A remark on stability and restrictions of vector bundles to hypersurfaces

We prove that a vector bundle on a smooth projective variety is (semi)stable if the restriction on a fixed ample smooth subvariety is (semi)stable.


Introduction
The purpose of this work is to show a property of slope-stability of vector bundles with respect to restriction to a given ample subvariety.
Given a slope-stable vector bundle E on a projective variety X, it is rather difficult to prove that the restriction of E to an ample subvariety is stable. This can be done for general subvarieties of sufficiently high degree (cf. [4,6]).
On the converse, it is easy to show that, if the restriction of E on a general ample subvariety of high degree is slope-stable, then E is slope-stable as well.
The purpose of this work is to show that if the restriction of E jY to one given smooth ample hypersurface Y & X (of any degree) is (semi)stable, then the vector bundle E is (semi)stable on X. This result only appears in the literature as its generic version, which is almost elementary to prove, so we think it is useful to write it here, as we need it as a reference for future use, and for the interest of the result in itself.
Furthermore, we notice that, contrarily to its generic version, the result is not granted on slightly broader hypotheses, for example if we consider stability of a torsion free sheaf instead of a locally trivial one, or if we consider the restriction to a given singular hypersurface instead of a smooth one. See questions below.

Notation and main definitions
Throughout this work X is a smooth projective variety of dimension n over an algebraically closed field k.
Intersection products, pull-backs, push-forwards, and Chern classes will be considered in the Chow ring A Ã ðXÞ with integral coefficients, and we will identify divisors in A 1 ðXÞ with line bundles when useful. For a 0-cycle W ¼ P k i p i , with p i 2 X, k i 2 Z we will denote by its degree. For any higher dimensional cycle Z, we will set \W [ X ¼ 0. Therefore, the intersection number of two classes a; b 2 A Ã ðXÞ with complementary dimensions will be denoted \a:b [ X .
We will say that two cycles a; b 2 A Ã ðXÞ are numerically equivalent when \a:c [ X ¼ \b:c [ X for every cycle c 2 A Ã ðXÞ. Definition 1.1 Let H be an ample divisor on X. Let F be a torsion free sheaf on X. Let n be the dimension of X.

Lemmata
The following is the first useful observation: 1 Let F be a torsion free sheaf on X. Suppose that F is locally free outside of a subset Z & X of codimension at least 3 in X. If Y & X is a hypersurface and H is its class in A 1 ðXÞ, then Proof Let us first remark that, if i : Y,!X is the immersion of Y in X, then In fact, this is obvious when F is a vector bundle. In general notice that c 1 ðF jY Þ and i Ã c 1 ðFÞ are two line bundles on Y, isomorphic on an open subset U ¼ YnZ whose complementary Y \ Z has codimension at least 2 in Y, therefore they are isomorphic. Hence c 1 ðF jY Þ:ðH jY Þ nÀ2 ¼ i Ã ðc 1 ðFÞ:H nÀ2 Þ.
To complete the proof observe that \w [ Y ¼ \i Ã w [ X for any cycle w 2 A Ã ðYÞ, and that by projection formula: where [Y] is the identity class in A Ã ðYÞ, and clearly i Ã ½Y ¼ H. h Lemma 2.2 Let E be a vector bundle on X.
where the dual sheaf of F is defined as the sheaf of homomorphisms Corresponding statements can be made on semistability.
Proof To prove stability, we have to consider the slope of any subsheaf F & E.
Suppose that condition (i) holds. Given any subsheaf F & E consider the exact sequence 0 ! F ! E ! G ! 0. If G ¼ E=F is not torsion free consider its torsion T ¼ TðGÞ and its torsion free quotient GG 0 ¼ G=T. Then Now, observe that c 1 ðTÞ is effective, in fact c 1 ðTÞ ¼ c 1 ðF 0 Þ c 1 ðFÞ Ã , and as F & F 0 then the line bundle c 1 ðTÞ has a section. Therefore and l H ðFÞ 6 l H ðF 0 Þ\l H ðEÞ: Suppose now that condition (ii) holds. Given any subsheaf F & E consider the injection of F in its bidual: F,!F ÃÃ . Now, F ÃÃ is a subsheaf of E as well. In fact, morphisms from F to E factor through F,!F ÃÃ , as it can be seen from the computation of homomorphism from F to E, recalling that E is a vector bundle: has codimension at least 3 (respectively at least 2).
Proof Suppose that F is a reflexive sheaf, let Z & X be an integral subscheme of codimension at most 2, and let z 2 X be the generic point of Z. Then O X;z has dimension at most 2. Therefore However, as X is a smooth variety, then O X;x is a regular ring for all x 2 X, so it satisfies the Auslander-Buchsbaum formula (cf. [1]): for any coherent sheaf F and any schematic point where dhðF x Þ is the minimal length of projective resolutions of F x . In particular, for z 2 X above, we have dhðF z Þ ¼ 0, therefore F z is a free O X;z -module. Therefore, singðFÞ cannot contain any 2-codimensional subscheme. The same argument applies to the codimension of the singular locus of a torsionfree sheaf. h

