Sheaves of t-structures and valuative criteria for stable complexes

Abstract

1.1.1. Results on sheaves of t-structures. In section 2 we are given a nondegenerate t-structure (D(X) ≤0 , D(X) ≥0 ) with heart C = D(X) ≤0 ∩ D(X) ≥0 (see section 1.2 for definitions and a brief introduction). For a smooth projective variety S with ample line bundle L we defineAssuming that C is noetherian we prove

Bridgeland-stable moduli spaces for $K$-trivial surfaces

“…, and so by the Hodge index theorem, we have Re(Z(E)) > 0, Re(Z(E[1])) < 0, and Z(E [1]) ∈ H, as desired.…”

“…Below the critical "wall" value t = 1 2 , we will have µ Z (O S (H)) > µ Z (i * O C (H)), exhibiting i * O C (H) as an unstable object of A # (!). The "replacement" stable object(s) will be of the form: Our moduli functor is based upon the generalized notion of a flat family we learned from Abramovich and Polishchuk [AP06]. One would, of course, like to have an a priori construction of moduli spaces of Bridgeland-stable objects via some sort of invariant theory argument, but the fact that we are not working exclusively with coherent sheaves makes it difficult to see how to make such a construction.…”

Lectures on Bridgeland Stability

“…The recent works [AS12, AHR15] on good quotients for Artin stacks may lead to a general answer to this question, though. Once a coarse moduli space exists, then separatedness and properness of the moduli space is a general result by Abramovich and Polishchuk [AP06], which we will review in Section 8.…”

“…It is interesting to note that the technique in [AP06] is also very useful in the study of the geometry of the moduli space itself. This will also be reviewed in Section 8.…”

“…The idea to show boundedness is to reduce to semistable sheaves and use boundedness from them. For openness, the key technical result is a construction by Abramovich-Polishchuk [AP06], which we will recall in Theorem 8.2. Then openness for Bridgeland semistable objects follows from the existence of relative Harder-Narasimhan filtrations for sheaves.…”