there exists a uniquely associated short exact sequence, 0→ A→ G→ B → 0 where A ∈ Ppi and B ∈ Qpi. By the above, we know that A ∈ SS( ≥ ∞2 ). If A itself was semistable, then we could conclude immediately that µpi(A) ≥ ∞ 2 . Using this argument, and inducting on the length of the HN filtration of A, we may prove that µpi(Qi) ≥ µpi(Ai) ≥ µpiQi+1, where Qi denotes the ith quotient object of the HN filtration of A, and Ai denotes the ith filtered object. This implies that µpi(A) ≥ ∞2 in general. 31 We know that µpi(G) ≥ ∞2 = (+∞, 0) so χ(G) ≥ 0 by non-negativity of denominators in µpi. Similarly, χ(A) ≥ 0. By the see-saw property and the semi-stability of G, we know that µpi(A) ≤ µpi(G) ≤ µpi(B), hence µpi(B) ≥ ∞2 and as before we have χ(B) ≥ 0. Notice that G must be supported on a fibre of pi because µpi(G) ∈ {+∞} × (0,+∞]. Hence B is also supported on a fibre. We claim that this forces H0(B) = 0. For suppose there was a non-zero map OY → B. This would yield non-trivial 0→ OC → B where C is the support of the map OY → B. From the proof of proposition 2.11, we know that R1pi∗OC = 0, which implies that OC ∈ Ppi which contradicts the definition of Qpi. Hence H0(B) = 0. However, χ(B) ≥ 0 so dimH1(B) ≤ 0. This implies that H1(B) = 0, which implies that R1pi∗B = 0 (by the theorem of cohomology and base- change) and hence B ∈ Ppi. Since B ∈ Qpi we conclude that B = 0 and G = A ∈ Ppi. Case 2: Let G ∈ SS( ≥ ∞2 ) be general. We need to show that G ∈ Ppi. Let 0 = G0 ⊂ G1 ⊂ · · · ⊂ GN−1 ⊂ GN = G be the HN filtration of G, and let Qi = GiGi−1 denote the corresponding semistable quotients. By assumption, µpi(Qi) ≥ ∞2 ; notice that G1 = Q1, so we have that G1 is semistable and µpi(G1) ≥ ∞2 . The previous case then proves that for all i, Qi ∈ Ppi, so in particular Q1 = G1 ∈ Ppi. Now consider the following short exact sequence 0→ G1 → G2 → Q2 → 0 We know G1 ∈ Ppi and Q2 ∈ Ppi hence G2 ∈ Ppi. Proceeding inductively, we conclude G ∈ Ppi. This completes the proof that Ppi = SS( ≥ ∞2 ). Now we prove that Qpi = SS( < ∞2 ). First, we will show that SS( < ∞ 2 ) ⊂ Qpi. Case 1: Let G ∈ SS( < ∞2 ) be semi-stable. There is a unique short 32 exact sequence associated to G 0→ A→ G→ B → 0 such that A ∈ Ppi and B ∈ Qpi. We will show that A = 0. Since G is semi-stable, either A = 0 or µpi(A) ≤ µpi(G). Suppose that µpi(A) ≤ µpi(G). Notice µpi(G) < ∞2 so µpi(A) < ∞ 2 in this case. However, the proof that Ppi = SS( ≥ ∞2 ) tells us that if A ∈ Ppi then either A = 0 or µpi(A) ≥ ∞2 . Thus, A = 0. Case 2: Let G ∈ SS( < ∞2 ) be general. As above, we have that each of the quotients associated to the HN filtration is an element of Qpi. Since Qpi is closed under extension, we then get that each Gi is in Qpi hence G ∈ Qpi. Next, we will show that Qpi ⊂ SS( < ∞2 ). Case 1: Let B ∈ Qpi be semi-stable. Assume B 6= 0. We need to show that µpi(B) < ∞2 . This is equivalent to showing that either B is not supported on a fibre, or that χ(B) < 0. Assume that B is supported on a fibre. As before, in this case H0(B) = 0. Notice that if H1(B) = 0 then B ∈ Ppi and so B = 0. Since we have assumed B 6= 0, this forces us to assume H1(B) 6= 0. Hence χ(B) < 0 and µpi(B) < ∞2 . Case 2: Let B ∈ Qpi be general. Let 0 = B0 ⊂ B1 ⊂ · · · ⊂ BN−1 ⊂ BN = B be the HN filtration of B, and let Qi be the associated quotients. By defini- tion of the HN filtration, B1 = Q1 is semi-stable. Since Qpi is closed under taking sub-objects, we know that B1 ∈ Qpi. By the previous case, we know that µpi(B1) < ∞2 . However, by definition of the HN filtration, we know that µpi(Q1) > µpi(Q2) > · · · > µpi(QN ). Hence, µpi(Qi) < ∞2 for all i, i.e., B ∈ SS( < ∞2 ). Before moving on, let us collect a functorial property of the Harder- Narasimhan filtration (see [42]). It will tell us something about how it is 33 preserved by morphisms. LetG be a coherentOY -module, letGi ⊂ G denote the Harder-Narasimhan subsheaves of G. For α ∈ (−∞,+∞]× (−∞,+∞], we define G(α) = Gk if µpi(QGk ) ≥ α > µpi(QGk−1) G if µpi(QGi ) ≥ α for all i 0 if µpi(QGi ) < α for all i. Lemma 4.5. If f : A→ G is a map of sheaves on Y , then f(A(α)) ⊂ G(α). Proof: First, we prove by induction that if f(Ak) ⊂ G` and ` is minimal with this property, then µpi(QG` ) ≥ µpi(QAk ). For k = 0, there is nothing to prove. For k = 1, suppose we have f(A1) ⊂ G` where ` is minimal with respect to this property. Then we get