t-stabilities and t-structures on triangulated categories

Abstract

We propose the notion of stability on a triangulated category that is a generalization of the T. Bridgeland's stability data. We establish connections between stabilities and t-structures on a category and as application we get the classification of bounded tstructures on the category D b (Coh P 1 ).

Polynomial Bridgeland stability conditions and the large volume limit

“…This was called "stability data" or "t-stability" in [14]. If S D Z, this notion is equivalent to a bounded t-structure (see Bridgeland [10, Lemma 3.1]), and for S D R, it is a "slicing" as defined in [9].…”

Stability conditions on curves

“…For the curve of genus greater than one there is a simple solution, applying a technical lemma of Gorodentsev, Kuleshov and Rudakov ( [12], Lemma 7.2).…”

A Lie theoretical construction of a Landau–Ginzburg model without projective mirrors

“…Hence the only simple coherent sheaves on P 1 are the line bundles O P 1 (t ), t ∈ Z, and the sheaves O p , p ∈ P 1 . No pair of them, not even after a shift L 0 [−i ], L 1 [− j ] may be of this form: if p, q ∈ P 1 and p = q, then Ext i (O p , O q ) = 0 for all i , either Hom(R, L) = 0 or Ext 1 (L, R) = 0 for any line bundles L, R, Hom(O p , L) = 0 and dim Ext 1 (O p , L) = 1 for all p ∈ P 1 and any line bundle L. Since P 1 is a smooth curve, [10,Prop. 6.3] gives that every simple element of D b (P 1 ) is isomorphic to some F[−i ] with F a simple coherent sheaf on P 1 .…”