Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture

Abstract

ABSTRACT. We construct new t-structures on the derived category of coherent sheaves on smooth projective threefolds. We conjecture that they give Bridgeland stability conditions near the large volume limit. We show that this conjecture is equivalent to a BogomolovGieseker type inequality for the third Chern character of certain stable complexes. We also conjecture a stronger inequality, and prove it in the case of projective space, and for various examples.Finally, we prove a version of the classical Bogomolov-Gieseker inequality, not involving the third Chern character, for stable complexes.

A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold

“…If L[i] ∈ B α,β holds for a line bundle L and i ∈ {0, 1}, then L[i] is tilt-stable (see [5,Proposition 7.4.1]). By Lemma 4.3, we get immediately O(−1) [3], O [1], O(1) ∈ A α,β , A α,β [1] .…”

“…Moreover, C is isomorphic to the category of finitely generated modules over some finite dimensional algebra A and O(−1) [3], S(−1) [2], O [1], O(1) are the simple objects.…”

“…Interestingly, there are other applications of this inequality besides the construction of Bridgeland stability conditions. One of the most interesting consequences is Fujita's conjecture (see [BBMT11]). Macrì was able to prove the inequality in the case of P 3 in [Mac12], while Maciocia and Piyaratne managed to show it for principally polarized abelian threefolds of Picard rank one in [MP13a,MP13b].…”

The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

“…It depends on the choice of two divisor classes ω, B ∈ NS(X ) R with ω ample. It was shown that this conjecture would imply the existence of Bridgeland stability conditions, 2 and, in the companion paper [6], a version of an open case of Fujita's conjecture, on the very ampleness of adjoint line bundles on threefolds.…”

“…Until recent work by Maciocia and Piyaratne [29,30], they were only known to exist on threefolds whose derived category admits a full exceptional collection. Possible applications of stability conditions range from modularity properties of generating functions of Donaldson-Thomas invariants [43,45] to Reider-type theorems for adjoint linear series [6].…”