stability of restrictions of Lazarsfeld-Mukai bundles

Stability of restrictions of Lazarsfeld-Mukai bundles via Wall-crossing, and Mercat’s conjecture

Soheyla Feyzbakhsh School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, Scotland EH9 3JZ, United Kingdom
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.

1. Introduction

Lazarsfeld-Mukai bundles on a K3 surface, and their restriction to curves, have been used for many different applications. Recently, they have been appeared as counterexamples to Mercat’s conjecture for vector bundles of rank and which requires slope-stability of restrictions of these bundles, see  [FO12, MAO14]. In this paper, we extend these results to any rank greater than by using Bridgeland stability conditions.

Overview.

Let be the set of semistable vector bundles of rank and degree on a smooth curve . Then for , Clifford index is defined as

 Cliff(E)=μ(E)−2nh0(C,E)+2≥0.

The rank Clifford index of is defined as

 (1) Cliffn(C)=min{Cliff(E):E∈UC(n,d),d≤n(g−1),h0(C,E)≥2n}.

Clearly, we have . However, Mercat conjectured that we have equality for any smooth curve  [Mer02]:

 (Mn):Cliffn(C)=Cliff1(C)

Assume is a smooth complex algebraic K3 surface, and let be a smooth curve on the surface. For a globally generated line bundle on the curve , the Lazarsfeld-Mukai bundle is defined via the exact sequence

 (2) 0→E∨C,A→H0(C,A)⊗OX→A→0.

The bundles have been appeared, for example, in Lazarsfeld’s proof of Brill-Noether-Petri Theorem  [Laz86] or in Voisin’s proof of Green’s canonical syzygy conjecture  [Voi05]; see  [Apr13] for a survey of applications. In addition, the restriction of Lazarsfeld-Mukai bundle to smooth curves on a K3 surface has led to counterexamples for Mercat’s conjecture  [FO12, MAO14].

Main result

Let be a smooth polarized K3 surface over . We say the pair satisfies condition if

 for any curveC′⊂X,(H2)|(H.C′).(∗)

For instance, a polarized K3 surface satisfies condition if . Recall that a vector bundle on is -stable if for each proper quotient sheaf we have , where is the slope of .

Theorem 1.1.

Let be a smooth polarized K3 surface satisfying condition . Let be a -stable locally free sheaf on with Mukai vector

 v(F)=(rk(F),c1(F),ch2(F)+rk(F)),

where and are coprime and . Then the restriction sheaf, is slope stable for any curve if

 (3) H2+H2(rk(F)−2)(rk(F)−1)2−2rk(F)2>ΔH(F),

where .

Corollary 1.2.

Let the smooth curves have genus , and let be a globally generated line bundle on the curve with and . Then, the restriction of the Lazarsfeld-Mukai bundle of is slope-stable if

 (4) 1+r2(r+1)r2(r+1)+(r−1)d

In particular, for and , the vector bundle is stable. Moreover, by Lazarsfeld’s Brill-Noether theorem  [Laz86] existence of a line bundle on the smooth curve with global sections and degree is equivalent to an upper bound for :

 (5) ρ(r,d,g)≥0⇒g≤r+1rd−(r+1).
Corollary 1.3.

Assume the pair and the smooth curves are as above. Then restriction of Lazarsfeld-Mukai bundle invalidates Mercat’s conjecture if

1. and

1. is odd, and

 43d−3
2. or is even and

 43d−2≤g≤32d−3;or
2. and

 1+d≤g≤r+1rd−(r+1).

For any smooth curve and given integers and which satisfy the assumption in the corollary, there exists a line bundle on the curve with and such that the corresponding sheaf is a counterexample for the Mercat conjecture . In fact, Corollary 1.3 provides all the possible cases where the restriction of Lazarsfeld-Mukai bundles invalidate Mercat’s conjecture.

Relation to Previous work.

It has been proved that holds for a general curve and for a smooth curve on a K3 surface with  [BF15]. However, counterexamples to have been found using curves on K3 surfaces of higher Picard rank, see  [FO12],  [MAO14], and  [LN11].
As proven in  [FO12], for a K3 surface with , if is a line bundle on with , then the restriction of Lazarsfeld-Mukai bundle is stable if

 degC(A)=⌊2g+83⌋andg=7,9org≥11.

Also, it invalidates the Mercat conjecture if or .
However, Corollary 1.2 and inequality (5) show that for and any value of (which there exists at least one) satisfying

 23g+2≤d<1312g−1312,

the bundle is stable. It is also a counterexample to under the assumption in Corollary 1.3.
It has been also shown in  [MAO14] that for a K3 surface with and line bundle on with whenever

 d+2≤g≤43d−4,

the bundle is slope-stable. Corollary 1.2 gives a better lower bound for .
There are also some other results which use different techniques, such as taking evaluation map on the curve instead of the surface to find counterexample for  [LMN12], or restricting the bundle to a curve of higher degree to show existence of a counterexample for when  [Sen16]. But, we show that for any smooth curve with genus , the Mercat’s conjecture fails for and fails where or .