a non-zero map A1 → QG` . By lemma 3.5, we have that µpi(A1) ≤ µpi(QG` ). For general k, suppose we have f(Ak) ⊂ G`, ` minimal. Consider the following short exact sequences 0 −−−−→ Ak−1 i−−−−→ Ak −−−−→ QAk −−−−→ 0 f y 0 −−−−→ G`−1 −−−−→ G` q−−−−→ QG` −−−−→ 0. If the composition q ◦ f ◦ i is zero, then we can define a non-zero map QAk → QG` which implies that µpi(QAk ) ≤ µpi(QG` ), as desired. If the composition q ◦ f ◦ i is non-zero, then we know that f(Ak−1) ⊂ G` and f(Ak−1) 6⊂ G`; by induction, we conclude that µpi(QG` ) ≥ µpi(QAk−1) > µpi(QAk ), as desired. So: f(Ak) ⊂ G`, where ` is minimal with this property, implies that µpi(QG` ) ≥ µpi(QAk ). We will finish proving the lemma using this fact. Fix an α ∈ (−∞,+∞] × (−∞,+∞]. By definition, A(α) = Ak for all indices k such that (QAk ) ≥ α > µpi(QAk−1). Fix such an index k. Choose the ` minimal such that f(Ak) ⊂ G`. Notice that we now have µpi(QG` ) ≥ µpi(QAk ) ≥ α, and thus G` ⊂ G(α) by definition. 34 The following lemma will be useful later. Lemma 4.6. SS(a ≤ ≤ b) is closed under extension. Proof: Let A and B ∈ SS(a ≤ ≤ b). Consider 0→ A i→ G j→ B → 0. First we show that G ∈ SS(a ≤ ). We know that A = A(a), so i(A(a)) = i(A) ⊂ G(a). If we choose k minimal such that µpi(QGk ) < a, then we also know that G(a) = Gk−1. Notice that the possibilities for GGk−1 are limited: either it is the zero sheaf, or it is an element of SS( < a). The inclusion i(A) ⊂ Gk−1 implies that B = Gi(A) maps onto GGk−1 ; denote the quotient map by ρ. Now consider the chain of inclusions and identities ρ(B(a)) ⊂ ( G Gk−1 )(a) ⊂ G Gk−1 = ρ(B); by assumption, ρ(B(a)) = ρ(B), so all of the inclusions are actually equali- ties. In particular, ( G Gk−1 )(a) = G Gk−1 , which means either GGk−1 is zero, or it is an element of SS(a ≤ ). By the above, we see that the only possibility is GGk−1 = 0 which means that G = G(a), i.e. G ∈ SS(a ≤ ). Now we prove that G ∈ SS( ≤ b). Let 0 = G0 ⊂ G1 ⊂ · · · ⊂ GN−1 ⊂ GN = G denote the Harder-Narasimhan filtration of G, and let Qi be the associated quotients. Since G1 = Q1 and µpi(Q1) > µpi(Qi) for all i > 1, it suffices to prove that µpi(Q1) ≤ b. Consider the short-exact sequence 0→ A ∩G1 → G1 → j(G1)→ 0. Since the quotients associated to the Harder-Narasimhan filtration are semi- 35 stable, and since G1 = Q1, we know that G1 is semi-stable. This implies that µpi(G1) ≤ µpi(j(G1)), hence it suffices to prove that µpi(j(G1)) ≤ b. Let B′ be any subsheaf of B; we will show that µpi(B′) ≤ b. We know that for any c > b, B′(c) ⊂ B(c) = 0, which implies that for all c > b, B′(c) = 0. Hence B′ ∈ SS( ≤ b) and µpi(B′) ≤ b. This allows us to conclude that µpi(G1) ≤ b. 36 Chapter 5 The motivic Hall algebra Here we provide a quick summary of the constructions and results of Bridge- land’s papers [11], [12] (which came into existence as a gentle introduction to part of Joyce’s theory of motivic Hall algebras [24], [25], [26], [27] ). Let S be a stack, locally of finite type over C and with affine stabilizers. Definition 5.1. The relative Grothendieck group K(St /S) of stacks over S is the Q-vector space spanned by symbols [ T m→ S] (where T is a finite-type stack andm is a morphism), subject to the following relations. a) [ T → S] = [U → S] + [F → S] where U is an open substack of T and F is the corresponding closed complement. b) [ T1 s◦f→ S] = [T2 s◦g→ S], if T1 f→ B and T2 g→ B are Zariski fibrations1 over B with identical fibres, and B s→ S is a morphism of stacks. c) [ T a→ S] = [T ′ b→ S] if there exists a commutative diagram T c // a 44 44 44 T ′ b S such that the associated map on C-points T (C) c→ T ′(C) is an equiv- alence of categories. 1a Zariski-local product space 37 The vector space K(St /S) is a K(St /SpecC)-module, whose action we now describe. Let [ A → SpecC] = [A] ∈ K(St /SpecC), and let[ T m→ S] ∈ K(St /S). Then we define [ A ] · [T m→ S] = [A×SpecC T f−→ S], where f is the composition of the projection A×SpecC T → T and the map T → S. We are most interested in the case where S = C, the stack of coherent sheaves on Y supported in dimension one or less. We denote this stack by M, and we denote K(St /M) by H(C). The vector space H(C) is the motivic Hall algebra; let us justify the name by endowing it with the structure of an algebra. First, we define M(2) to be the stack of short-exact sequence of sheaves on M. Now, given [A → M] and [B → M] we define the convolution product [ A→M] ∗ [B →M] to be [Z →M] where Z c◦g→ M is defined by the following Cartesian diagram: Z g−−−−→ M(2) c−−−−→ My y(l,r) A×B −−−−→ M×M. The morphisms l, r are the “left hand” and “right hand” morphisms, which project a short exact sequence to its left-most (resp. right-most) non-zero entry. The morphism c is the “centre” morphism. Intuitively, given families of sheaves A → M and B → M their product in the Hall algebra is the family Z →M parametrizing extensions of objects of B by objects of A. The motivic Hall algebra is useful tool. It holds enough information to allow us to retrieve Euler characteristics, yet is flexible enough to produce decompositions of elements in terms of extensions. We will describe an “in- tegration” map on H(C) taking values in the ring of polynomials. Equations among elements of H(C) will be integrated to yield equations of polynomials. This entire framework will then be souped-up to incorporate Laurent series, and our theorem will be the result of applying the souped-up integration 38 map to equations in the souped-up Hall algebra. In [11], Bridgeland introduces regular elements. Let K(Var/C) denote the relative Grothendieck group of varieties over C (cf. definition 5.1). Let L denote the element [A1 → C], the Tate motive. Consider the maps of commutative rings, K(Var/C)→ K(Var/C)[L−1]→ K(St/C), and recall from [11] that H(C) is algebra overK(St/C). Define aK(Var/C)[L−1]- module Hreg(C) ⊂ H(C) to be the span of classes of maps [V f→ M] with V a variety. We call an element of H(C) regular if it lies in this submodule. The following result is theorem 5.1 of [11]. Theorem 5.2. The submodule of regular elements is closed under the con- volution product: Hreg(C) ∗Hreg(C) ⊂ Hreg(C), and is therefore a K(Var/C)[L−1]-algebra. Moreover the quotient Hsc(C) = Hreg(C)/(L− 1)Hreg(C), is a commutative K(Var/C)-algebra. Bridgeland equips Hsc(C) with a Poisson bracket, defined by {f, g} = f ∗ g − g ∗ f L− 1 . The integration map I is defined on Hsc(C). Now we work toward the poly- nomial ring in which it takes values. Recall that K(Y ) is the numerical K-theory of Y . Recall ∆ ⊂ F1K(Y ) is the effective cone of F1K(Y ), that is, the collection of elements of the form [F ] where F is a one-dimensional sheaf. Define a ring C[Γ] to be the vector 39 space spanned by symbols xα for α ∈ ∆ and defining the multiplication by xα · xβ = xα+β. We equip C[∆] with the trivial Poisson bracket. We are now ready for the following theorem. Theorem 5.3 (5.1 of [12]). There exists a Poisson algebra homomorphism I : Hsc(C)→ C[∆] such that I( [ Z f→Mα ] ) = χ(Z, f∗(ν))xα, where ν :M→ Z is Behrend’s microlocal function of M, and Mα denotes the component of M with fixed Chern character α. 40 Chapter 6 Equations in the infinite-type Hall algebra and the fake proof For the sake of exposition only, we follow [12] and [17] by introducing an infinite-type version of the Hall algebra. This has the benefit of allowing non-finite-type stacks, but the devastating draw-back of not admitting an integration map. We use it because it will allow us to temporarily work without having to think about convergence of power series. Also, many of the arguments will be used again later. We end this chapter with a fake proof of our main result. It is our hope that this fake proof helps the reader to navigate the true one in the following chapter. The infinite-type Hall algebra is defined by considering symbols as in definition 5.1, but with T assumed only to be locally of finite type over C, and use relations as before, except that we do not use relation (a). (Admitting relation (a) in this case would make every infinite-type Hall algebra trivial). We denote it by H∞(C). Given a substack N ⊂M, we let 1N = [N i→M] denote the inclusion i : N → M. Pulling back the morphism (2.13) to N ⊂ M gives a stack denoted N (O) with a morphism N (O) q→ N , and hence an element 1NO = [N (O) q→M] ∈ H∞(C). 41 For example, Ppi and Qpi are full subcategories of C, and define substacks of M (see lemma 2.16), which we abusively denote with the same letters, Ppi,Qpi ⊂ M. These substacks define elements of the infinite-type Hall algebra, 1Ppi , 1Qpi ∈ H∞(C). Other examples include H = [Hilb(Y )→M], Hexc = [Hilbexc →M], and Hpi = [pi-Hilb(Y )→M] ∈ H∞(C), where Hilbexc denotes the Hilbert scheme of curves supported on fibres of pi, and the map to M is given by taking OY → G to G. Note that all Hilbert schemes are restricted to the components parametrizing sheaves G of dimension one. Lemma 6.1. 