Vector bundles of finite rank on complete intersections of finite codimension in ind-Grassmannians

In this article we establish an analogue of the Barth-Van de Ven-Tyurin-Sato theorem. We prove that a finite rank vector bundle on a complete intersection of finite codimension in a linear ind-Grassmannian is isomorphic to a direct sum of line bundles.


Introduction
The Barth-Van de Ven-Tyurin-Sato theorem claims that any finite rank vector bundle on the infinite complex projective space P ∞ is isomorphic to a direct sum of line bundles. For rank two bundles this was established by Barth and Van de Ven in [1], and for finite rank bundle it was proved by Tyurin in [10] and Sato in [6]. In particular, the Barth-Van de Ven-Tyurin-Sato theorem holds for linear ind-Grassmannians and their linear sections [2,9,14,8].
In this work we will extend these results on the case of a complete intersections in linear ind-Grassmannians. The ground field in this work is C.
First, we recall the definition of linear ind-varieties in general and linear ind-Grassmanninans studied by I. Penkov and A. Tikhomirov in [14,8,7] in particular.
Definition 1. An ind-variety X = lim − → X m is the direct limit of a chain of embeddings: where X m is a smooth algebraic variety for any m ≥ 1.

Definition 2.
A vector bundle E of rank r > 0 on X is the inverse limit of an inverse system of vector bundles {E m } m≥1 of rank r on X (i.e., a system of vector bundles E m with fixed isomorphisms E m ∼ = φ * m E m+1 ).
In particular, the structure sheaf O X = lim ← − O Xm of an ind-variety X is well defined. By the Picard group P icX we understand the group of isomorphism classes of line bundles on X.

Linear ind-Grassmannians and complete intersections in them
For integers m ≥ 1, n m and k m satisfying 1 ≤ k m ≤ n m consider the vector space V nm of dimension n m and the Grassmannian G(k m , n m ) of k mdimensional vector subspaces in V nm . Consider as well the Plücker embedding of G(k m , n m ): G(k m , n m )֒→P Nm−1 = P(Λ Nm ), where N m = nm km . We define the ind-Grassmannian G : = G(∞) as the direct limit lim − → G(k m , n m ) of a chain of embeddings: with conditions lim Further on we assume that the ind-Grassmannian G is linear, i.e., it has a line bundle O G (1) = lim ← − O G(km,nm) (1), where the class of the line bundle O G(km,nm) (1) generates P ic(G(k m , n m )) for all m ≥ 1.
Since G is linear the embeddings can be extended to linear embeddings of Plücker spaces Next, for l ≥ 1 and d 1 , d 2 , ..., d l ≥ 1 consider the linear ind-subvariety X of the linear ind-Grassmannian G that is the direct limit X = lim − → X m of the following chain of embeddings Here X m is the intersection of the Grassmannian G(k m , n m ) .., l, m ≥ 1: Definition 4. The constructed ind-variety X is called a complete intersection of codimension l in the linear ind-Grassmannian G.
In other words, the ind-variety X is an intersection of the ind-Grassmannian G with ind-hypersufaces Y 1 , Y 2 , . . . , Y l in the linear ind-space P ∞ = lim − → P Nm−1 : The main result of this work is the following theorem.
Theorem 1. Any vector bundle of finite rank on a complete intersection X ⊂ G of finite codimension is isomorphic to a direct sum of line bundles.
Acknowledgement. I would like to thank my PhD advisor A.S. Tikhomirov and A. Kuznetsov and D. Panov as well.

