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Vector bundles on projective varieties which split along q-ample subvarieties

Let Y be a subvariety of a smooth projective variety X, and V a vector bundle on X. Given that the restriction of V to Y splits into a direct sum of line bundles, we ask whether V splits on X. I answer this question in affirmative if holds: Y is a q-ample subvariety of X (for appropriate q), it admits sufficiently many embedded deformations, and is very general within its own deformation space. The result goes beyond the previously known splitting criteria for vector bundles corresponding to restrictions. It allows to treat in a unified way examples arising in totally different situations. I discuss the particular cases of zero loci of sections in globally generated vector bundles, on one hand, and sources of multiplicative group actions (corresponding to Bialynicki-Birula decompositions), on the other hand. Finally, I elaborate on the symplectic and orthogonal Grassmannians; I prove that the splitting of any vector bundle on them can be read off from the restriction to a low dimensional `sub'-Grassmannian.


Introduction
For a vector bundle V on the irreducible projective variety X, I consider its restriction V Y to an irreducible subvariety Y ⊂ X, and investigate the following: Question. Assuming that V Y splits, under what assumptions on Y does V split too?
To my knowledge, there is not a single reference which addresses the question in this generality. Yet, the problem is interesting because it allows to probe the splitting of vector bundles on (high dimensional) varieties by restricting them to (possibly low dimensional) subvarieties. The goal of this article is to give a tentative answer to the question.
Horrocks' criterion [15] is the most widely known result of this type. Splitting criteria, corresponding to restrictions to ample divisors, have been obtained in [5]. Surprisingly, restrictions to higher codimensional subvarieties have not been considered. For this reason, I studied in [13] the case when Y is the zero locus of a regular section in a globally generated, ample vector bundle N on X.
Unfortunately, this setting discards very basic situations, e.g. X is a product X ′ × V and N = O X ′ (1). Second, in perfect analogy with Horrocks' criterion, I proved that the splitting of a vector bundle on any Grassmannian can be verified by restricting it to an arbitrary, standardly embedded Grass(2; 4); this is not an ample subvariety. These observations are the motivations to consider 'sufficiently positive' subvarieties of X, which posses 'many' embedded deformations. Loosely speaking, the main result of this article is the following: Theorem. Let V be an arbitrary vector bundle on a projective variety X, defined over an uncountable, algebraically closed field of characteristic zero. Let Y be a subvariety of X satisfying the following properties: -Y is q-ample for appropriate q (e.g. q+1 dim X −3codim X (Y ), but not necessarily); (The definition of a q-ample subvariety is inspired from [22,25,2], also [11].); -The embedded deformations of Y cover an open subset of X, and their intersection pattern is sufficiently non-trivial. Then V splits if and only if it splits along a very general deformation of Y .
The precise statement can be found in theorem 2.8; applications to the cases of globally generated vector bundles and multiplicative group actions, with emphasis on homogeneous varieties, are stated in theorems 4.1 and 6.3 respectively.
The q-ampleness condition is asymptotic in nature, and involves thickenings of subvarieties. For this reason, the price to pay for the generality of the theorem is that of testing the splitting along subvarieties which are very general within their deformation space. Thus the q-ample case studied here can be characterized as probabilistic, in contrast with the case of ample vector bundles studied in [13], which is deterministic. Although the result is formulated algebraically, the proof uses complex analytic techniques; the final statement is deduced by base change.
The essential feature of the criterion is that of being intrinsic to the subvariety; it avoids additional data. This allows to treat in a unified way examples arising in totally different situations: zero loci of globally generated (not necessarily ample) vector bundles (cf. sections 3, 4), on one hand; homogeneous subvarieties of homogeneous varieties (cf. sections 5, 6), on the other hand. As a by-product, we obtain many examples of q-ample subvarieties which are not zero loci of regular sections in globally generated q-ample vector bundles.
The case of isotropic Grassmannians is detailed in section 7. As I already mentioned, a vector bundle on Gr(u; w), u 2, w 4, splits if and only if its restriction to any embedded Gr(2; 4) splits. Unfortunately, the same proof breaks down in the case of the symplectic and orthogonal Grassmannians. However, our 'probabilistic' approach works very well.
Theorem. Let V be a vector bundle either on the symplectic Grassmannian sp Gr(u; w), with u 2, w 2u, or the orthogonal Grassmannian o Gr(u; w), with u 3, w 2u. Then V splits if and only if it splits along a very generally embedded: − sp Gr (2; 4), in the symplectic case; 5. Positivity properties of sources of G m -actions 17 6. Splitting criteria for vector bundles on homogeneous varieties 20 7. Splitting criteria for vector bundles on Grassmannians 23 References 28 1. Short digression on q-ampleness 1.1. Why q-ample subvarieties? Before reviewing the relevant definitions, I briefly present the reasons which led to consider the notion of q-ampleness in this context. One says that V splits on X if V = j∈J L j ⊗ C m j with L j ∈ Pic(X) pairwise non-isomorphic, and j∈J m j = r.
It is easy to see that V splits if and only if it admits an endomorphism with rk(V ) distinct eigenvalues. The main tool to attack the splitting problem is the following observation.
Lemma 1.1 Let S ⊂ X be a connected, projective subscheme, such that V S splits. If the restriction homomorphism res S : Γ(X, E ) → Γ(S, E S ) is surjective, then V splits too.
The subschemes S will usually be thickenings of smooth subvarieties of X. ; with this convention, Y 0 = Y . The structure sheaves of two consecutive thickenings of Y fit into the exact sequence The formal completion of X along Y is defined as lim − → Y m , and is denoted byX Y .
To apply lemma 1.1, I adopt a two-step strategy: (i) Prove that Γ(X, E ) → Γ(X Y , EX Y ) is surjective; this step requires the (dim Y − 1)ampleness of Y . (Actually, we will consider also more general thickenings.) (ii) Prove that Γ(X Y , EX Y ) → Γ(Y, E Y ) is surjective. This step is more involved, requires less ampleness (more positivity) of Y , and also its genericity in the space of embedded deformations. If Y has the property that H 1 (X, F ⊗ I m Y ) = 0 for all vector bundles F on X (m depending on F), then the splitting of V along a sufficiently high order thickening of Y implies the splitting on X. Indeed, simply take F = E . (For ample subvarieties, explicit lower bounds for m are obtained in [13].) . Then V splits on X if and only if it does so along the formal completion of X along Y .
The difficulty consists in proving that the splitting of V along a very general subvariety Y ⊂ X implies the splitting of V alongX Y . This step constitutes the body of the article.

