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

Abstract

We prove an effective restriction theorem for stable vector bundles E on a smooth projective variety: $$E|_D$$ E | D is (semi)stable for all irreducible divisors $$D \in |kH|$$ D ∈ | k H | for all k greater than an explicit constant. As an application, we show that Mercat’s conjecture in any rank greater than 2 fails for curves lying on K3 surfaces. Our technique is to use wall-crossing with respect to (weak) Bridgeland stability conditions which we also use to reprove Camere’s result on slope stability of the tangent bundle of $${\mathbb {P}}^n$$ P n restricted to a K3 surface.

Stability condition on Calabi–Yau threefold of complete intersection of quadratic and quartic hypersurfaces

“…Then by the wall-crossing technique on the Bridgeland stability space on 𝑆 2,2 as in [Fey20], one gets a Clifford type inequality of F. 2. Using Feyzbakhsh's restriction theorem in [Fey22a], one recovers a Bogomolov-Gieseker type inequality on 𝑆 2,5 . Using Feyzbakhsh's restriction theorem again, a Bogomolov-Gieseker type inequality on 𝑋 5 is obtained.…”

“…For a stable vector bundle F on , by pushing forward along the embedding , one regards F as a torsion sheaf on . Then by the wall-crossing technique on the Bridgeland stability space on as in [Fey20], one gets a Clifford type inequality of F .Using Feyzbakhsh’s restriction theorem in [Fey22a], one recovers a Bogomolov–Gieseker type inequality on . Using Feyzbakhsh’s restriction theorem again, a Bogomolov–Gieseker type inequality on is obtained.By applying the Bogomolov–Gieseker type inequality on in step (2), one proves that for a Brill–Noether stable object E .…”

“…Using Feyzbakhsh’s restriction theorem in [Fey22a], one recovers a Bogomolov–Gieseker type inequality on . Using Feyzbakhsh’s restriction theorem again, a Bogomolov–Gieseker type inequality on is obtained.…”

Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

“…Since Pic(X) = Z.H, Lemma 3.5 of [Fey22] implies that I (H) is σ α,β=0 -stable for any α > 0. Thus I (H) is σ α,β -semistable along the numerical wall W (I (H), O X [1]) where I (H) and O X [1] have the same phase.…”

“…Since µ H (E) = 0 and E is Gieseker semistable, we get E ∈ Coh β (X) for β < 0, and by [BMS16, Lemma 2.7], the Ulrich bundle E is σ α,β -semistable for α 0. Theorem 3.1 of [Fey22] implies that E is σ α,β 0 -semistable for any α > 0 and…”