1C = 1Ppi ∗ 1Qpi This lemma reflects the fact that (Ppi,Qpi) is a torsion pair. Proof: Form the following Cartesian diagram: Z f−−−−→ M(2) b−−−−→ My y(a1,a2) Ppi ×Qpi i−−−−→ M×M. (6.2) By lemma A.1 [11], the groupoid of T -valued points of Z can be described as follows. The objects are short exact sequences of T -flat sheaves on T ×Y of the form 0→ A→ G→ B → 0 such that A and B define families of sheaves on Y lying in the subcategories Ppi and Qpi respectively. The morphisms are isomorphisms of short exact 42 sequences. The composition, g = b ◦ f : Z →M sends a short exact sequence to the object G. Since Ppi and Qpi are sub- categories of C, it follows that the composition factors through C. This morphism induces an equivalence on C-valued points because of the torsion pair property: every object G of C fits into a unique short exact sequence of the form (6.2). Thus, the identity follows from the relations in the infinite Hall algebra. We will need a framed version of the previous lemma. Lemma 6.3. 1OC = 1 O Ppi ∗ 1OQpi . Proof: Form the following Cartesian diagram: U p // V j // M(2) b // (a1,a2) M Ppi(O)×Qpi(O) q×id // Ppi ×Qpi(O) //M×M. Then 1OPpi ∗ 1OQpi is represented by the composite map b ◦ j ◦ p : U →M. Note that, by lemma 2.5 of [12], the map Ppi(O)→ Ppi is a Zariski fibration with fibre H0(P ) over a point P ∈ Ppi. By pullback, the same is true of the morphism p. The groupoid of T -valued points of V can be described as follows. The objects are short exact sequences of T -flat sheaves on T × Y of the form 0→ P → G→ B → 0 such that P and B define flat families of objects in Ppi and Qpi respectively, together with a map OT×Y → B. We represent the objects of V as diagrams 43 of the form: OT×Yy 0 −−−−→ P −−−−→ G −−−−→ B −−−−→ 0. Consider the stack Z from lemma 6.1 with its map Z →M. Form the diagram W h−−−−→ C(O)y qy Z g−−−−→ C. Since g induces an equivalence on C-valued points, so does h, so that the element 1OC can be represented by the map q ◦ h. We represent the objects of W as diagrams of the form: OT×Yyδ 0 −−−−→ P −−−−→ G β−−−−→ B −−−−→ 0. Setting γ = β◦δ defines a map of stacksW → V , which is a Zariski fibration with fibre an affine model of the vector space H0(P ) over a sheaf P . Now, since H1(P ) = 0, U → V is a Zariski fibration with fibre H0(P ), hence they represent the same element of the Hall algebra, namely Qpi. Lemma 6.4. 1OC = H ∗ 1C This is lemma 4.3 of [12]. Intuitively, this amounts to the fact that every map O γ→ G factors uniquely into a surjection O → im(γ) and an inclusion im(γ) → G. The following lemma is a restriction of the previous to the substack Ppi. Lemma 6.5. 1OPpi = Hexc ∗ 1Ppi 44 Proof: Form the Cartesian diagram: Z −−−−→ M(2) b−−−−→ My y(a1,a2) Hexc × Ppi i−−−−→ M×M. The groupoid of T -valued points of Z may be described as follows. The objects are short exact sequences of T -flat sheaves of T × Y 0→ A→ G→ B → 0 such that B ∈ Ppi, and A is supported on exceptional fibres, together with an epimorphism OY → A. We can represent these objects as diagrams of the form OT×Y γ 0 // A α // G β // B // 0. Since OY → A is an epimorphism, we know that A is of the form OC for some one-dimensional subscheme C of Y . By the proof of proposition 2.11, we know that Rpi∗OC = 0. Since A = OC has exceptional support, it follows that pi∗A is a zero-dimensional sheaf, hence A ∈ Ppi. Since B ∈ Ppi by design, and since Ppi is closed under taking extensions, we conclude that in any such short exact sequence, G ∈ Ppi. There is a map h : Z → Ppi(O) sending the above diagram to the composite map δ = α ◦ γ : OT×Y → G. This morphism h fits into a commuting diagram of stacks Z h // b◦f 66 66 66 6 Ppi(O) q M 45 We argue that the map h then induces an equivalence on C-valued points. Suppose OT×Y δ→ G is an arbitrary map of sheaves, with G defining a family of sheaves in Ppi. Then we get the following diagram: OT×Y γ 0 // im(δ) // G // coker(δ) // 0. Since G ∈ Ppi we know that the one-dimensional component of its sup- port is exceptional, hence im(δ) is also exceptional, so that OT×Y → im(δ) defines a family of objects in Hilbexc. As well, we know that Ppi is closed under taking quotients, so coker(δ) is in Ppi. This completes the proof. Morally, the next lemma is similar to lemma 6.4 sinceHpi may be thought of as the surjections OY → G in a tilt of the abelian category generated by O and C. We provide a direct proof since we have not constructed this tilt. Lemma 6.6. 