1.2.
Definition and first properties. The concept of q-ampleness for globally generated vector bundles was introduced in [23]; the q-ampleness in defined intrinsically in [2,25] through cohomology vanishing properties. We recall the latter definition, the case of globally generated vector bundles being detailed in section 3.
Proposition 1.5 (cf. [25,Section 7], [22,Definition 3.1]). Let Y ⊂ X be an equidimensional subscheme of codimension δ,X := Bl Y (X) be the blow-up of the ideal of Y , and E Y ⊂ X be the exceptional divisor. Assume thatX is Gorenstein. The following statements are equivalent: (i) For any locally free (hence also for any coherent) sheafF onX holds For all locally free sheaves (that is vector bundles) F on X holds Proof. Is identical with [22, Lemma 2.2 and Proposition 6.2].
If X is smooth and Y ⊂ X is a locally complete intersection (lci for short), thenX is automatically Gorenstein. We are primarily interested in the cohomology vanishing property (1.3); for this reason, we introduce an ad hoc terminology. Definition 1. 6 We say that a lci subvariety Y ⊂ X is (has the property) p >0 if for any vector bundle F on X holds: Intuitively, Y is p >0 if its normal bundle at each point contains at least p 'positive' directions.
Probably the appropriate name for this property of Y would be 'q-ample subvariety', for q = dim Y − p. (The case of ample subvarieties [22] corresponds to q = 0.) This choice is made to emphasize the amount of positivity of the various objects which which appear subsequently.
then D is connected. (Indeed, takeF = KX in 1.4 and apply Serre duality.) Hence, if Y ⊂ X is lci such that E Y is 1 >0 , then Y is connected and also equidimensional. In particular, Y is smooth implies that it is irreducible.
, and are both irreducible lci, then Proof. (i) (⇒) Let F be a vector bundle on X. Since E Y = P(N), the exact sequence implies H t (P(N), F ⊗ O P(N) (−m)) = 0, for all t dim X − p and m ≫ 0. The equality ); according to [22, (5.1)], the right hand side The lack of sufficient positivity of the normal bundle N Z/X prevents to conclude that Z ⊂ X is min{r, p} >0 . However, the next proposition shows that it is close to be so. .
The set U is allowed to be either a Zariski or an analytic (tubular) open neighbourhood of Z.
(ii) We say that a subscheme Y ⊂ X is p 0 (approximately p >0 ) if there is a decreasing sequence of sheaves of ideals {J n } n such that the following hold: -an isomorphism, for t p − 1, -injective, for t = p.
(1.9) Proposition 1.10 Let Z ⊂ Y, Y ⊂ X be lci, and both p >0 . Then Z ⊂ X is p 0 , and for all vector bundles F on X holds: Proof. Since Z ⊂ Y ⊂ X are lci, for any l a one has the exact sequence: Indeed, just use (I l Z + I a Y )/I a Y ∼ = I l Z /I a Y · I l−a Z . Note that the left hand side is an O Y -module; I Z /I Y = I Z⊂Y is the ideal of Z ⊂ Y , and that I a Y /I a+1 Y = Sym a N ∨ Y /X is the symmetric power of co-normal bundle. The p >0 assumption implies that there are integers k F , l F , and a linear function l(k) = λk + µ (λ, µ independent of F) with the following properties: The last claim uses the uniform q-ampleness property [25,Theorem 6.4] and the subadditivity of the regularity [25,Theorem 3.4]: first, there is a linear function l(r) such that for any vector bundle F with regularity reg(F Y ) r holds H t (F Y ⊗ I l Z⊂Y ) = 0, for t p, l l(r); second, for a k, reg(F Y ⊗ Sym a N ∨ Y /X ) linear(k). Recursively, for a = 1, . . . , k, the exact sequence (1.10) yields: → 0 (tensored by F), and deduce that . We denote by Z l,k the subscheme defined by I k Y + I l Z ; it is an 'asymmetric' thickening of Z in X. For any l as above, there is m > l such that Notice that res l k • ξ = res X l,k , which is injective, so ξ is injective. It remains to prove that ξ maps ontoH t (F Z l ), for t p−1: assume α = res U Y l (α), where U ⊃ Z is open andα ∈ H t (F U ); then α = res m l ( res U m,l (α) ), and res U m,l (α) is in the image of res X m,l .
Proof. Since f is flat, Y is still lci in X and codim , and m ≫ 0. Then the Leray spectral sequence implies that E Y is (dim X ′ − p) + d − 1 -ample.  (Sommese) q-ample vector bundles. (This will be detailed in section 3.) However, is necessary to have a test for the q-ampleness of a subvariety in more general circumstances. Fortunately, an elementary criterion was already used in [13]. Proposition 1.12 Let the situation be as in 1.5. Assume there is an irreducible variety V and a morphism b : Proof. LetF be a coherent sheaf onX, and j dim X − p > dim b(X). Then