Preliminary notions and the idea of proof
To explain the idea of proof of Theorem 1 we will need to give some definitions and recall the main results of articles [12] and [13].
In the case dim(M ) = 1 we call M a projective line or just a line in X.
Using Definition 5 we can give the definition of projective subspace in an ind-variety X.
Definition 7. A path p n (x, y) of length n on an ind-variety X connecting points x and y, is a collection of points x = x 0 , x 1 , ..., x n = y in X and a collection of projective lines l 0 , ..., l n−1 в X such that The variety of all length n paths connecting x and y is denoted by P n (x, y).
Definition 8. Linear ind-variety X is called 1-connected, if for any two points x, y ∈ X there exists a path connecting x with y.
Definition 9. We will say that a vector bundle E on X is trivial on lines if for any projective line l on X the restriction E| l is trivial.
Definition 10. Let E be a rank r bundle on a linear variety X. The splitting type of the bundle E on a projective line l ⊂ X is a collection of numbers r i > 0 and a i ∈ Z, i = 1, ..., s such that A bundle E is called uniform, if its restriction to all projective lines has the same splitting type.
We will need to use several results of articles [12] and [13] on complete intersections X ⊂ G, that we recall now for convenience. Theorem 2 ( [12]). Let X be a complete intersection of G(n, 2n) embedded by Plücker with a collection of hypersurfaces of degrees d 1 , ..., d l : , then the variety P n (u, v) of length n paths connecting any two points u, v in X is non-empty and connected.
We will also need a corollary of this theorem for the case of complete intersection in G(k, n). We will assume k ≤ [ n 2 ]. Corollary 1. Let X be a complete intersection of G(k, n) embedded by Plücker with a collection of hypersurfaces of degrees d 1 , ..., d l : ] then the variety P k (u, v) of length k paths connecting any two points u, v in X is non-empty and connected.
Proof. We will assume that points u and v in G(k, n) are generic 1 ; in the case when the points are not generic the proof goes in the same way the proof of Theorem 2, see [12].
Let U and V be the k-dimensional spaces corresponding to the points u and v of G(k, n). To construct a path p k (u, v) connecting u and v consider the Y i is non-empty and connected. It follows that the variety of paths P k (u, v) in X is non-empty and connected as well.
Рroposition 1 ( [13]). Consider the Segre embedding of P l ×P m in P (l+1)(m+1)−1 . Chose natural numbers (k, d) such that 2kd < min(l, m). Then for any variety Y in P (l+1)(m+1)−1 whose irreducible components have codimension at most k and degree at most d the variety Y ∩ P l × P m is 1-connected.
There exists a projective subspace P k+1 ⊂ Y containing P k if the following holds:

Theorem 3 ([13]
). Any finite rank vector bundle E on X is uniform.
Finally we list the steps in our proof of Theorem 1.
1 i.e., the intersection of the corresponding k-planes is 0.
• We prove first that any finite rank vector bundle E contains a flag of • Next we prove that every finite rank bundle on X trivial on lines is trivial.
• Finally, using Kodaira vanishing theorem we prove that the bundle E splits as a sum

Constructing a flag of subbundles in E
In this section we will construct a flag of subbundles in a rank r vector bundle E on a complete intersection X ⊂ G of finite codimension.
Before doing this formally we will describe the main idea. Chose a point x ∈ X and consider the fibre E x of E over x. According to Grothendieck's theorem ([4] Theorem 2.1.1), for any projective line l passing through x the restriction E| l has a canonical flag of subbundles 0 Hence, we get in E x a flag of subspaces that we denote by F(x, l). A priori the flag F(x, l) might depend on the choice of l passing through x but we will show that this is not the case.
Let B m (x) be the base of the family of lines on X m passing through x, considered as a reduced scheme. We will show that the map from B m (x) to the space of flags E x that associates to each line l ⊂ B m (x) the flag F(x, l) is a morphism for all m. After that we will apply the following theorem.

Proof of Theorem 4
We will start by analysing B m (x). Recall that for i = 1, ..., l the number Proof of Theorem 4. Note that for any d there is a number M such that for any m > M the following two conditions hold: 1) The variety B m (x) is 1-connected. This follows from Proposition 1 together with Proposition 2.
2) Any projective line on B m (x) is contained in a projective subspaces P d ⊂ B m (x) of dimension d. This follows from Lemma 1.
From conditions 1) and 2) it follows that any morphism from B m (x) to any variety of dimension less than d is a morphism to a point. Indeed any line in B m (x) is mapped to a point since it is contained in some P d (which in its turn has to be mapped to a point by [11], section II, § 7, exercise 7.3a). Since any two points in B m (x) are connected by a chain of lines, the whole variety B m (x) is mapped to a point as well.

A standard lemma
Further on we will need the following standard fact. Lemma 2. Let X, Y , Z be projective varieties and suppose that Y is smooth. Suppose we have morphisms p : X → Y and π : X → Z such that the fibres of the morphism p are contained in the fibres of the morphism π. Then there is a morphism f : Y → Z such that f • p = π.
Note that the projection p ′ 1 : φ(X) → Y is an isomorphism ( [15], section II.4, Theorem 2) since by our assumptions the projection p ′ is a bijective morphism and Y is smooth. The desired morphism f : Y → Z is given by the composition f = π ′ • p ′−1 .