Strategy of the proof

In order to prove Theorem 1.1, we use stability conditions on the bounded derived category of coherent sheaves on and wall-crossing, see  [Bri08, BM14a, BM14b].
The slope-stability of vector bundle shows that there are stability conditions and such that and are and -stable, respectively. Also, if inequality (3) satisfies, there exists a stability condition such that and have the same phase. Then, we show that and remain stable on the paths which connect stability conditions and to . Hence, they both are -stable. Now, the distinguished triangle

 F→F|C→F(−H)[1]

gives -semistability of for . Finally, by changing in the right direction, we can reach strict stability of . Then a general argument immediately implies that is slope-stable.

Slope stability of tangent bundle of Pn restricted to a surface.

In the second part of the paper, we use similar methods, to reprove Camere’s result on the stability of the vector bundle , which is defined as follows. Let be an algebraic K3 surface over , which not necessarily satisfies condition , and be a globally generated ample line bundle on . Assume is the kernel of evaluation map on the global sections of :

 (6) 0→ML→H0(X,L)⊗OXev−→L→0.
Theorem 1.4.

[Cam12, Theorem 1] Assume is a complex algebraic K3 surface and is a globally generated ample line bundle on . Then the vector bundle is -stable.

Acknowledgements

I am thankful to my advisor Arend Bayer for the patient guidance, encouragement and support. I am grateful for comments by Gavril Farkas, Chunyi Li and Angela Ortega. The author was supported by the ERC starting grant WallXBirGeom .

2. Review: Geometric Stability Conditions

In this section, we give a brief review of stability conditions on derived category of coherent sheaves on a K3 surface, see  [Bri07, Bri08] for details.
Suppose is a complex algebraic K3 surfaces and is the bounded derived category of coherent sheaves on . The Mukai vector for is defined as

 v(E):=(rk(E),c1(E),ch2(E)+rk(E))=ch(E)√td(X)∈N(X)

where is the Chern character of and is the numerical Grothendieck group. Recall that the Mukai pairing for is given by

 ⟨v(E),v(E′)⟩=c1(E).c1(E′)−% rk(E).(ch2(E′)+rk(E′))−rk(E′).(ch2(E)+rk(E)).

The Riemann-Roch theorem shows that for two objects ,

 ⟨v(E),v(E′)⟩=−χ(E,E′)=−∑i(−1)idimExti(E,E′).

A numerical stability condition on consists of a group homomorphism (central charge)

 Z:N(X)→C,Z(E)=⟨Ω,v(E)⟩,

and a collection of abelian subcategories (semistable objects of phase ) for each which together satisfy some axioms.
A stability function on an abelian category is a group homomorphism such that for any non zero object ,

 Z(E)∈R>0exp(iπϕ(E))with0<ϕ(E)≤1.
Proposition 2.1.

[Bri08, Proposition 3.5] To give a stability condition on a triangulated category is equivalent to giving a bounded t-structure on and a stability function on its heart which has the Harder-Narasimhan property.

For a pair when is an ample divisor, one defines group homomorphism as

 Z(β,ω)(E)=⟨exp(β+iω),v(E)⟩

and the slope function as

 μω(E)=c1(E).ωrk(E).

Consider torsion pair on the category of Coh(X), where consists of sheaves whose torsion free parts have -semistable Harder-Narasimhan factors of slope and consists of torsion-free sheaves whose -semistable Harder-Narasimhan factors have slope . Tilting with respect to the torsion pair gives a bounded t-structure on with the heart

 A(β,ω)={E∈D(X):Hi(E)=0fori∉{0,1},H−1(E)∈FandH0(E)∈T}.

For any choice of , above construction will not give a stability condition on . Let

 W(X)={(β,ω):β,ω∈NS⊗R,ωis an ample devisor}and
 V(X)={(β,ω)∈W(X):for everyδ∈Δ(X)% withrk(δ)>0,⟨exp(β+iω),δ⟩∉R≤0},

where is the root system. Then we have following result.

Proposition 2.2.

[Bri08, Lemma 6.2] For any pair , the function is a stability function on which has the Harder-Narasimhan property. Therefore is a Bridgeland stability condition on .