Remark 1.13
In section 2, we will be interested in families of p >0 subvarieties of X; the proposition generalizes straightforwardly. Let Y ⊂ S × X be a smooth family of subvarieties of X indexed by some parameter space S. Assume that there is an S-variety V such that The criterion covers the situations discussed in sections 3 and 5, that is the case of zero sections in globally generated vector bundles, and the case of sources of G m -actions.
1.4. q-positive line bundles. Intuitively, the hypotheses in 1.12 allow OX (E Y ) to be 'negative' in at most dim b(X) < dim X −p directions. As we will see below, this stronger positivity property allows to control Pic(Y ). Definition 1.14 (cf. [11]) A line bundle L on a smooth projective varietyX is q-positive if it admits a Hermitian metric whose curvature form has at most q negative (or zero) eigenvalues.
It is generally true (cf. [1, Proposition 28], [11, Proposition 2.1]) that a q-positive line bundle L satisfies H q+1 (F ⊗ L m ) = 0, for any vector bundleF and m ≫ 0, that is L is q-positive ⇒ L is q-ample. Proposition 1.15 Let the situation be as in proposition 1.12, with X, Y smooth. Then Proof. One may assume that V is smooth; otherwise, take an embedding into a smooth variety. As OX(E Y ) is b-relatively ample, there is an embeddingX The maximal rank of dbx is dim b(X), attained on a dense open subset ofX, so rk(dbx) dim b(X) for allx ∈X. Since ι is an embedding, the curvature of the pull-back metric on OX (m 0 E Y ) is positive definite on Ker(dbx), and dim Ker(dbx) dim X − dim b(X). So, at each point ofX, there are at most dim b(X) negative eigenvectors. Lemma 1.16 Let ϕ : X → X ′ be a smooth morphism of relative dimension d. If L ′ is a q-positive line bundle on X ′ , then ϕ * L ′ is (q + d)-positive on X.
Proof. Take a metric in L ′ as in 1.14, and pull it back to L.

1.5.
Geometric properties of q-ample subvarieties. The next section deals with families of p >0 subvarieties. The non-emptiness and connectedness of their intersections, as well as their Picard groups will be essential; these issues are investigated below.
1.5.1. Non-emptiness and connectedness of the intersections. For a family of irreducible subvarieties Y ⊂ S × X of X indexed by some variety S, we denote (1.11) lci subvarieties of X such that their double and triple intersections are equidimensional lci (if non-empty). If 2δ + 1 p, then Y os and Y ost are indeed non-empty and connected.
Proof. The following diagram is commutative 12) and the exceptional divisors are obtained by restriction, The contradiction shows that the double intersections are non-empty.
If Y ost = ∅, then Y os ⊂ Y o \ Y ot , and we get another contradiction: The connectedness follows from the remark 1.7: the assumption p 2δ + 1 implies that all the exceptional divisors in (1.12) are at least 1 >0 , so they are connected.

1.5.2.
Picard group of q-ample subvarieties. Let Y ⊂ X be smooth. We are interested when is the pull-back res X Y : Pic(X) → Pic(Y ) an isomorphism.
. Now compare the exponential sequences for X and Y : (1.13) is an isomorphism for t p − 1, for any p >0 subvariety Y (cf. [22,Corollary 5.2]). Hence, if Y is 3 >0 , Pic 0 (X) → Pic 0 (Y ) is a finite morphism, and NS(X) → NS(Y ) has finite index. Theorem 1.20 If L is a q-positive line bundle on a smooth projective variety X and the zero locus

The splitting criterion
In this case, V j := L j ⊗ C m j , j ∈ J , are called the isotypical components of the splitting. If [4, Theorem 1 and 2].) However, the isotypical components are not uniquely defined, because the global automorphisms of V send a splitting into a new one. We consider the partial order '≺' on Pic(X): The isotypical components corresponding to the maximal elements, are canonically defined.
Lemma 2.1 Let M ⊂ J be the subset of maximal elements with respect to ≺. Then there is a natural, injective homomorphism of vector bundles Consequently, given a family of subvarieties {Y s } s∈S of X such that V splits along each of them, we can glue together the maximal isotypical components of V Ys (cf. lemma 2.6 below).