Constructing a flag of subbundles in E on X m
Finally, we start to construct the flag of subbundles. Let B m be the base of the family of lines on X m . Let us consider the following set as a reduced scheme. Denote by p m : Γ m → X m the morphism such that p m (x, l) = x.
Recall that E| Xm = E m and the rank of E m is r. The fibre at point x ∈ X m is denoted by E m (x).
According to Theorem 3 the bundle E m is uniform. So for any line l ∈ B m we have the following splitting where r i and a i do not depend on the choice of the line l.
Proof. It is sufficient to prove this statement for all m from Theorem 4 that are larger than M (2r). Let E (1) := E m . Let be the grassmannisation of the bundle E (1) with its natural projection ϕ 1 : . Let us show that the fibres of the morphism p m : (x, l) → x are contained in the fibres of the morphism π 1 . Indeed, for any point x ∈ X m the fibre p −1 m (x) of the morphism p m is isomorphic to the base of the family of lines passing though x on X m . The morphism π 1 sends p −1 m (x) to Gr(r 1 , E (1) (x)) and this map is a map to a point according to Theorem 4 since dim Gr(r 1 , E (1) (x)) < 2r.
Consider the following diagram: y y s s s s s The existence of the section f 1 of the projection ϕ 1 follows from Lemma 2 in which we set X = Γ m , Y = X m , Z = Gr(r 1 , E (1) ), p = p m , π = π 1 . Denote by S r1 the tautological r 1 -dimensional subbundle in ϕ * 1 E (1) . Applying the functor f * 1 to the monomoprphism of bundles we get the following monomorphism of bundles: Denote E (2) = E (1) / f * 1 Sr 1 , and consider the grassmannisation of the bundle E (2) . We get the following diagram where the existence of the morphism f 2 follows from Lemma 2 in the same way as the existence of the morphism f 1 . By the same considerations as before we get the embedding f * 2 τ 2 : f * 2 S r2 E (2) . Denote E (3) = E (2) / f * 2 Sr 2 . Repeating our previous considerations we get a family of epimorphisms of bundles: We associate to it a family of subbundles in E (1) : Indeed, from the construction of the flag of subbundles E m it follows that the restriction of F i /F i−1 to any line l ∈ B m is equal to Remark 1. Subbundles F i of the bundle E m in Theorem 5 are defined in a unique way. This follows from the fact that any vector bundle E on a projective line P 1 has a canonically defined filtration Using the fact that the flag of subbundles 0 = F 0 ⊂ F 1 ⊂ F 2 ⊂ ... ⊂ F s constructed in Theorem 5 does not depend on m, and using the linearity of the ind-variety X = lim − → X m we get the following corollary from Theorem 5.
Theorem 6. Let X be a complete intersection in the linear ind-Grassmannian G and let E be a uniform bundle on X. Then there exists a flag of subbundles such that any quotient bundle F i /F i−1 is a bundle trivial on lines twisted by a line bundle.

A criterion of triviality of bundles trivial on lines
The goal of this section is to prove Theorem 7 which gives a criterion for a bundle trivial on lines to be trivial.
Let X be a normal projective variety and E a be vector bundle on X. Let Y be the Fano scheme of lines on X. Let Z ⊂ X × Y be the universal line. Denote the projections to Y and to X by π and p respectively. Note that π : Z → Y is a P 1 -bundle. Consider the scheme It parameterizes lines on X with pair of points on them. Let p 1 , p 2 : Z 1 → X be the compositions of the projections p r1 , p r2 : Z 1 → Z with the map p : Z → X. Further, we define inductively the variety with projections p 1 : Z 1 → X и p n+2 : Z n+1 → X, using the following diagram: Finally, for a point x ∈ X we define Z n (x) := p −1 1 (x). The projection Z n (x) → X induced by the projection p n+1 : Z n → X will be denoted by f x,n .
Lemma 3. Let E be a vector bundle on X trivial on lines. Then on Z 1 we have an isomorphism p * 1 E ∼ = p * 2 E. Proof. First consider the vector bundle p * E on Z. Since E is trivial on lines, it is trivial on all fibres of π : Z → Y . Since the latter is a P 1 -bundle, it follows that p * E ∼ = π * F for some vector bundle F on Y . Now consider the diagram The next step is the following Lemma 4. If E is trivial on lines then for each n > 0 the bundle f * x,n E on Z n (x) is trivial.

