Stability conditions on higher dimensional projective varieties by Yucheng Liu B.S. in Mathematics, Zhejiang University M.S. in Mathematics, Zhejiang University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 15th, 2020 Dissertation directed by Emanuele Macr`ı Professor of Mathematics 1 Acknowledgements First and foremost, I would like to acknowledge my supervisor Emanuele Macr`ıfor his patient guidance and continued support. I am very grateful to him for taking me as a student when I barely know anything in algebraic geometry. Thanks go to Jonathan Mboyo Esole, Alina Marian, and Alexander Martsinkovsky for joining my defense committee and for their comments and suggestions for my dissertation. I am indebted to Alex Perry for his careful reading of Chapter 1 and first two sections of Chapter 3. His comments on grammar and mathematics are very beneficial. I would also like to thank Arend Bayer, Aaron Bertram, Chunyi Li, Paolo Stellari, Yukinobu Toda and Xiaolei Zhao for helpful discussions and suggestions. Special thanks to Hongyu Shi, without her love and support, this journey would be much less interesting and funny. Finally, I would like to thank my parents and brother for the consistent inspiring support and care. 2 Abstract of Dissertation In this thesis, we discuss the theory of stability conditions on higher dimensional projec- tive varieties. By the observation of polynomial structure on the global heart constructed by Abramovich and Polishchuk in [AP06] and [Pol07], we are able to establish some inequalities on the coefficients of the polynomial. These inequalities turn out to be useful in constructing stability conditions. ≤1 We provide a construction of stability conditions on DS (see 2.2.1 and 3.3.2 and for the notion). In particular, this gives us a family of stability conditions on X × C from a stability condition on X, where C is an integral curve. This establish the existence of stability conditions on product of curves over any algebraically closed field. 3 Table of Contents Acknowledgements2 Abstract of Dissertation3 Table of Contents4 Disclaimer 5 Introduction6 1 Stability conditions and sheaves of t-structures 10 1.1 Bridgeland Stability conditions........................... 10 1.2 Sheaf of t-structures and polynomial functions................... 16 2 Some mathematical structures on AS 30 2.1 Positivity of the coefficitens............................. 30 2.2 Abelian subcategories in AS ............................. 34 2.3 Weak stability conditions and torsion pairs..................... 42 3 Constructing stability conditions 50 3.1 Product over a curve................................. 50 3.2 Large volume limit and support property...................... 58 ≤1 3.3 Stability conditions on DS ............................. 68 4 Disclaimer Chapter 1 and the first two sections of Chapter 3 are adapted from the paper [Liu19]: \Stability conditions on product varieties". To appear in Journal f¨urdie reine und ange- wandte Mathematik. 5 Introduction The notion of stability conditions was introduced by Bridgeland in [Bri07], motivated by Douglas's work on D-branes and Π-stability (see [Dou02]). The theory was further studied by Kontsevich and Soibelman in [KS08]. In general, stability conditions are very difficult to construct: while we have a very good knowledge in the case of curves and surfaces (see [Mac07], [Bri08] and [AB13]), starting from 3-folds no example was known on varieties of general type or Calabi-Yau varieties in dimension 4 or higher (for Calabi-Yau threefolds, see [MP15], [BMS16] and [Li19]). In this thesis, we try to construct stability conditions for triangulated categories on higher dimensional projective varieties. In this thesis, all algebraic varieties are defined over a field k. The field k can be of any characteristics. We always assume that k is algebraically closed, but I believe this condition is not necessary, we will remark on how to prove the same results in the case when k is not algebraically closed, especially when k is a finite field. Let X be a projective variety, S be a smooth projective variety. And let σ = (A;Z) be a stability condition on the bounded derived category of coherent sheaves D(X). The following theorem is the main theorem of this thesis. Main Theorem. Assume that the image of the central charge Z is discrete. Then there ex- ≤1 ists a continuous family of stability conditions on DS , parametrized by R>0 ×R>0, associated with σ. ≤1 ≤1 For the notion of DS , please see Definition 3.3.2. In the case when S is a curve, DS = D(X ×S), the bounded derived category of coherent sheaves on X ×S, and we get the main 6 theorem in [Liu19]. There is one thing worth noticing, the main theorem includes the case when X is singular. In that case, though