2.1.
Gluing of split vector bundles. Let S be an irreducible quasi-affine variety, and Y ⊂ S × X be subvariety (denote by π, ρ the morphisms to S and X, respectively) with the following properties: Assume that the very general point o ∈ S has the following properties: , and the intersection is transverse for s ∈ S(o) generic; (b) For s ∈ S(o) generic, all the arrows below are isomorphisms: The (1-arm) condition: these configurations cover an open neighbourhood of Y o . (1-arm) The (no-△) condition: such configurations should not exist. However, for the symplectic and orthogonal Grassmannian studied in section 7, the generic double and triple intersections are empty. The analysis of these cases led to the weaker conditions above.
A very general point of S refers to a point outside a countable union of subvarieties of S. (ii) The double (resp. triple) geometric generic self-intersections of Y, denoted Y 2 and Y 3 respectively, are defined by means of the diagrams: Thus there are natural morphisms Y 2 ⇒ Y (the projections onto the first and second com- for very general (s, t) ∈ S 2 , and the morphisms are the inclusions into Y s , Y t respectively; similarly for Y 3 .) There is an open analytic subset (a ball) B ⊂ S, which can be chosen arbitrarily in some Zariski open subset of S, such that the following hold: This condition is satisfied in the particular case when Y ⊂ S × X is a δ-codimensional, p >0 family of subvarieties of X as in 1.13, with p δ + 3.
r ; the right hand side is defined over a finitely generated (algebraic) extension of C(S). Thus there is an open affine S • ⊂ S, and a finite morphism σ : The second statement is analogous.
In the situation 1.13, Henceforth, for o ∈ S very general, we replace S by S(o) (actually by an irreducible component containing o) and Y by Y × S S(o); the restrictions of π, ρ are denoted the same.
On the other hand, after restricting to Y os , the hypothesis (Pic) implies that this set is dense; thus it is the whole S. It follows that ℓ ∼ = π * l , withl ∈ Pic(S). But Pic(S) is trivial for S sufficiently small, so ℓ ∼ = O.
Let V be a vector bundle on X, such that ρ * V splits; by the previous lemma, For any s ∈ S, let M s ⊂ J be the subset of maximal elements, for V Ys . By semi-continuity, there is a neighbourhood S s ⊂ S of s such that M s ⊂ M s ′ for all s ′ ∈ S s ; thus there is a largest subset M ⊂ J, and an open subset S ′ ⊂ S such that M = M s , for all s ∈ S ′ . Hence, possibly after shrinking S, the set of maximal isotypical components of V Ys with respect to (2.1) is independent of s ∈ S.
Lemma 2.6 Let the situation be as above, and B ⊂ S be a standard (analytic) ball. We consider the (analytic) open subset U := ρ(Y B ) ⊂ X. Then the following statements hold: Now we proceed with the proof of the lemma. For all s ∈ B, the restriction of ev : , ∀ µ ∈ M, and ev is pointwise injective. We claim that, after suitable choices of bases in π * ρ * (L −1 µ ⊗ V ), the homomorphism ev descends to U . We deal with each µ ∈ M separately, the overall basis being the direct sum of the individual ones.
Consider µ ∈ M , and a base point o ∈ B. Then V ′ := L −1 µ ⊗ V has the properties: Indeed, for any s ∈ B, we consider the diagram (recall that Y os = ∅): (the second equality uses Im α o | Yos = T Yos = Im α s | Yos , and α s β s | T = 1l), and similarly a s a ′ s = 1l. Thus a s ∈ Gl(m; C) for all s ∈ B, and the new trivializationα := α • a of T satisfies Also, for all s, t ∈ B such that Y st = ∅, the trivializations of T Yst induced byα from Y s and Y t , coincide; equivalently, the following diagram commutes: Indeed, Y ost is non-empty and connected by (no-△), so is enough to prove that the restriction of (2.7) to Y ost is the identity. After restricting (2.6) to Y ost , we deducẽ Apply repeatedly the first part.
Lemma 2.7 Let the situation be as in lemma 2.6, and assume Y o ⊂ U is 2 0 (cf. 1.9). Let A, B, C be vector bundles on X, whose restriction to 2.2. The splitting criterion. Let F ֒→ C be a finitely generated extension of Q, such that X, Y, V are defined over F; its algebraic closureF ⊂ C is countable.
Theorem 2.8 Let X be a a smooth, projective variety, and assume the following: (i) the situation is as in (2.3); (ii) V Y splits on Y; alternatively, V Ys splits, for a very general s ∈ S; (iii) Y ⊂ X ⊗ CĪ k is 2 0 (in particular, it is 2 >0 ). Then V is obtained by successive extensions of line bundles on X. If, moreover, X has the property that H 1 (X, L) = 0 for all L ∈ Pic(X), then V splits.
The very same statements remain valid if X is defined over an uncountable, algebraically closed field, rather than over C.
Proof. First assume that V Y splits. By lemma 2.4, there is a ball B ⊂ S, such that (ρ * V ) Y B splits; lemma 2.6 implies that V is a successive extension of line bundles on a tubular neighbourhood of Y o , o ∈ B. It remains to apply lemma 2.7.
Let τ : S → SF be the trace morphism; for s ∈ S, let k 0 :=F τ (s) be the residue field of τ (s) ∈ SF. For s very general, τ (s) is the generic point of SF, so k 0 =F SF . Assume V s splits; in the Cartesian diagram below V s = Vk 0 ×k 0 C: The splitting of a vector bundle commutes with base change, for varieties defined over algebraically closed fields (cf. [13]). The previous discussion implies that Vk 0 splits on Yk 0 (the geometric generic fibre of YF → SF); hence the same holds for V Y on Y = Yk 0 × C. This brings us back to the previous case. For the last statement, we use once more that the splitting property commutes with base change.
The proof of the theorem even precises the meaning of the term 'very general': if F is the field of definition of X, Y, V , then, in local affine coordinates coming from S F , the coordinates of s ∈ S should be algebraically independent over F.

Definition 2.9
We say that the variety X is 1-splitting if H 1 (X, L) = 0, for all L ∈ P ic(X).
The simplest examples of 1-splitting varieties are the Fano varieties of dimension at least two with cyclic Picard group, and products of such. In 6.4 are obtained examples of homogeneous 1-splitting varieties with larger Picard groups.

Remark 2.10
The genericity assumption in 2.8 plays an essential role. If one is interested in the same result for arbitrary Y , one needs stronger positivity hypotheses, in order to apply effective cohomology vanishing results; in [13] I obtained a similar result for ample, globally generated vector bundles. In this case, the situation (2.3) holds automatically. Theorem 2.8 will be illustrated in the following sections with concrete examples.

Positivity properties of zero loci of sections in vector bundles
Throughout this section, N stands for a globally generated vector bundle on X of rank ν.