1OQpi = Hpi ∗ 1Qpi . Proof: Form the following Cartesian diagram: Z f−−−−→ M(2) b−−−−→ My y(a1,a2) Hpi ×Qpi i−−−−→ M×M The groupoid of T -valued points of Z is described as follows. The objects are short exact sequences of T -flat sheaves on T × Y 0→ A→ G→ B → 0 with the property that G ∈ Qpi, together with a map OT×Y → A that pulls back to a pi-stable pair OY → At for every t ∈ T . We can represent these objects as diagrams of the form: 46 OT×Y γ 0 // A α // G β // B // 0. (6.7) Since A and B are objects of Qpi, and Qpi is closed under extensions, we conclude G ∈ Qpi. Thus, there is a map h : Z → Qpi(O) sending the above diagram to the composite map δ = α ◦ γ : OT×Y → G. This map h fits into a commuting diagram of stacks Z h // b◦f 66 66 66 6 Qpi(O) q M (6.8) The map h then induces an equivalence on C-valued points because of the following argument. Let OY σ→ G be an arbitrary map, with G ∈ Qpi. We need to produce a diagram, OY γ 0 // A α // G β // B // 0, with OY → A a pi-stable pair, B ∈ Qpi, and α ◦ γ = σ. Consider the cokernel K of σ. Since (Ppi,Qpi) is a torsion pair, we know that K fits into a short exact sequence 0→ P → K → Q→ 0 where P ∈ Ppi and Q ∈ Qpi. Let G c→ K be the canonical map from G to the cokernel K of σ, and let G d→ Q be the composition of G → K and 47 K → Q. Define A to be the kernel of d. Consider the following diagram. 0 // OY = // OY // σ 0 0 // A // G d // c Q = // 0 0 // P // K // Q // 0 0 0 0. We know that Q ∈ Qpi, and a diagram chase proves that the dotted vertical morphisms exist and that P is the cokernel of OY → A. The sheaf A is a subsheaf of G ∈ Qpi, and Qpi is closed under taking subsheaves, so A ∈ Qpi. This proves that OY → A is a pi-stable pair. This proves that the map h is surjective on C-valued points. It remains to show that h is injective on C-valued points. Now let γ : OY → G be an arbitrary map with G ∈ Qpi. Since Qpi is closed under subobjects, the image of γ is an object of Qpi, im(γ) ∈ Qpi. Now we can use the description of Qpi in terms of stability to show that G im(γ) is also in Qpi. This proves that the map h is injective on C-valued points. This completes the proof. We end this section by giving a fake proof of theorem 1.6 that depends on a fake integration map. In fact, no such integration map is known to exist, but if there was one, the proof our theorem would be simpler. As it stands, we have a chapter dedicated to convergence issues to get around the fact that no such integration map exists on the infinite type Hall algebra. It is our hope that this fake proof will make the true one easier to follow. Fake proof: By lemma 6.6, we have Hpi ∗ 1Qpi = 1OQpi . (6.9) 48 Lemma 4.3 of [12] proves H ∗ 1C = 1OC . Using lemma 6.1, we may rewrite H ∗ 1C : H ∗ 1C = H ∗ 1Ppi ∗ 1Qpi . Lemma 6.3 allows us to write 1OC = 1 O Ppi ∗ 1OQpi . Putting these together yields H ∗ 1Ppi ∗ 1Qpi = 1OPpi ∗ 1OQpi . Applying lemma 6.5 H ∗ 1Ppi ∗ 1Qpi = Hexc ∗ 1Ppi ∗ 1OQpi . Using equation 6.9, we get H ∗ 1Ppi ∗ 1Qpi = Hexc ∗ 1Ppi ∗ Hpi ∗ 1Qpi . Now, for reasons we will explain in the next section, 1Ppi and 1Qpi are in- vertible in the Hall algebra. We may therefore cancel the copies of 1Qpi and isolate H. H = Hexc ∗ 1Ppi ∗ Hpi ∗ 1−1Ppi . The elements H,Hexc,Hpi all lie in the subalgebra Hreg(C) since they are represented by schemes. As we will see in the next section, conjugation by 1Ppi induces a Poisson homomorphism of Hreg(C) of the form: identity + terms expressed in the Poisson bracket. Since the Poisson bracket of the polynomial ring is trivial, these terms vanish when we apply the fake integration map, and we are left with I(H) = I(Hexc) · I(Hpi). 