the triangulated category of perfect complexes P erf(X × S) is a very interesting subject to study, We will focus on D(X × S), the bounded derived categories of coherent sheaves on X × S. This is because of the obstruction of existence of bounded t-structure on P erf(X × S) (see [AGH19]). Theorem holds in more general situations. For instance, the triangulated category D(X) is replaced by an admissible subcategory D ⊂ D(X) and D(X × C) is replaced by the base + change category DC (see [BLMS17] and [BLM 19]). Special cases in dimension three were studied in [Kos18]. One of the interesting Corollaries is the following one. Main Corollary. Let C1; ··· ;Cn be smooth projective curves. Then stability conditions exist on D(C1 × · · · × Cn). In the case when n = 3, some related results appeared in [Sun19a] and [Sun19b]. The techniques are completely different. An important special case of Corollary is when all curves are elliptic curves. This gives examples of stability conditions on Calabi-Yau varieties of any dimension. In this case, the mirror symmetry version of this statement, for Fukaya categories of products of elliptic curves has been announced by Kontsevich in [Kon15]. The starting point of our construction is the construction by Abramovich and Polishchuk in [AP06] and [Pol07] of a heart of bounded t-structure on D(X × S), where S is any quasi- projective variety of finite type. Then we observed that there exists polynomial structure on the global heart. According to the degree of the polynomial, we find some abelian subcategories in the global heart. The next step is using the global slicing constructed by Bayer and Macr`ıin [BM14a]. This will give us some quadratic inequalities, which can be viewed as a generalization of Bogomolov-Gieseker inequalities on product varieties. In fact, we can use that method to 7 establish some Bogomolov-Gieseker inequalities in new cases. But unfortunately, I do not have enough time to include this application in the thesis, because I need to graduate by May this year. Besides the generalized Bogomolov-Gieseker inequalities, there are some other interesting mathematical structures on the global heart, for example, some abelian subcategories and torsion pairs in the global heart. In particular, we provide some interesting cuts of the global heart into torsion pairs. We can get some interesting inequalities after tilting with respect to that kind of torsion pairs. This is the content of chapter 2. The last part of this thesis is devoted to constructing stability conditions. We firstly give a construction of stability conditions in the case when S is a curve. In this case, the polynomial is a linear polynomial. Using the quadratic inequality in Chapter2, we can construct stability conditions on X × S. Then we apply the construction on some triangulated subcategories on the product variety when S is of higher dimension. Outline of the thesis The first chapter is almost identical to the first sections of paper [Liu19], it is devoted to introduce some basic notions and results on stability conditions, sheaves of t-structures and the polynomial structure. In the second chapter, we discuss some mathematical structures on the global heart and the generalized Hilbert polynomials. This includes some positivity of the coefficients of the polynomial, abelian subcategories of the global heart and torsion pairs. The construction of stability conditions is included in the last chapter. The main con- struction is included in Section 3.1. We prove a family of quadratic inequalities in Section 3.2, these quadratic inequalities are stronger than what we used in the construction. 8 Notations and Conventions In this paper, all varieties are integral algebraic varieties over a field k, a curve is such a variety of dimension 1. We will use D(X) rather than the usual notation Db(CohX) to denote the bounded derived categories of coherent sheaves on X. We set H = fz 2 C j Im(z) > 0g. All functors are derived unless otherwise specified. We use k(x) to denote the skyscraper sheaf supported on the point x. We set Arg(z), Im(z), and Re(z) to be the argument, the imaginary part, and the real part of a complex number z respectively. 9 Chapter 1 Stability conditions and sheaves of t-structures 1.1 Bridgeland Stability conditions In this section, we review the definition and some basic results on weak stability conditions (See [BBD82], [Pol07], [Bri08], [KS08] and [BMT14]). Definition 1.1.1. Let D be an triangulated category. A t-structure on D is a pair of full subcategories (D≤0; D≥0) satisfying the condition (i) and (ii) below. We denote D≤n = D≤0[−n], D≥n = D≥0[−n] for every n 2 Z. Then the conditions are: (i) Hom(X; Y ) = 0 for every X 2 D≤0 and Y 2 D≥1; (ii) D≤−1 ⊂ D≤0 and D≥1 ⊂ D≥0.