3.1.
Sommese's q-ampleness for globally generated vector bundles. We briefly review the q-ampleness concept introduced in [23]. A vector bundle with these properties is called q-ample.
Proof. The first statement is proved in [19,Theorem 1.4]. Since the section is regular, codim X (Y ) = ν. For any coherent sheaf F on X holds so the H ν+q -cohomology on P(N) vanishes; the same holds forX ⊂ P(N).
For ν = 1, the line bundle N is q-ample if an only if the morphism X → |N| has at most q-dimensional fibres. This property is easy to check, and convenient for concrete applications. In contrast, for ν 2, the criterion is not effective; the q-ampleness test for N is too restrictive to check the positivity of the zero loci of its sections (cf. remarks 3.4, 7.4.) 3.2. The positivity criterion 1.12. Suppose Y ⊂ X is lci of codimension δ, and the zero locus of a section s in N, of rank ν 2; we do not assume that s is regular, so we allow δ ν.
In this context, the situation 1.12 arises as follows: since Y is the zero locus of s ∈ Γ(N), the blow-upX fits intõ and holds The assumptions of the proposition 1.12 are satisfied, and we deduce the following. Let W ⊆ Γ(N) be a vector subspace which generates N; let dim W = ν + u + 1. A globally generated vector bundle N on X is equivalent to a morphism f : X→ Gr(W ; ν) into the Grassmannian of ν-dimensional quotients of W ; det(N) is ample if and only if ϕ is finite onto its image.
For Gr(W ; ν) and N the universal quotient bundle on it, we can write the morphism b in (3.2) explicitly: P(N) → P is defined by
Corollary 3.5 Let ϕ : X → Gr(W ; ν), ν 2, be a morphism finite onto its image. If the general fibre of g univ • ϕ : P(W ) × X Flag(W ; ν + u; ν − 1) is at least (p + 1)-dimensional, then Y s is p >0 , for all s ∈ W such that Y s is irreducible lci in X.

Picard groups, and the diagram (2.3). The sheaf K defined by
is locally free, and the incidence variety is isomorphic to P(K). We denote π and ρ the projections onto P(W ) and X respectively; actually we restrict ourselves to a sufficiently small open subset S ⊂ P(W ). The vector bundles N ⊕2 and N ⊕3 are generated by W ⊕2 and W ⊕3 respectively, and determine the double and triple self-intersection diagrams (2.4).

Splitting along zero loci of globally generated vector bundles
Let V be an arbitrary vector bundle on X. The previous discussion immediately yields the following criterion. Theorem 4.1 Let X be 1-splitting variety (cf. 2.9), N be a globally generated vector bundle of rank ν on X such that det(N) is ample, and W ⊂ Γ(X, N) be a generating vector subspace. Assume that N satisfies one of the conditions in proposition 3.6.
Then V splits on X, if its restriction to the zero locus of a very general s ∈ W splits. (If X is not 1-splitting, V is a successive extension of line bundles.) Is natural to ask what happens if one drops the hypothesis that det(N) is ample. The Stein factorization of ϕ : X → Gr(W ; ν) decomposes it into a morphism with connected fibres followed by a finite map.
Theorem 4.2 Let ϕ : X → X ′ be a smooth morphism of relative dimension d, N ′ a globally generated vector bundle on X ′ of rank ν with det(N ′ ) ample, and N = ϕ * N ′ .
Assume that X is 1-splitting, and N ′ satisfies proposition 3.6. Then V splits on X as soon as V splits along Y s , for s very general.
Proof. The conditions of theorem 2.8 are fulfilled. Indeed, consider the family Y ′ of subvarieties of X ′ as above and let Y := ϕ −1 (Y ′ ). The positivity is preserved under pull-back (cf. lemma 1.11, 1.16). For the condition (Pic), the morphismφ :X →X ′ induced at the level of the blows-up is still smooth of relative dimension d, so OX(E Y ) =φ * OX ′ (E Y ′ ) is (q + d)-ample; it remains to apply 1.20.
It is interesting that this yields new results even in the simplest case X = X ′ × V , with X ′ , V smooth, and N ′ = O X ′ (1) is globally generated, ample. If X is 1-splitting, then X ′ , V are both 1-splitting; if either X ′ or V are simply connected, the converse is true.
is ample, globally generated. Then V splits on X, if it splits along a very general divisor in |ϕ * O X ′ (1)|. Example 4.4 (i) (Vector bundles on multi-projective spaces). A cohomological splitting criterion for vector bundles V on X := P n 1 × . . . × P nr is obtained in [10,Theorem 4.7]. It generalizes Horrocks' criterion, and involves the vanishing of (n 1 + 1)·. . . ·(n r + 1) cohomology groups. By repeatedly applying corollary 4.3 for the pull-back of O P n (1), we deduce that V splits if and only if it does so along a very general Y := P 2 × . . . × P 2 . (As formulated, the corollary yields the restriction to a product of copies of P 4 . We can restrict to a product of copies of P 2 because the condition (Pic) in the diagram (2.3) is fulfilled; see also [5,13].) Thus the number of cohomological tests for the splitting of V is reduced to 3 r . (ii) (Vector bundles on products of projective spaces and quadrics). The results of [10] have been extended in [6, Theorem 2.14, 2.15] for vector bundles on the product X 1 ×X 2 , where X 1 is as above and X 2 is a product of hyper-quadrics Q n ⊂ P n+1 . Again, the splitting criterion involves a very large number of cohomological conditions.
Our result implies that a vector bundle on X 1 × X 2 splits if and only if it splits when restricted to a very general X ′ 1 × X ′ 2 ⊂ X 1 × X 2 , where X ′ 1 is a product of projective planes P 2 and X ′ 2 is a product of copies of Q 3 . (The reduction from Q 4 to Q 3 is implied by [5,13].) Hence, the number of necessary cohomological tests is dramatically reduced again.

Positivity properties of sources of G m -actions
Another (totally different) framework which leads to the situation (2.3) arises in the context of actions of the multiplicative group on (almost) homogeneous varieties.