Two-dimensional subspace of stability conditions

Let be a fixed primitive ample divisor on an algebraic K3 surface . Consider following projection maps:

 P1:N(X)→R3,P1(r,C,s)=(r,C.HH2,s),
 P2:R3∖{s=0}→R2,P2(r,c,s)=(cs,rs),

where . Also is their composition.
In this paper, we only focus on a two dimensional subspace which consists of numerical Bridgeland stability conditions such that skyscraper sheaves at every point are -stable of phase one, and the central charge factors via .
Thus, every stability condition is of the form for some in the upper half plane . For simplicity, we denote such a stability condition by .
Consider the isomorphism

 k:StabH(X)→UH(X),where
 k(σ(b,w))=Pr(Ker(Z(b,w)))=(bd′(b2+w2),1d′(b2+w2))=:k(b,w),

and with the standard topology on .
Therefore we can work with the space instead of and apply all known results about the space of stability conditions to this space. By abuse of notations, in the figures we always denote by the corresponding point in . For instance, if is large enough, is a stability condition which is on the -axis (). Similarly, for any fixed value of , the stability conditions are on the line . When gets bigger, the corresponding point in gets nearer to the point , see Figure 1.

Lemma 2.3.

Given a root . Let be the set

 {k(b,w):(b,w)∈H,Z(b,w)(δ)∈R≤0}⊆R2.

Assume is the intersection point of the parabola with the line through origin and . Then is the line segment between and .

Proof.

For any ,

 Im(Z(b,w)(r,C,s))=wC.H−2d′bwr=0⇒b=C.H2d′r.

Thus, the point is on the line with equation which passes the point . In addition,

 Rel(Z(b,w)(r,C,s))=bC.H−s−rd′(b2−w2)≤0⇒d′w2≤s/r−(C.H)2/(4r2d′),

and we have equality if , which makes the claim clear. ∎

Therefore, is the open subset minus the line segments that pass roots (see Figure 2).

Lemma 2.4.

The interior of the ellipse with equation and a sufficiecnly small punctured disk around the point do not contain any projection of a root .

Proof.

Let be a root, i.e. . If , then clearly is outside of the required area. Therefore, we assume . By the Hodge index theorem,

 d′(C.H2d′s)2+r2s2−rs≥C22s2−rs−r2s2=r2−1s2≥0

which shows is not inside the ellipse.
For the second part of lemma, since we only care about an open neighbourhood around , we can assume that and . Thus

 14d′

and the claim follows. ∎

Remark 2.5.

It follows from [Bri08, Proposition 9.3] that for any object , the space and therefore admit a well-behaved wall and chamber structure controlling stability of . There exists a locally finite set of walls of dimension one with following properties:

1. When varies within a chamber, stability or instability of does not change.

2. When lies on a single wall , then is -semistable, and if is stable in one of the adjacent chamber, then it is unstable in the other adjacent chamber.

The next lemma describes walls in .

Lemma 2.6.

Let be a stability condition and are two -semistable objects. Then, and have the same phase if and only if the points , and are collinear. In particular, the walls of stability for any , are segments of the lines passing through , see Figure 3.

Proof.

The two semistable objects have the same phase if and only if for some , which means

 αP1(v(E))−P1(v(E′))∈P1(Ker(Z(b,w))).

Or, equivalently, the points , and are collinear. Indeed, is a one dimensional subspace that can be generated by the vector .
Therefore, the set of points that any fixed object can be -stable factor of , is precisely segment of the line that connects to , see  [Bri08] for details. ∎

Relation to slope-stability

Definition of stability conditions for some are based on slope-stability of torsion free sheaves. The following well-known Lemma makes clear the relation between these two notions of stability.

Lemma 2.7.

Let be a locally-free sheaf of positive rank and be a stability condition on . Then is -stable with slope if and only if is -stable of phase 1.

Proof.

Assume is -stable of phase one.

 Z(β,ω)(E[1])=⟨v(E[1]),eβ+iω⟩=⟨(r,C,s),(1,β,β2−ω22)⟩+i⟨(r,C,s),(0,ω,β.ω)⟩.

Since the imaginary part vanishes,

 Im(Z(E[1]))=C.ω−rβ.ω=0⇒μω(E)=C.ωr=β.ω.

Moreover, by definition of the heart, and all -semistable HN factors of have slope less than or equal to . Therefore is -semistable.
Now assume for a contradiction there exists a proper torsion free quotient sheaf () in with the same -slope. So, we have exact sequence in Coh where all three sheaves , and are -semistable torsion free sheaf of slope . By definition, all these three sheaves are in and we have following exact sequence in the abelian category .

 0→F[1]→E[1]→F′[1]→0.

