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

Mathematische Zeitschrift https://doi.org/10.1007/s00209-022-03036-1 Mathematische Zeitschrift An effective restriction theorem via wall-crossing and Mercat’s conjecture Soheyla Feyzbakhsh1 Received: 25 May 2021 / Accepted: 31 March 2022 © The Author(s) 2022 Abstract We prove an effective restriction theorem for stable vector bundles E on a smooth projective variety: E| D is (semi)stable for all irreducible divisors 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 Pn restricted to a K3 surface. 1 Introduction Inspired by the construction of Bridgeland stability conditions on K3 surfaces [7] Bayer et al. [5, 6] studied weak stability conditions on any smooth complex projective variety. In this paper, we use wall-crossing with respect to these weak stability conditions to prove an effective restriction theorem that expresses sufficient conditions on a slope-stable reflexive sheaf such that its restriction to a hypersurface remains stable. Restriction theorems provide us with the possibility of studying higher dimensional varieties via the geometry of their subvarieties. That is why they have been long-studied via different approaches; see [16, Chapter 7] for a survey. Let X be a smooth complex projective variety of dimension n ≥ 2 with an ample divisor H . For a μ-stable coherent sheaf E of positive rank on X , we define  2 ch1 (E).H n−1 ch2 (E).H n−2 − 2 , ch0 (E)H n ch0 (E)H n μmax (E) := max {μ(F) : F is a subsheaf of E with μ(F) < μ(E)} ,   μmin (E) := min μ(F  ) : F  is a proper quotient sheaf of E ,  (E) := (1) (2)   and δ(E):= min μmin (E) − μ(E), μ(E) − μmax (E) . B Soheyla Feyzbakhsh 1 Department of Mathematics, Imperial College, London SW7 2AZ, UK 123 S. Feyzbakhsh Theorem 1.1 Let E be a μ-stable reflexive sheaf on X of rank rk > 0. The restricted sheaf E| D for any irreducible divisor1 D ∈ |k H | is μ-semistable on D if   k2 k rk +2 (E)  (E) and k ≥ √ + −  (E) ≥ . (3) 2 4 δ(E) rk +1 Moreover, E| D is μ-stable on D if the inequalities in (3) are both strict. The μ-slope of a coherent sheaf and the notion of μ-(semi)stability are defined in Section 2. Note that if k ≥ 2δ(E), the conditions in (3) are equivalent to   (E) rk +2 k ≥ max √ (E) , δ(E) + . (4) δ(E) rk +1 1 When rk > 1, we always have δ(E) ≥ H n rk(rk −1) . If we substitute this lower bound, we obtain one of Langer’s restriction theorems [19], see Corollary 4.4 and Remark 4.5 for more details. Clifford indices The Clifford index Cliff(C) of a smooth curve C is the second important invariant of C after the genus g, which carries the information of special line bundles on C. Lange and Newstead [22] proposed a generalisation of Cliff(C) to higher rank Clifford index Cliffr (C) which depends on rank r -semistable vector bundles on C. Take a vector bundle E of rank r and degree d on C. The Clifford index of E is defined as Cliff(E) = r1 (d − 2(h 0 (E) − r )). We say E contributes to the rank r -Clifford index of C if E is μ-semistable with degree d ≤ r (g − 1) and h 0 (C, E) ≥ 2r . Then the rank r - Clifford index of C is defined as the quantity   Cliff r (C):= min Cliff(E) : E contributes to the rank r −Clifford index of C . Note that Cliff1 (C) = Cliff(C) is the classical Clifford index of C. In terms of these new invariants, Mercat’s conjecture [24] can be expressed as Mr (C) : Cliffr (C) = Cliff(C) which says higher rank Clifford indices are equal to the rank one Clifford index. Curves over K3 surfaces have played an important role in the Brill-Noether theory of vector bundles on curves. In [4], the conjecture M2 (C) was proved for any smooth curve C ∈ |H | on a K3 surface S when Pic(S) = Z.H . Using this, M2 (C) was shown for generic curves of every genus. However, the restriction of Lazarsfeld-Mukai bundles on S to the curve C (see Section 5 for a definition) have led to counterexamples to M3 (C) [12] and M4 (C) [1]. As a consequence of Theorem 1.2, we generalise these results to higher ranks and show Mr (C) fails for r ≥ 3 and any smooth curve C ∈ |H |. Theorem 1.2 Let (S, H ) be a smooth polarised K3 surface such that H 2 divides H .D for all curve classes D on S, (5) e.g. Pic(S) = Z.H . Take a μ-stable vector bundle E on S with Chern character ch(E) = (r , H , ch2 ) such that r ≥ 2 and ch2 ≥ 0. Then for any smooth curve C ∈ |H | of genus g, 1 D can be singular. 123 An effective restriction theorem... the restricted bundle E|C contributes to the rank r -Clifford index of C. If r ≥ 4, or, r = 3 and g − ch2 < 4 + 23  g−1 2 , then Cliff(E|C ) < Cliff(C) = g−1 . 2 In particular, Mr (C) does not hold if either (i) r ≥ 4 and g ≥ r 2 , or (ii) r = 3 and g = 7, 9 or g ≥ 11. Slope stability of the restriction of tangent bundle of Pn to a K3 surface. Let S be a smooth projective K3 surface and let L be an ample line bundle generated by its global sections on S. Then there is a well-defined morphism φ L : S → P(H 0 (L)∗ ) ∼ = Pn . In the final part of the paper, we apply wall-crossing with respect to Bridgeland stability conditions on K3 surfaces to reprove Camere’s result on slope-stability of φ L∗ TPn . The restriction of the Euler exact sequence to the K3 surface S and tensoring with L ∗ gives the short exact sequence 0 → φ L∗ TPn ∗ ev ⊗ L → H 0 (S, L) ⊗ O S − → L →0. Theorem 1.3 [8, Theorem 1] Let S be a smooth projective K3 surface over C, and let L be a globally generated ample line bundle on S. Then the kernel M L of the evaluation map on the global sections of L ev 0 → M L → H 0 (S, L) ⊗ O S − → L → 0, (6) is μ-stable with respect to L. Method of the proof of Theorem 1.1 There is an abelian subcategory A ⊂ D(X ) of the bounded derived category of coherent sheaves on X which includes E and E(−k H )[1] for k ∈ N. Thus for any irreducible divisor D ∈ |k H |, the restricted sheaf E| D lies in an exact sequence E → E| D → E(−k H )[1] in A. The slope-stability of the reflexive sheaf E implies that there is a weak stability condition σ on A such that both E and E(−k H )[1] are stable with respect to σ . If k is large enough, we may apply wall-crossing techniques to show that we can deform the weak stability condition σ , while keeping E and E(−k H )[1] stable, to reach a weak stability condition σ  where E and E(−k H )[1] have the same slope (phase). Thus their extension E| D is σ  -semistable of the same slope. Then a general argument immediately implies that E| D is slope-stable. The main advantage of this method is the possibility to strengthen effective restriction theorems for special sheaves E as soon as we have more control over the position of the walls for E and E(−k H )[1]; see for instance Proposition 4.6. 123 S. Feyzbakhsh Related wor (...truncated)