49 Up to signs arising from lemma 2.18, the “polynomials” I(H), I(Hexc), and I(Hpi) are the generating series of DT(Y ), DTexc(Y ), and pi-PT(Y ), respectively, and we see that the above equation is the formula claimed in theorem 1.6. or I(H) I(Hexc) = I(H pi) which is exactly the statement of our theorem. unionsq The true proof will follow precisely these steps, fully justified, and with the appropriate convergence arguments. The next chapter describes the Laurent Hall algebra, which does have an integration map. 50 Chapter 7 Equations in the Laurent Hall algebra and the true proof Laurent subsets In this section, we will formally modify the algebra H(C) and its integration map so that the modified integration map on the modified algebra takes valued in power series. This section is a summary of sections 5.2 and 5.3 of [12]. Definition 7.1. A subset S ⊂ ∆ is Laurent if for all β ∈ N1(Y ), the collection of elements of the form (β, n) ∈ S is such that n is bounded below. Let Φ denote the set of all Laurent subsets. It has the following proper- ties: 1. if S, T ∈ Φ then so it S + T = {α+ β : α ∈ S, β ∈ T} 2. if S, T ∈ Φ, and α ∈ ∆ then there are only finitely many ways to write α = β + γ such that β ∈ S and γ ∈ T . Given a ring A graded by ∆, A = ⊕ γ∈∆Aγ , we can use the Laurent subset to define a new algebra, which we will denote AΦ. Elements of AΦ are of the form a = ∑ γ∈S aγ 51 where S ∈ Φ, and aγ ∈ Aγ ⊂ A. Given an element a ∈ AΦ as above, we define ργ(a) = aγ ∈ A. (Here, our notation differs from [12], since we are using the symbol pi for the map pi : Y → X.) The projection operator ρ allows us to define a product ∗ on AΦ by: ργ(a ∗ b) = ∑ γ=α+β ρα(a) ∗ ρβ(b). AΦ admits a natural topology that may be identified by declaring a sequence (aj)j∈N ⊂ AΦ to be convergent if for any (β, n) ∈ ∆, there exists an integer K such that for all m < n i, j > K ⇒ ρ(β,m)(ai) = ρ(β,m)(aj). Lemma 7.2. If A is a C-algebra and a ∈ AΦ satisfies ρ0(a) = 0 then any series ∑ j≥1 cja j with coefficients cj ∈ C is convergent in the topological ring AΦ. See Lemma 5.3 of [12] for a proof. Given two ∆-graded algebras A and B, and a morphism f : A→ B that preserves the ∆-grading, we get an induced continuous map fΦ : AΦ → BΦ by defining ργ(fΦ(a)) = f(ργ(a)). Applying this process to the map of ∆-graded algebras I : Hsc(C)→ C[∆] yields a continuous map IΦ : Hsc(C)Φ → C[∆]Φ. We call Hsc(C)Φ the Laurent Hall algebra; it, too, is equipped with a Poisson bracket. Definition 7.3. A morphism of stacks f :W → C is Φ-finite if a) Wα = f−1(Cα) is of finite type for all α ∈ ∆, and b) there is a Laurent subset S ⊂ ∆ such that Wα is empty unless α ∈ S. 52 A Φ-finite morphism of stacks f :W → C defines an element of Hsc(C)Φ by the formal sum ∑ α∈S [Wα f→ C]. In [12], it is shown that [Hilb → M] is Φ-finite, and hence defines an element in Hsc(C)Φ. Lemma 7.4. The maps Hilbpi → C, Hilb→ C, Hilbexc → C, are Φ-finite. The corresponding elements Hpi,H and Hexc of Hsc(C)Φ satisfy IΦ(Hpi) = ∑ (β,n)∈∆ (−1)n pi-PT(β, n)xβqn = pi-PT(Y )(x,−q), where we have written xβ = q(β,0) and q = q(0,1). Similarly, IΦ(H) = ∑ (β,n)∈∆ (−1)nDT(β, n)xβqn = DT(Y )(x,−q) IΦ(Hexc) = ∑ (β,n)∈∆,pi∗β=0 (−1)nDT (β, n)xβqn = DTexc(Y )(x,−q). Proof: Lemma 2.16 proves that pi-Hilb is locally of finite type, and is of finite type once the Chern character is fixed. As well, the set of elements α ∈ ∆ for which pi-Hilbα is non-empty is Laurent. To prove this, it suffices to show that for any curve class β, there exists an integer N such that for any n < N , the moduli space pi-Hilb(β,n) is empty. Fix a curve class β, and consider all pi-stablepairs OY → G in that class. There is the associated short exact sequence, 0→ OC → G→ P → 0 and since there are only finitely many decompositions β = β1+ β2 of β into a sum of effective curve classes, we lose no generality in fixing the curve class of OC and P . Now the structure sheaf OC lives in a Hilbert scheme, and the set of elements (β1, n1) ∈ ∆ for which Hilb(β1,n1) is non-empty is Laurent, so 53 there is a “minimal” Euler characteristic of OC , which we denote by N1. As for P , the Leray spectral sequence shows that χ(P ) ≥ 0 (see lemma 3.16). This proves that we may take N = N1, and for any n < N , pi-Hilb(β,n) is empty. The formulae then follow from lemma 2.18 and Behrend’s description of DT invariants as a weighted Euler characteristic. Hilbexc is a subscheme of Hilb, so the desired properties follow from [12, lemma 5.5]. Lemma 7.5. Let I ⊂ (−∞,+∞]× (−∞,+∞] be an interval bounded from below. Then 1SS(I) → C is Φ-finite. Proof: Since this is holds for Gieseker stability ([23, theorem 3.3.7]), it suffices to prove that for any b = (b1, b2) ∈ (−∞,+∞] × (−∞,+∞] there exists a number M such that the family of all G with µpi(G) ≥ b satisfies µ(G) ≥M . Here, µ stands for Gieseker slope stability, namely µ(G) = χ(G) β · L . Case 1: χ(G) > 0 Here, we have 0 < χ(G) β · L = µ(G), so we may take M = 0 in this case. Case 2: χ(G) < 0 Now, since β · H̃ ≤ β · L and χ(G) < 0, we have b1 ≤ χ(G) β · H̃ ≤ χ(G) β · L . In this case, we may take M = b1. 