For the converse, assume is a -slope-stable locally-free sheaf of slope , so and . Therefore and is -semistable object. Assume for a contradiction that is strictly -semistable. [Bri08, Lemma 10.1] implies that the every stable object of phase one is a skyscraper sheaf or shift of a locally-free sheaf. Since is a locally-free sheaf,

 HomD(X)(Ox,E[1])=HomD(X)(E[1],Ox)=0

Therefore, all stable factors of are shift of locally-free sheaves which implies that has a subsheaf in Coh with the same slope and smaller rank, a contradiction. ∎

3. Mercat’s conjecture

In this section, we always assume is a smooth polarized K3 surface over which satisfies condition . Like before, we use the notation . Let be a locally free sheaf which is -stable and has Mukai vector . Therefore, Lemma 2.7 gives -stability of , where

 b1=C.H2d′randd′w21>1.

Hence, the corresponding point is on the line segment that connects to origin, see Figure 4. Similarly, the shift of twisted sheaf with the Mukai vector

 v(F(−H)[1])=(−r,−C+rH,−s−d′r+C.H),

is -stable where

 b2=H.(C−rH)2d′r=b1−1andd′w22>1.
Lemma 3.1.

Given real number where , if

 (7) d′w2b=−d′b2+b(−d′+C.Hr)+C.H2r−12d′(C.Hr)2+sr>1,

the pair gives a stability condition such that and have the same phase.

Proof.

To find the expression for , it is enough to check the intersection point of the line through and with the line . If , Lemma 2.4 implies that we have stability condition . In addition, since , the objects and are in the heart and Lemma 2.6 implies that and have the same phase in the stability condition . ∎

Equation (7) shows that for the critical point , we have the maximum value

 d′w23=d′4−ΔH(F)2r2,

which is greater than one if the sheaf satisfies inequality (3). Hence, we have stability condition such that and have the same phase. The next step towards proof of Theorem 1.1 is to show and are both -stable. Therefore we must check the walls of stability of and .

Lemma 3.2.

Let be an object in and be a stability condition. Let be a sublattice such that the quotient can be generated by . Then cannot be strictly -semistable if .

Proof.

Assume for a contradiction that is strictly -semistable and is one of its -stable factors. By the assumptions, there are and such that any element of the lattice , and so , can be written as

 P1(v(E))=xP1(v(F))+yv′+zv′′,

for some and . The objects and have the same phase. Hence and , which is in contradiction to . ∎

Proof of Theorem 1.1.

By the assumption, the equation

 (8) mr−n(C.H2d′)=−1m,n∈N

has always a solution with .
Now, consider a straight line path in that starts at the stability condition and go to the stability condition . If this path hits any wall , then as it is shown in Figure 5, that wall would also intersect the line segment for

 b4=C.H2d′r−1rn0=m0n0.

Inequality (3) implies that the intersection point is always of form when

 d′w24≥d′r(r−1)−d′r2(r−1)2−ΔH(F)2r2>1.

Thus, the intersection point is always a stability condition .

Consider the sublattice which is generated by and . Then, clearly and can generate the quotient . Hence, Lemma 3.2 implies that cannot be strictly -semistable which is in contradiction to . Thus cannot be destabilized along the path and it is -stable.
Similarly, for proving -stability of , it is enough to consider the path which starts at go straight to .

As it is shown in Figure 6, if the path hits any wall , then that wall has intersection with the line for

 b5=m1n1−1=C.H2d′r−1−1rn1,

where is a solution for the equation (8) with . Inequality (3) implies that the intersection point is of form when

 d′w25≥d′r(r−1)−d′r2(r−1)2−ΔH(F)2r2>1.

Again, by considering the sublattice which is generated by and , Lemma 3.2 leads to a contradiction to . Therefore, is also -stable.
Now, consider the following distinguished triangle in for ,

 F→F|C→F(−H)[1].

The objects and are -stable of the same phase. Therefore, is -semistable with and as its -stable factors. Moreover,

 Rel[Z(b3,w3)(F(−H)[1])]=Rel[Z(b3,w3)(F)]=%Rel[Z(b3,w3)(F|C)]=0

and if ,

 Rel[Z(b3,w)(F(−H)[1])]<0,Rel[Z(b3,w)(F)]>0,

and

 Rel[Z(b3,w)(F|C)]=0,

which shows -strict stability of where and is a sufficiently small positive number.
Now assume is a subsheaf of . By definition, the torsion sheaves and on the surface are in the heart . Assume the Mukai vector of as a sheaf on is . Then -stability of gives

 Rel[Z(b3,w)(~F)]=b3~C.H−~s>0⇒μ(~F)<μ(F|C),

because

 μ(~F)=degC′(~F)rk(~F)=(g−1)(2~s~C.H+1).

Therefore, is a slope-stable vector bundle on the curve . ∎

Let