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California State University, Northridge Torsion CALIFORNIA STATE UNIVERSITY, NORTHRIDGE TORSION CLASSES OF COHERENT SHEAVES ON AN ELLIPTIC OR PRODUCT THREEFOLD A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics By Jeremy Keat-Wah Khoo August 2020 The thesis of Jeremy Keat-Wah Khoo is approved: Katherine Stevenson, Ph.D. Date Jerry D. Rosen, Ph.D. Date Jason Lo, Ph.D., Chair Date California State University, Northridge ii Table of Contents Signature page ii Abstract iv 1 Introduction 1 1.1 Our Methods . .1 1.2 Main Results . .2 2 Background Concepts 3 2.1 Concepts from Homological Algebra and Category Theory . .3 2.2 Concepts from Algebraic Geometry . .8 2.3 Concepts from Scheme theory . 10 3 Main Definitions and “Axioms” 13 3.1 The Variety X ................................. 13 3.2 Supports of Coherent Sheaves . 13 3.3 Dimension Subcategories of AX ....................... 14 3.4 Torsion Pairs . 14 3.5 The Relative Fourier-Mukai Transforms Φ; Φ^ ................ 15 3.6 The Product Threefold and Chern Classes . 16 4 Preliminary Results 18 5 Properties Characterizing Tij 23 6 Generating More Torsion Classes 30 6.1 Generalizing Lemma 4.2 . 30 6.2 “Second Generation” Torsion Classes . 34 7 The Torsion Class hC00;C20i 39 References 42 iii ABSTRACT TORSION CLASSES OF COHERENT SHEAVES ON AN ELLIPTIC OR PRODUCT THREEFOLD By Jeremy Keat-Wah Khoo Master of Science in Mathematics Let X be an elliptic threefold admitting a Weierstrass elliptic fibration. We extend the main results of Angeles, Lo, and Van Der Linden in [1] by providing explicit properties charac- terizing the coherent sheaves contained in the torsion classes constructed there. We utilize the relative Fourier-Mukai transform Φ on the derived category of coherent sheaves to de- scribe the cohomology objects of the transforms of coherent sheaves, and show that this, along with information on the dimension of coherent sheaves, is sufficient information to completely describe the main torsion classes. Our method of proof to show these character- izations is then generalized to produce more examples of torsion classes, and gives another method to produce torsion classes in the category of coherent sheaves. iv Chapter 1 Introduction An active area of research in algebraic geometry is the study of stability conditions on a triangulated category. One of the primary motivations for this comes from mathematical physics, as the mathematical analogue to the concept of π-stability for Dirichlet branes in string theory. Formulating the problem in terms of homological algebra and category theory reduced much of the problem to studying certain subcategories of the bounded derived category of coherent sheaves on a complex manifold. Bridgeland showed in [4] that defining a stability condition on a triangulated category is equivalent to defining a t-structure on the triangulated category, and then defining a stability function on the heart of the t-structure satisfying the Harder-Narasimhan or HN property. Therefore, in order to construct a stability condition, one method of approach is to first construct a t-structure. In the case where the triangulated category is the derived category of coherent sheaves, one way to construct a t-structure is tilting at a torsion pair. Given a torsion pair, a pair of full subcategories (T ; F) in the category of coherent sheaves satisfying certain properties, there is a corresponding t-structure in the derived category, and the heart of that t-structure could potentially have a stability condition defined on it. It is an open problem to determine if every smooth projective threefold has a stability condition on it1. This suggests a strategy towards defining a stability condition on the derived category. By constructing a large number of torsion classes on the category of coherent sheaves, each of which defines a distinct t-structure, we gain a larger number of candidates for defining a stability function, and therefore a stability condition. In turn, giving a process by which we can define more torsion classes could eventually lead to more progress. 1.1 Our Methods As suggested in [1], we will work with a smooth projective threefold X, admitting a Weierstrass elliptic fibration π : X ! B to a smooth projective surface. We will use a set of properties which encode information from homological algebra and algebraic geometry and treat them as “axioms”. Most of the work then reduces down to the application of these axioms to a given problem. One of our main methods is to take an instance of a problem, for example identifying properties of a coherent sheaf, and transform it into a problem in the derived category. We then push the problem back down into one of coherent sheaves using cohomology. While it may seem that a significant detour took place in trying to solve the problem, the process simplified much of the analysis so that little else aside from homological algebra 1See for example [3, Section 4] for a brief remark on the non-trivial nature of defining a stability condition . 