54 Case 3: χ(G) = 0 In this case, µpi(G) is either 0 or +∞, so we may take M = 0 in this case. Case-by-case analysis reveals that we may use M = min{0, b1}. Equations in the Laurent Hall algebra In this section, following [12] we establish equations in Hsc(C)Φ, and ulti- mately prove theorem 1.6. Lemma 7.6. Let µ ∈ (−∞,+∞]× (−∞,+∞] such that µ < ∞2 . Then the following equality holds in Hsc(C)Φ: 1SS(µ≤≤∞) = 1Ppi ∗ 1SS(µ≤<∞2 ). Proof: Form the following Cartesian diagram: Z f // M(2) c // (l,r) M Ppi × SS(µ ≤ < ∞2 ) //M×M T -valued points of Z are short exact sequences 0 → A → G → B → 0 of T -flat sheaves on T × Y such that A defines a family of objects in Ppi and B a family in SS(µ ≤ < ∞2 ). By lemma 4.4, we know ∞2 ≤ µpi(A) ≤ ∞. Now by [12, lemma6.2], we know that G defines a family of objects in SS(µ ≤ ≤ ∞). Now let G ∈ SS(µ ≤ ≤ ∞). If G ∈ Ppi or SS(µ ≤ < ∞2 ) then we are done, since then G will be an extension where one term is zero (recall that all SS(a < < b) include the zero objects). Otherwise, let 0 = G0 ⊂ G1 ⊂ . . . ⊂ GN−1 ⊂ GN = G 55 be the Harder-Narasimhan filtration of G, let Qi be the associated quotients. Then there exists an index j ∈ N, 1 < j < N such that for all i < j, µpi(Qi) ≥ ∞ 2 and µpi(Qj) < ∞ 2 . Finally, by the uniqueness of the Harder-Narasimhan filtration, we have that Gj ∈ Ppi and G/Gj ∈ SS(µ ≤ < ∞2 ). Remark 7.7. The proof of this lemma is strikingly similar to the proof of Lemma 6.1. This is no coincidence. The above is actually just a minor refinement of Lemma 6.1 which says that we may cut off the tail end of Qpi and have the corresponding result still hold. As we go on, we will be less explicit about the proofs of lemmas when the argument has been already made in the infinite-type case. Lemma 7.8. Let µ ∈ (−∞,+∞]× (−∞,+∞]. Then, as µ→ −∞, we have H ∗ 1SS(µ≤≤∞) − 1OSS(µ≤≤∞) → 0. Proof: Fix (β, n) ∈ ∆, and consider (β,m) for m < n. There are only finitely many decompositions (β, n) = (β1, n1) + (β2, n2) such that both ρ(β1,n1)(H) and ρ(β2,n2)(1SS(µ≤≤∞)) are non-zero. This follows from the fact that there are only finitely many decompositions β = β1+β2 with both βi effective. Now for each fixed β, there exist finitely many n such that both ρ(β1,n1)(H) and ρ(β2,n2)(1SS(µ≤≤∞)) are non-zero). By the boundedness of the Hilbert scheme, we may assume that µ is small enough so that for any of the decompositions, β = β1 + β2, all points OY → A of Hilb(β1,n1) satisfy A ∈ SS(µ ≤ ≤ ∞). Consider a diagram of sheaves, OY γ 0 // A α // G β // B // 0 (7.9) with OY → A in Hilb(β1,n1) and ch([G]) = (β, n). Now G ∈ SS(µ ≤ ≤ ∞) if and only if B ∈ SS(µ ≤ ≤ ∞). Since 56 Bridgeland proves [12, prop 6.5] ρ(β,n)(H ∗ 1SS(µ≤≤∞)) = ρ(β,n)(1OSS(µ≤≤∞)), the claim is proven. Lemma 7.10. Let µ ∈ (−∞,+∞] × (−∞,+∞). Then, as µ → −∞, we have Hpi ∗ 1SS(µ≤<∞ 2 ) − 1OSS(µ≤<∞ 2 ) → 0. Proof: Fix (β, n) ∈ ∆, and consider (β,m) for m < n. There are only finitely many decompositions (β, n) = (β1, n1) + (β2, n2) such that both ρ(β1,n1)(Hpi) and ρ(β2,n2)(1SS(µ≤<∞2 )) are non-zero. By the boundedness of the moduli space of pi-stable pairs, we may assume that µ is small enough so that for any decompositions, β = β1+β2, all points OY → A of pi-Hilb(β1,n1) satisfy A ∈ SS(µ ≤ < ∞2 ). Consider a diagram of sheaves, OY γ 0 // A α // G β // B // 0 (7.11) with OY → A in pi-Hilb(β1, n1) and [G] = (β, n). Using that SS(I)∗SS(I) ⊂ SS(I), we see that B ∈ SS(µ ≤ < ∞2 ) if and only if G ∈ SS(µ ≤ < ∞2 ). Now since A ∈ Qpi and B ∈ Qpi, we have that G ∈ Qpi. Composing the map OY → A with the map A → G, yields a map OY → G; this represents an object of 1OQpi . The proof of lemma 6.6, that 1OQpi = Hpi ∗ 1Qpi , can be easily adapted to now prove that ρ(β,n)(Hpi ∗ 1SS(µ≤<∞ 2 )) = ρ(β,n)(1 O SS(µ≤<∞ 2 )), 57 using lemma 4.6. This completes the proof that Hpi ∗ 1SS(µ≤<∞ 2 ) − 1OSS(µ≤<∞ 2 ) → 0 as µ→ −∞. Proposition 7.12. We have the following equality in the Laurent Hall algebra, Hsc(C)Φ: H ∗ 1Ppi = Hexc ∗ 1Ppi ∗ Hpi. Proof: Using 1SS(µ≤≤∞) = 1Ppi ∗ 1SS(µ≤<∞2 ) and 1OSS(µ≤≤∞) = 1OP1pi ∗ 1OSS(µ≤<∞ 2 ), we