Now we can finish by the following argument
Theorem 7. Assume that X is normal and for some n > 0 and some point x ∈ X the map f x,n : Z n (x) → X is dominant and has connected fibers. Then any vector bundle on X trivial on all lines is trivial.
Proof. Assume that f x,n is dominant and has connected fibers. Then (f x,n ) * O Zn(x) ∼ = O X since X is normal. Hence, by projection formula we have Finally, by Lemma 3 we have f * n, Comparing these two equalities we see that E is trivial.

Splitting of the bundle E
To finish the proof of splitting of the bundle E on the variety X we apply Kodaira vanishing theorem [3].
Theorem 8. Let X be a smooth projective variety and let L be an ample line bundle on it. Then for any q > 0 we have H q (X, K X ⊗ L) = 0.
Recall the following standard fact.
Theorem 9. Let X be a complex projective variety and let E be a vector bundle on X with a subbundle F . Suppose that H 1 ((E/F ) * ⊗F ) = 0 then E ∼ = F ⊕E/F . Recall as well the formula for the canonical class of the Grassmannian G(k, n):

Recall the adjunction formula
Theorem 10. Let X be a smooth variety that is a complete intersection of G(k, n) with a finite collection of smooth hypersurfaces Y 1 , . . . , Y l of degrees d 1 , ..., d l correspondingly, and let d = Recall finally that in the case when the number l of hypersurfaces is less than dim G(k, n) − 2 (i.e., dim X > 2) by Lefschetz hyperplane theorem ( [5], Theorem 3.1.17) we have P ic(X) = P ic(G(k, n)) = Z. H 1 (O X (d − n + r)) = 0 for all r > 0. In particular for n ≫ 1, d = 2 we have H 1 (O X (a)) for all a > 0.
Theorem 11. Let X be an ind-variety and E be a vector bundle on it. Let 0 = F 0 ⊂ F 1 ⊂ ... ⊂ F s = E be a flag of subbundles such that each bundle F i /F i−1 is a twist of a bundle trivial on lines by a line bundle. Then E = ⊕ i F i /F i−1 .
Proof. Let us show that there is M ∈ Z + such that for any m > M the restriction of the bundle E on X m splits as a sum ⊕ i F i /F i−1 | Xm of corresponding subbundles on X m . Namely, chose M such that d − n M < 0. We will establish this splitting by induction in i. Suppose it is proven that F i−1 = 1≤j≤i−1 r j O(a j ), a j−1 > a j . Then F i is an extension of the bundle F i−1 by r i O ai . To prove that F i splits it is enough to know that The latter holds by Corollary 3 since a j − a i > 0 > d − n M for any j ≤ i.

Proof of Theorem 1
In this section we prove Theorem 1. Since the bundle E is uniform we can apply Theorem 6. It follows that each quotient bundle F i /F i−1 for 1 ≤ i ≤ s is a twist of a bundle trivial on lines by a line bundle.
Set in Theorem 7 n = k m and apply it to X = X m (a complete intersection in G(k m , n m )). Then the fibre of the morphism f x,km over y ∈ X m is the space of paths of length k m on X m that start at x and finish at y. In Corollary 1 set X = X m (recall that X m = G(k m , n m ) ∩ l i=1 Y i,m ). Then for a sufficiently large m such that k m and n m satisfy the inequality 2 i (d i + 1) ≤ km 2 ≤ [ nm 4 ] where d i = deg Y i,m , we deduce that the space of paths P km (x, y) connecting x with y on X m is non-empty and connected. Hence the fibres of the morphism f x,km are non-empty and connected, in particular f x,km is dominant.
So the conditions of Theorem 7 hold for X m . It follows that any vector bundle on X trivial on lines is trivial. So each F i /F i−1 is a twist of a trivial bundle by a line bundle.
Finally, using Theorem 11 we deduce that the bundle E is a direct sum E = ⊕ i F i /F i−1 . This proves Theorem 1.