An effective restriction theorem via wall-crossing and Mercat's conjecture

Abstract

We use wall-crossing with respect to Bridgeland stability conditions to prove slope-stability of restrictions of locally free sheaves to curves on the K3 surfaces. As a result, we find many new counterexamples to Mercat's conjecture for vector bundles of rank greater than two.

Walls and asymptotics for Bridgeland stability conditions on 3-folds

“…Figure 8 illustrates the typical situation described in Lemma 6. 13, and shows that it does arise. It is easy to see that the intersection with α = 0 must happen to the right of Γ − v,s because Γ − v,s is monotonic.…”

“…If E is asymptotically λ α,β,s -semistable along Γ − v,s , then E is a sheaf by Lemma 7.4 or Proposition 5. 13.…”

“…Bridgeland's notion of stability conditions on triangulated categories, introduced in [9] and [10], provides a new set of tools to study moduli spaces of sheaves on smooth projective varieties. Such tools have been successfully applied by many authors first to the study of sheaves on surfaces, for example, [1,2,4,5,13,14,15,21,39], and more recently on three-folds (especially P 3 ), see for instance [16,25,34,36]. One way to study moduli spaces of sheaves using Bridgeland stability spaces is to restrict attention to the so called geometric stability conditions parametrized by (a subset of) the upper half plane H. Once we know that the moduli space of Bridgeland stable objects is asymptotically given by the Gieseker semistable moduli space along an unbounded path we can try to locate all the points where the moduli space changes along this path (these isolated points are called walls) and compute the change to the moduli space.…”

Higher rank Clifford indices of curves on a K3 surface

“…We assume X is a K3 surface with Pic(X) = ZH and C ∈ |H| is a smooth curve of genus g. Denote by ι : C ֒→ X the embedding of the curve C into X. We first briefly recall the main result in [Fey16], which constructs semistable vector bundles on C by restricting vector bundles on X with low discriminant. By [BM14, Theorem 2.15], there exists a slope stable sheaf Ẽr on X with Chern character (r, H, g r − r).…”

Categorical Torelli theorems: results and open problems

“…simply-connected) Calabi-Yau threefold. The involved strategy makes use (among all) of a restriction lemma, first appeared in [47], which allows to reduce to show a Clifford type bound for the dimension of global sections of stable vector bundles on curves defined as complete intersections of two quadrics and a quintic hypersurface in P 4 . The existence of stability conditions is then obtained by using the stronger Bogomolov inequality to prove the following statement (see [10,Conjecture 4.1]): if E is σ s,q -semistable for a certain choice of (s, q) ∈ ∆, then E satisfies…”