can rewrite H ∗ 1SS(µ≤≤∞) − 1OSS(µ≤≤∞) → 0 as H ∗ 1Ppi ∗ 1SS(µ≤<∞2 ) − 1 O Ppi ∗ 1OSS(µ≤<∞2 ) → 0, as µ→ −∞. Multiplying Hpi ∗ 1SS(µ≤<∞ 2 )− 1OSS(µ≤<∞ 2 ) → 0 on the left by 1OPpi , and rewriting using 1OPpi = Hexc ∗ 1Ppi yields Hexc ∗ 1Ppi ∗ Hpi ∗ 1SS(µ≤<∞2 ) − 1 O Ppi ∗ 1OSS(µ≤<∞2 ) → 0 as µ→ −∞. Hence Hexc ∗ 1Ppi ∗ Hpi ∗ 1SS(µ≤<∞2 ) −H ∗ 1Ppi ∗ 1SS(µ≤<∞2 ) → 0 as µ → −∞. Since 1SS(µ≤<∞ 2 ) is invertible, we can cancel it from both sides: H ∗ 1Ppi = Hexc ∗ 1Ppi ∗ Hpi. 58 The proof of theorem 1.6 We first collect results. The next proposition is theorem 3.11 of [29], and is a very deep result whose proof depends on all the full power of the formalism of [24, 25, 26, 27, 28]. Proposition 7.13. For each slope µ ∈ ((−∞,−∞), (+∞,+∞)], we can write 1SS(µ) = exp(µ) ∈ H(C)Φ with νµ = [C∗] · µ ∈ Hreg(C)Φ a regular element. Proof: The proof is identical to that of [12, theorem 6.3]. Bridgeland uses Joyce’s machinery, which applies in our case just as it does in his. The following corollary corresponds to Bridgeland’s 6.4 [12]. Corollary 7.14. For any µ ∈ ((−∞,−∞), (+∞,+∞)], the element 1SS(µ) ∈ H(C)Φ is invertible, and the automorphism Ad1SS(µ) : H(C)Φ → H(C)Φ preserves the subring of regular elements. The induced Poisson automor- phism of Hsc(C) is given by Ad1SS(µ) = exp{ηµ,−}. Proof: The proof of this is identical to that of corollary 6.4 of [12]. Now we can prove theorem 1.6. We have H ∗ 1Ppi = Hexc ∗ 1Ppi ∗ Hpi. Rearranging yields H = Hexc ∗ 1Ppi ∗ Hpi ∗ (1Ppi)−1. By lemma 4.4, we can write 1Ppi = SS( ∞ 2 ≤ ≤ ∞), and by lemma 6.2 of 59 [12], we can write SS( ∞ 2 ≤ ≤ ∞) = ∏ ∞ 2 ≤µ≤∞ 1SS(µ). In [12, lemma 6.2], it is explained that given an interval J ⊂ (−∞,+∞] × (−∞,+∞] that is bounded below, and an increasing sequence of finite sub- sets V1 ⊂ V2 ⊂ . . . ⊂ J the sequence 1SS(Vj) in converges to 1SS(J), where 1SS(Vj) is defined to be∏ v∈Vj 1SS(v), where the product is taken in descending order of slope. So, letting J denote the interval of slopes between ∞2 and ∞, including ∞2 and excluding ∞, we can write H = Hexc ∗ lim finite V⊂J 1SS(µN ) ∗ . . . ∗ 1SS(µ1) ∗ Hpi ∗ (1SS(µ1))−1 ∗ · · · ∗ (1SS(µN ))−1, where µi enumerate all the elements of V . Using proposition and corollary , we can rewrite H = Hexc ∗ lim finite V⊂J exp({ηµN , exp{ηµN−1 , . . . exp{ηµ1 ,−} . . .}(Hpi). Now hitting this equation with the integration map yields IΦ(H) = IΦ(Hexc) · IΦ ( lim finite V⊂J exp({ηµN , exp{ηµN−1 , . . . exp{ηµ1 ,−} . . .}(Hpi) ) . The integration map commutes with limits since it is continuous, thus IΦ(H) = 60 IΦ(Hexc) · lim finite V⊂J IΦ ( exp({ηµN , exp({ηµN−1 , . . . exp({ηµ1 ,−}) . . .})(Hpi) ) . Now, the Poisson bracket is a commutator which is trivial in the ring of Laurent series, so it vanishes after applying the integration map, and we are left with IΦ(H) = IΦ(Hexc) · IΦ(Hpi). Applying lemma 7.4, we get DT(Y )(x,−q) = DTexc(Y )(x,−q) · pi-PT(Y )(x,−q) and substituting q for −q yields DT(Y ) = DTexc(Y ) · pi-PT(Y ), which is what we set out to prove. 61 Chapter 8 Conclusion Summary of results In this dissertation, we introduced the notion of a pi-stable pair, which may be thought of as a stable pair whose cokernel is allowed to be one-dimensional provided it is contained in the exceptional divisor of Y pi→ X and has vanish- ing higher derived push-forwards.. Following ideas from [12], we described pi-stable pairs in terms of a torsion pair (Ppi,Qpi). Then, we constructed a stability condition, and established a relationship between the stability con- dition and the torsion pair. Finally, after reviewing the construction of the various Hall algebras, we established equations of elements in the Laurent Hall algebra which integrate to give us the desired equality of generating series. Future research This dissertation is progress toward proving the DT crepant resolution con- jecture. The complete proof might take the following form. Let us consider the case of global quotients for simplicity, i.e., stacks of the form X =M/G. In this case, the derived category of X , Db(X ), is isomorphic to the G- equivariant derived category of the master space M , DbG(M). By the cel- ebrated results of Bridgeland, King, and Reid [13], we know that there is an equivalence Ψ between the derived category of Y and the G-equivariant derived category of M . 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