1 and the axioms was necessary. In fact, the process allowed us to write down a small set of properties that coherent sheaves in certain torsion classes satisfy, which are both necessary and sufficient. 1.2 Main Results We consider a smooth projective threefold X admitting a Weierstrass elliptic fibration π : X ! B, and consider the category of coherent sheaves on X. We define a collection of subcategories Cij, and from these construct a collection of torsion classes Tij in the category of coherent sheaves. The first main result we present here is a classification scheme for these torsion classes Theorem 1.1. Let X = C × B be a product of a smooth elliptic curve and a K3 surface of Picard rank 1. Let AX be the category of coherent sheaves on X. The torsion classes Tij can be divided into three distinct types 1. T00; T12; T20; T32; T40; T52, which are subcategories of AX that are dimension subcat- egories; 2. T10, which is the intersection of a dimension subcategory and an existing torsion class; 3. T11; T30; T31; T50; T51, which are intersections of dimension subcategories. The torsion classes of type 3 are determined by the cohomology objects of transforms of their coherent sheaves. In particular, each Tij is determined by the dimensions of 0 1 H (ΦE);H (ΦE), for E 2 Tij, where Φ is the relative Fourier-Mukai transform defined b on D (AX ). We also give a smaller result that generalizes the construction used to give the classifi- cations for the torsion classes of type 2, give some consequences of the construction, and conjecture another method for constructing new torsion classes in the category of coherent sheaves on X. 2 Chapter 2 Background Concepts In this section we collect some of the relevant basic definitions and concepts that provide the background for the definitions and results that will follow later. The reader is assumed to have an understanding of basic algebra concepts such as groups, rings, modules, and homomorphisms of such structures. The reader is also assumed to have some exposure to more advanced concepts in ring theory such as noetherian rings, integral domains, ideals generated by a set, radical ideals, and other concepts from commutative algebra. 2.1 Concepts from Homological Algebra and Category Theory A chain complex (of abelian groups) C is a collection of abelian groups fCng, with n 2 Z, and group homormorphisms called differentials dn : Cn ! Cn−1 such that dn−1 ◦dn = 0 for all n 2 Z. A chain complex can be represented as a diagram dn+2 dn+1 dn dn−1 ::: Cn+1 Cn Cn−1 ::: which may extend infinitely in both directions. The condition on the differentials dn imply that im (dn) ⊂ ker(dn−1), and we define the degree n homology of C to be the quotient group Hn(C) = ker(dn)=im (dn+1). Similarly, a cochain complex (of abelian groups) C is a collection of abelian groups fCng with n 2 Z, and differentials dn : Cn ! Cn+1 such that dn+1 ◦ dn = 0 for all n 2 Z. A cochain complex has the same diagram representation but with the arrows reversed. We define the degree n cohomology of C to be the quotient n group H (C) = ker(dn)=im (dn−1). For the purposes of this paper, we will primarily be working with cochain complexes. A category C is determined by the following data: • A class of objects, Obj(C), which by abuse of notation is sometimes denoted C • For every pair of objects A; B 2 C, a class of morphisms, denoted HomC(A; B), whose elements are denoted by arrows A ! B. We will call the object A the source and B the target of the morphism. • For every A 2 C, an identity morphism idA 2 HomC(A; A) • For every triple of objects A; B; C 2 C, a composition of morphisms ◦ : Hom(A; B) × Hom(B; C) ! Hom(A; C) that is associative and satisfies the following property: if f 2 HomC(A; B), then idB ◦ f = f = f ◦ idA A subcategory of a category C is a collection of some of the objects of C, along with some 3 of the morphisms in C, that is itself a category. A subcategory B of C is full if for every pair of objects A; B 2 B, HomB(A; B) = HomC(A; B). In other words, B has “all” of the morphisms between any two objects A; B in B. Given two categories C and D, a covariant functor F is a rule assigning to each object A 2 C a corresponding object F (A) 2 D, and to each morphism f 2 HomC(A; B), a mor- phism F (f) 2 HomD(F (A);F (B)).
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