Bounded T-Structures On The Bounded Derived Category Of Coherent Sheaves Over A Weighted Projective Line

This article is on the classification of certain t-structures.The main aim is to de-scribe bounded t-structures on the bounded derived category Db(X)of coherent sheaves over a weighted projective line X of virtual genus≤1.We will see from our descrip-tion that the combinatorics in the classification of bounded t-structures on Db(X)can be reduced to that in the classification of bounded t-structures on the bounded derived cate-gories of finite dimensional right modules over representation-finite finite dimensional hereditary algebras.The essential part consists of Chapters 2-5.Chapter 2 contains some preparatory materials and results on t-structures.In §2.1,we recall basic definitions and properties of t-structures.In §2.2,we introduce width-bounded t-structures and then HRS-tilt,which is an important way of constructing t-structures.In §3.2-3.5,we recall recollements of triangulated categories,admissible subcategories,gluing t-structures and properties of glued t-structures.In §2.6,we recall Ext-projective objects,and use an exceptional Ext-projective object to establish a rec-ollement with which the given t-structure is compatible.In §2.7,we recall some facts on hereditary categories,including Happel-Ringel Lemma,and relate a silting object in the bounded derived category of a hereditary category to an excpetional sequence.In§2.8,we recall and prove some facts on t-structures on the bounded derived category of finitely generated modules over a finite dimensional algebra,in particular,we recall the part of Konig-Yang correspondence that algebraic t-structures(i.e.,bounded t-structures with length heart)are in bijection with equivalence classes of silting objects(for the bounded derived category of finite dimensional modules over a finite dimensional al?gebra).In §2.9,we describe bounded t-structures on the bounded derived category of finite dimensional nilpotent representations of a cyclic quiver.Chapter 3 contains preparatory materials and results on weighted projective lines.In §3.1,we recall basic definitions and facts on weighted projective lines.In §3.2,we recap Auslander-Reiten theory.In §3.3,we recall the classification and important prop?erties of vector bundles over a weighted projective line of virtual genus≤1.In §3.4,we recall descriptions of perpendicular categories of some exceptional sequence.In §3.5,we prove the non-vanishing of some morphism spaces in the category coh X of coher-ent sheaves over a weighted projective line X.In §3.5,we investigate full exceptional sequences in coh X,and prove the existence of certain nice terms in some situations.In§3.7,we give some preliminary descriptions of some torsion pairs in coh X,and estab-lish bijections between isoclasses of basic tilting sheaves,certain torsion pairs in coh X and certain bounded t-structures on the bounded derived category Db(X)of coh X,and finally we investigate the Noetherianness and the Artinanness of tilted hearts given by certain torsion pairs in coh X.Chapter 4 aims to describe bounded t-structures on the bounded derived category Db(X)of coherent sheaves over a weighted projective line X of virtual genus≤1.In§4.1,we investigate and describe bounded t-structures that can be restricted to bounded t-structures on the bounded derived category Db(coh0X)of the category coh0X of tor-sion sheaves.In §4.2,we investigate those bounded t-structures on Db(X)that can-not restrict to t-structures on Db(coh0X)even up to the action of the group of exact autoequivalences of Db(X).In particular,we prove that the heart of such a bounded t-structure is length and possesses only finitely many indecomposable objects,all of which are exceptional.In §4.3,we prove some properties possessed by silting objects in Db(X).This is mainly acquired via properties of full exceptional sequences obtained earlier and further yields information on bounded t-structures by virtue of Konig-Yang correspondence.In §4.4,we complete our description of bounded t-structures on Db(X),in which we mainly use HRS-tilt and recollement.In §4.5,we use our description of bounded t-structures to give a description of torsion pairs in coh X.Chapter 5 aims to prove a characterization of when the heart of a bounded t-structure(D≤0,D≥0)on Db(X)is derived equivalent to coh X using the right t-exactness of the Serre functor of Db(X),which gives an application of our main result(i.e,descrip-tion of bounded t-structures).We conjecture that this result holds for arbitrary weighted projective line and propose a potential approach at the end of this chapter.