5.1.
Basic properties of the BB-decomposition. We start with general considerations which should justify the appearance of the homogeneous varieties in the next section. Let G be a connected reductive linear algebraic group, and T ⊂ B be a maximal torus and a Borel subgroup of it. Also, consider a 1-parameter subgroup (1-PS for short) λ : G m → G, where G m = C * is the multiplicative group; we assume λ(G m ) ⊂ T . Finally, let X be a smooth projective G-variety with an effective G-action µ : G × X → G × X.
x is a locally trivial, affine space fibration. It is not necessarily a vector bundle, that is the transition functions may be non-linear. • The source Y := Y source and the sink Y sink of the action are uniquely characterized by the fact that Y + = Y + source ⊂ X is open and Y + sink = Y sink . We denote: Then G(λ) is a connected, reductive subgroup of G, P (±λ) ⊂ G are parabolic subgroups, G(λ) is their Levi-component, and U (±λ) the unipotent radical (cf. [24, §13.4]). The adjoint action of λ on Lie(G) decomposes it into the direct sum of its weight spaces; we group them into the zero, strictly positive, and negative weight spaces: Proof. (i) First we prove that G(λ) leaves Y invariant; for y ∈ Y , and c ∈ G(λ) holds: But X λ is the disjoint union of its components, G(λ)y is connected, and contains y ∈ Y ; hence G(λ)Y = Y . We claim that Y is also P (−λ)-invariant; for g ∈ P (−λ) holds: We claim that gy ∈ Y ; otherwise gy ∈ X \ Y is 'repelled' from Y , since Y is the source of λ, and the limit belongs to some other component of X λ . (ii) The previous argument shows also that Y s is G(λ)-invariant, for any s ∈ S BB . Now consider x ∈ X with lim t→0 λ(t)x = y ∈ Y s ⊂ X λ , g ∈ P (λ):

One may check that, lim
is projective, hence Im(µ) is a closed in X.

5.2.
The positivity criterion 1.12. Now we follow the same steps as in section 3.

Proposition 5.4
Assume G m acts effectively on the smooth projective variety X, and Y is the source of the action. Then Y is p >0 , with p : Proof. Take a G m -equivariant embedding of X into some P N , such that X is not contained in a hyperplane. (Linearize the G m -action in a very ample line bundle on X.) In coordinates z N 1 ∈ C N 1 , . . . , z Nr ∈ C Nr , the G m -action on P N is: . . , t mr z Nr ], with 0 < m 2 < . . . < m r .
The source and sink of P N , X are respectively: . . , z Nr ], which can be resolved by a simple blow-up. By restricting to X, we get the situation 1.12: is relatively ample. It remains to understand (bι)(X); we claim that it is a component of X \ Y + . Indeed, (P N 0 ) + = {z = [z N 1 , z N 2 , . . . , z Nr ] | z N 1 = 0} and P N \ (P N 0 ) + is the set of limit points lim t→∞ t × z, where z ∈ (P N 0 ) + and t acts by scalar multiplication (not (5.4)) on the coordinates z N 2 , . . . , z Nr . The situation on X is similar: Y + = X ∩ (P N 0 ) + and Y + → Y is an affine space fibration (cf. [7,Theorem 2.5]); the scalar multiplication on fibres make sense (at least generically), and the corresponding limit points (for t → ∞) are contained in X \ Y + . 5.3. The Picard group, and the diagram (2.3). Theorem 5.5 Let the situation be as in proposition 5.4, with dim X − dim(X \ Y + ) 2. Then Pic(X) → Pic(Y ) is an isomorphism.
Proof. Proposition 5.4 implies that Y is 1 >0 , so the cohomological dimension of X \ Y is at most dim X − 2, cf. 1.8(i); in particular, X \ Y contains no divisors. On the other hand, the inclusion Y + ⊂ X yields the exact sequence 0 → Pic(X \ Y + ) = Pic On the right hand side, the isomorphism holds because Y + → Y is an affine space fibration. The left hand side is the free abelian group generated by the divisors Y + s , s ∈ S BB \ {source}; but we saw that such divisors, contained in X \ Y + , do not exist.
Note that the inequality in theorem 1.20 is 'twice weaker' than in 5.5 above. This is due to the fact that G m -actions yield complex deformation retracts of X \ Y onto subvarieties of X, in contrast with the real retracts arising in the Morse theoretic proof of 1.20. Proof. By lemma 1.17, the double and triple self-intersections (2.4) of Y are non-empty and connected; the generic intersections are also smooth. It remains to prove the isomorphisms of the Picard groups. For Y (so also Y g = gY , g ∈ G) this follows from 5.5; for the double intersections Y ∩ Y g , g ∈ G generic, the isomorphism is implied by 1.20.

Splitting criteria for vector bundles on homogeneous varieties
In this section we apply the conclusions of the previous section to homogeneous varieties. Assume X = G/P , where G is connected, reductive, and P is a parabolic subgroup, and consider a 1-PS λ of G. For any parabolic subgroup Q of G, denote Weyl(Q) := Weyl Levi(Q) . Proof. (i) The differential of the multiplication dµ y : Lie(G) → T y X is surjective at any y ∈ X λ , and is λ-equivariant for the adjoint action on Lie(G). Both sides decompose into direct sums of weight spaces; in particular, dµ y (Lie(G) 0 λ ) = (T y X) 0 , so the differential of µ : G(λ) → Y is surjective at any y ∈ X λ . Hence all the G(λ)-orbits are open in X λ , therefore the components of X λ are homogeneous under G(λ). (ii) The pointê belongs to Y if and only if: -ê is fixed by λ, that is λ ⊂ P ; -the weights of λ on TêX ∼ = Lie(G)/ Lie(P ) are positive, that is Lie(P ) ⊃ Lie(G) − λ . The maximal torus T acts with isolated fixed points on G/P ; they are precisely wP , with w ∈ Weyl(G)/ Weyl(P ). The components Y s ⊂ X λ , s ∈ S BB , are invariant under G(λ) and Now we recall some classical facts about Bruhat decompositions (cf. [20, §8], [24, §8]). If Q, P ⊂ G are two parabolic subgroups, G/P decomposes into the following finite disjoint union of locally closed orbits under the action of Q: Actually, S Bruhat parameterizes the Weyl(Q)-orbits in (G/P ) T . Each double coset in S Bruhat contains a unique representative of minimal length; for each w ∈ S Bruhat of minimal length, dim(QwP ) = length(w) + dim Levi(Q) Levi(Q) ∩ wP w −1 .