Main theorem
We give in this section the proof of the main theorem stated in the introduction. Proof Let us prove the statement concerning stability, the argument being essentially the same for semistability. We want to prove that given a subsheaf F,!E, we have l H ðFÞ\l H ðEÞ, knowing that this property holds on Y.
According to Lemma 2.2 we can suppose that F is reflexive and that G :¼ E=F is torsion-free, so by Corollary 2.4 the singular locus of F singðFÞ ¼ fx 2 XjF x is not a free O X;x -module g; has codimension at least 3, and singðGÞ has codimension at least 2 in X.
Let us restrict the exact sequence 0 ! F ! E ! G ! 0 to the hypersurface Y: and let us remark that, as codim X singðFÞ > 3, then codim Y singðF jY Þ > 2.
As suppðT or 1 O X ðG; O Y ÞÞ & singðGÞ \ Y, and codim X singðGÞ > 2 in X, then T or 1 O X ðG; O Y Þ is a torsion sheaf on Y, injecting in F jY . Therefore, its support must be contained in singðF jY Þ.
Hence, c 1 ðF jY Þ ¼ c 1 ðF jY =T or 1 O X ðG; O Y ÞÞ, because quotienting by subsheaves concentrated on high codimension subsets does not affect the first Chern class.
On Y, we have the following exact sequence Hence by stability of E jY we have that where dim X ¼ n. As codim X singðFÞ > 3, and E is a vector bundle, then by Lemma 2.1 we have: Therefore, we get stability on X. h

Applications and questions
By recursive restriction to hypersurfaces it is immediate to prove the following consequence of the main theorem: Restriction to smooth curves obtained in such a way is used in various ways in the literature: in some works of the author (cf. [7,8]), some complete intersection curves are constructed to prove the stability of vector bundles on higher dimensional varieties. Also, in [3], stability of Picard bundles is proven by restriction to curves which are intersection of theta divisors. Even though it is not needed to fix one subvariety in these cases, Corollary 4.1 could be used in these works.
Question 4.2 Since stability is obtained knowing stability on a fixed smooth subvariety, it is natural to ask about weakening the hypothesis to stability on a fixed singular hypersurface, possibly limiting the kind of singularities. We leave this question to further investigations.

Question 4.3
In addition, we can consider weaker hypothesis on E. We see from the proof that we can consider restrictions of a reflexive sheaf E to a fixed smooth ample hypersurface Y & X. Then same result hold for stability of E, knowing stability of the restriction of E to Y. However the same cannot be said for torsion free sheaves, and therefore, we cannot apply this recursively to a fixed complete intersection of ample hypersurfaces with the same numerical class. A first obstruction in the proof lies in the fact that the restriction of a torsion free sheaf to a hypersurface is not necessarily torsion free. In addition, the dimensional properties used in the proof cannot be used for torsion free sheaves.

Remark 4.4
In the recent works (cf. [2,[9][10][11]), we consider asymptotic base loci of vector bundles, to get positivity properties, and construct Iitaka fibrations. It would be interesting to consider restrictions of stable vector bundles to (smooth subvarieties in) their asymptotic base loci as well. Any relationship between asymptotic base loci and stability would be surprising.