Bounded t-Structures on the Bounded Derived Category of Coherent Sheaves over a Weighted Projective Line

Abstract

We use recollement and HRS-tilt to describe bounded t-structures on the bounded derived category  ( ) of coherent sheaves over a weighted projective line of domestic or tubular type. We will see from our description that the combinatorics in the classification of bounded t-structures on  ( ) can be reduced to that in the classification of bounded t-structures on the bounded derived categories of finite dimensional right modules over representation-finite finite dimensional hereditary algebras.We obtain from the two theorems above certain bijective correspondence for those bounded t-structures whose heart is not of finite length. Note that any group of exact autoequivalences of  ( ) acts on the set of bounded t-structures on  ( ) by Φ(( ≤0 ,  ≥0 )) ∶= (Φ( ≤0 ), Φ( ≥0 )) for Φ ∈ and a bounded t-structure ( ≤0 ,  ≥0 ) on  ( ). In the following corollary, we deem ℤ as the group of exact autoequivalences generated by the translation functor of  ( ), which acts freely on the set of bounded t-structures on  ( ). Corollary 1.3 (Corollary 4.21).(1) If is of domestic type then there is a bijectionwhere  runs through all equivalence classes of proper collections of simple sheaves.(2) If is of tubular type then there is a bijectionwhere  runs through all equivalence classes of proper collections of simple sheaves.

Exchange graphs and Ext-quivers of hearts of tube categories

“…([31, Prop. 2.30]) Given a bounded t-structure (D ≤0 , D ≥0 ) on D = D b (T p ), there is unique (up to equivalence) proper collection S of simple objects in T p such that…”