Proposition 6.2
The Bialynicki-Birula decomposition of G/P for the action of λ coincides with the Bruhat decomposition for the action of P (λ).
Proof. First, each Bruhat cell is the P (λ)-orbit of some x ∈ (G/P ) T ; second, any such x belongs to a component Y s ⊂ (G/P ) λ ; finally, the BB-cells Y + s are P (λ)-invariant. Hence each Bruhat cell is contained in a unique BB-cell. But the union of the former is G/B, and the latter cells are pairwise disjoint. It follows that each Bruhat cell equals some BB-cell.
For homogeneous varieties, the criterion 2.8 yields the following. Theorem 6.3 Let V be an arbitrary vector bundle on G/P . Consider a 1-parameter subgroup of T , such that Lie(G) − λ ⊂ Lie(P ) (soê ∈ G/P belongs to the source Y ). We assume that dim(X) − dim(X \ Y + ) 2codim X (Y ) + 2 6. (6.1) and g * V splits on Y , for a very general g ∈ G. Then V is a successive extension of line bundles. If the variety G/P is 1-splitting, then V splits.
Since the BB-and the Bruhat-decompositions coincide, (6.1) can be expressed in terms of the root datum of G. A pleasant feature of the criterion is that the splitting of V is reduced to the splitting along a homogeneous subvariety of G/P , so the procedure can be iterated (see 6.5 below). Explicit calculations are performed in section 7.
-Assume α ∈ I. Since any two vectors in I make an angle of at least 90 • (they are simple roots), and (−β ⊥ , c) = −(β, c) 0, for all c ∈ I, we deduce that −β ⊥ is in the cone c∈I R 0 c. Thus holds: For the last step: the angle between a vector and its projection to any plane is at most 90 • . Hence the only relevant condition in (6.2) is the first one. However, (β ⊥ , β) 0 from the very construction, so we must eliminate the case (β ⊥ , β) = 0. This happens precisely when 0 ∈ β + Λ + (I) ⇔ β ∈ Λ + (I) ⇔ β ⊥ α, ∀ α ∈ I. Corollary 6.5 Assume T ⊂ B ⊂ P I ⊂ G, the variety X = G/P I is 1-splitting, and 2 ·#I 1 + #∆ (here # stands for the cardinality).
Then there is λ : G m → T such that the source Y of the action has the properties: (iii) G(λ) ∩ P I corresponds to the simple roots I \ {α 0 }, α 0 ∈ I.
Proof. If Ψ I stands for the roots generated by I, the (positive) roots of the unipotent radical U I ⊂ P I are Ψ + \ Ψ + I (cf. [24,Theorem 8.4.3]). It holds: ). The simplest way to get this situation is if the simple roots of Lie(G(λ)) are ∆ \ {α 0 }, with α 0 ∈ I. It remains to impose that Y is 1-splitting, that is: ∃ β ∈ ∆ \ I such that β is adjacent only to α 0 . Let us prove that there is such an α 0 ∈ I. Otherwise, for all α ∈ I there is β ∈ ∆ \ I adjacent only to α; this yields an injective function I → ∆ \ I, so #I #(∆ \ I), which contradicts the hypothesis.

Splitting criteria for vector bundles on Grassmannians
The Grassmannian plays a central role because is homogeneous, and is the 'universal target' for pairs (X, N) consisting of a variety and a globally generated vector bundle on it. Hence the results of both sections 4, 6 apply. In this section we obtain splitting criteria for vector bundles on isotropic (symplectic and orthogonal) Grassmannians. The degenerate case, when the bilinear form has kernel, is also included, to demonstrate that theorem 2.8 is not restricted only to the situations discussed in sections 3, 5.
Cohomological splitting criteria have been obtained in [21,18,3,17]; however, they involve a large number of conditions. The results below are interesting for their simplicity: indeed, the problem of deciding the splitting of a vector bundle on a Grassmann variety, which is a high dimensional object, is reduced to the splitting along a (very) low dimensional subvariety. Throughout this section, W stands for a w + 1 = ν + u + 1-dimensional vector space. 7.1. The Grassmannian of linear subspaces. This case is discussed in [13], where is proved that things are as good as possible, without any genericity assumptions.
Theorem 7.1 The vector bundle V on Gr(u; C w ), u 2, w u + 2, splits if and only if its restriction to an arbitrary Gr(2; C 4 ) ⊂ Gr(u; C w ) does so. This is in perfect analogy with Horrocks' criterion. However, the proof uses a (fortunate) cohomology vanishing, and can not be extended directly.
7.2. The symplectic-isotropic Grassmannian. Let ω be a skew-symmetric bilinear form on W . If dim W is even, we assume that ω is non-degenerate (ω is a usual symplectic form), while for dim W odd, we assume that dim Ker(ω) = 1 (ω is symplectic on W/Ker(ω) of dimension w). Let X := sp Gr(u + 1; W ) be the variety of ω-isotropic, (u + 1)-dimensional subspaces of W . It is a Fano variety with dim(X) = (u + 1)(2w − 3u) 2 . The quotient bundle N and the tautological bundle U := Ker(W ⊗ O X → N) on X are the pull-back by ϕ of their counterparts on the Grassmannian. An element s ∈ W \ Ker(ω) defines a section in N, whose zero set is the 'smaller' isotropic Grassmannian: In particular, s ∈ W/Ker(ω) \{0} determines σ s (·) := ω(s, ·) ∈ W ∨ ; in this case, σ ⊥ s = s ⊥ . For ω non-degenerate, s → σ s defines an isomorphism W → W ∨ ; however, if dim Ker(ω) = 1, the image of this map is a hyperplane in W ∨ . In this latter case, for generic σ ∈ W ∨ , ω| σ ⊥ is non-degenerate, so σ ⊥ is a symplectic subspace of W . In general, it holds dim Ker(ω| σ ⊥ ) = 1 − dim Ker(ω), for generic σ. We start by explicitly computing the positivity of some subvarieties of X. Deliberately, we consider both zero loci of sections and sources of G m -actions, to illustrate the general theory developed in the previous sections.
The complement of the open BB-cell is X \ Y + = U ∈ X | s ∈ lim t→0 λ(t)U . Consider a basis in U such that the corresponding column matrix is lower triangular; then one can see that X \ Y + = {U | U ⊂ C Remark 7.4 (i) Sommese's criterion 3.1 implies that the quotient bundle on sp Gr(u + 1; W ) is q-ample, for q = dim sp Gr(u + 1; W ) − (u + 1). Hence Gr(u; s ⊥ / s ) is p >0 , with p = (u + 1) − codim sp Gr(u; s ⊥ / s ) = 2u − w + 1 < 0. This shows that this test is week compared with proposition 3.3. (ii) The conclusion of 7.3(i), for ω non-degenerate, can not be obtained by using a 1-PS of G, because sp Gr(u + 1; s ⊥ ) is not homogeneous. (iii) At 7.3(ii), the section s is neither generic (since s = Ker(ω)), nor regular (transverse to the zero section): indeed sp Gr(u; W/ s ) is (w − 2u)-codimensional in sp Gr(u + 1; W ), rather than rk(N) = w − u (cf. subsection 3.2.) Proposition 7.5 Let ω be a skew-symmetric bilinear form on W , dim W = w + 1, and κ := dim Ker(ω) 1. Let X := sp Gr(u + 1; W ), and V be an arbitrary vector bundle on X. In the following cases, V splits if and only if V Ys splits.
Proof. In both cases, we verify (2.3) and apply theorem 2.8 directly. (i) Take S ⊂ W ∨ \ {0} open subset such that (7.3) holds, and Y ⊂ S × X be the zero locus of the universal section in (pr S×X X ) * U ∨ . The morphisms Y π → S and Y ρ → X are both open, and π −1 (s) = Y s is projective, connected; also, dim Ker(ω| s ⊥ ) = 1 − κ.
-The condition (no-△) reads: [s ⊥ o, t ⊥ o, s ⊥ t] ⇒ ∃ V ∈ X, o, s, t ⊂ V. The left hand side implies that o, s, t ⊂ W is isotropic subspace; then V exists, since u 2. -Finally, by lemma 7.3(ii), Y s ⊂ X is u >0 , so at least 2 >0 . Theorem 7.6 Let ω be a skew-symmetric bilinear form on C w , with κ = dim Ker(ω) 1. We consider the isotropic Grassmannian X = sp Gr(u; C w ), with u 2, and an arbitrary vector bundle V on it. Then V splits if and only if it does so along a very general subvariety Y ∼ = sp Gr(2; 4 + κ) of X.
Note that sp Gr(2; 4), the Lagrangian 2-planes in C 4 , is isomorphic through the Plücker embedding to the 3-dimensional quadric.
Proof. By applying repeatedly the first part of previous proposition (after replacing w+1 w and u + 1 u), we deduce that V splits if and only if V Z splits, for some very general subvariety Z ∼ = sp Gr(u, 2u + κ) of X. Now apply the second part to deduce reduce the splitting problem from Z to Y ∼ = sp Gr(2, 4 + κ). (For this latter, the process can not be iterated anymore.) 7.3. The orthogonal-isotropic Grassmannian. The situation is similar to the previous case: let β be a symmetric, non-degenerate, bilinear symmetric form on W , and consider X := o Gr(u + 1; W ) be the variety of isotropic (u + 1)-dimensional subspaces of W ; assume u 1, dim W 5. (If w + 1 = 2(u + 1), the full space of Lagrangian planes in C 2(u+1) has two connected components, and we consider only one of them.) It is a homogeneous variety for G = SO(β), with dim X = (u + 1)(2w − 3u − 2) 2 , and Pic(X) = Z · O X (1).
Similar arguments as before yield the following. The form β| s ⊥ is non-degenerate if and only if s is a non-isotropic vector.
-W os := o, s ⊥ / o, s has an induced non-degenerate orthogonal form, so Y os is connected.
-The diagram (Pic) consists of isomorphisms.
-The condition (1-arm), that is ρ(Y S(o) ) = X is: ∀ U ∈ X ∃ s ∈ S(o) ∃ V ∈ X such that s ∈ U, o, s ⊂ V. Indeed, take s ∈ U ∩ o ⊥ non-zero, so o, s ⊂ W is isotropic; now take any V containing it.
-By lemma 7.7(ii), Y s ⊂ X is at least 2 >0 . Theorem 7.9 Let ω be a non-degenerate bilinear form on C w . We consider the isotropic Grassmannian X = sp Gr(u; C w ), with u 3, and an arbitrary vector bundle V on it. Then Proof. Assume that w 2u + 2. Then the first part of previous proposition implies (after replacing w + 1 w and u + 1 u) that V splits if and only if V Z splits, for some very general subvariety Z ∼ = sp Gr(u, 2u + 2) of X. There remain three possibilities: o Gr(u, 2u), o Gr(u, 2u + 1), o Gr(u, 2u + 2). The theorem follows now from the second part of the proposition.
The somewhat non-uniform formulation of the theorem, compared to 7.6, is due to the lack of sufficient positivity of Y = o Gr(u; 2u + 1) ⊂ o Gr(u, 2u + 2) = X, which is only 1 >0 .