To Yuri Ivanovich Manin on the occasion of his 70^{th} birthday

RANK 2 VECTOR BUNDLES ON IND-GRASSMANNIANS

IVAN PENKOV AND ALEXANDER S. TIKHOMIROV

1. Introduction

The simplest example of an ind-grassmannian is the infinite projective space P^{∞}. The Barth-Van de Ven-Tyurin (BVT) Theorem, proved more than 30 years ago [BV], [T], [Sa1] (see also a recent proof by A. Coanda and G. Trautmann, [CT]), claims that any vector bundle of finite rank on P^{∞} is isomorphic to a direct sum of line bundles. In the last decade natural examples of infinite flag varieties (or flag ind-varieties) have arisen as homogeneous spaces of locally linear ind-groups, [DPW], [DiP]. In the present paper we concentrate our attention to the special case of ind-grassmannians, i.e. to inductive limits of grassmannians of growing dimension. If V is a countable dimensional vector space, where ⋃_{n>k} V^{n} = V, then the ind-variety

G(k; V) = lim_{→} G(k; V^{n}) (or simply G(k; ∞)) of k-dimensional subspaces of V is of course an ind-grassmannian: this is the simplest example beyond P^{∞} = G(1; ∞). A significant difference between G(k; V) and a general ind-grassmannian X = lim_{→} G(k_{i}; V^{n_{i}}) defined via a sequence of closed immersions

(1)  G(k_{1}; V^{n_{1}}) ⟶^{φ_{1}} G(k_{2}; V^{n_{2}}) ⟶^{φ_{2}} ⋯ ⟶^{φ_{m−1}} G(k_{m}; V^{n_{m}}) ⟶^{φ_{m}} …,

is that in general the morphisms φ_{m} can have arbitrary degrees. We say that the indgrassmannian X is twisted if deg φ_{m} > 1 for infinitely many m, and that X is linear if deg φ_{m} = 1 for almost all m.

Conjecture 1.1. Let the ground field be ℂ and E be a vector bundle of rank r ∈ Z_{>0} on an ind-grasmannian X = lim_{→} G(k_{m}; V^{n_{m}}), i.e. E = lim_{←} E_{m}, where {E_{m}} is an inverse system of vector bundles of (fixed) rank r on G(k_{m}; V^{n_{m}}). Then

(i) E is semisimple: it is isomorphic to a direct sum of simple vector bundles on X, i.e. vector bundles on X with no non-trivial subbundles;

(ii) for m ≫ 0 the restriction of each simple bundle E to G(k_{m}, V^{n_{m}}) is a homogeneous

vector bundle; (iii) each simple bundle E′ has rank 1 unless X is isomorphic G(k; ∞) for some k: in the

latter case E′, twisted by a suitable line bundle, is isomorphic to a simple subbundle of the tensor algebra T^{·}(S), S being the tautological bundle of rank k on G(k; ∞);

(iv) each simple bundle E (and thus each vector bundle of finite rank on X) is trivial whenever X is a twisted ind-grassmannian.

The BVT Theorem and Sato’s theorem about finite rank bundles on G(k; ∞), [Sa1], [Sa2], as well as the results in [DP], are particular cases of the above conjecture. The purpose of the present note is to prove Conjecture 1.1 for vector bundles of rank 2, and also for vector bundles of arbitrary rank r on linear ind-grassmannians X.

In the 70’s and 80’s Yuri Ivanovich Manin taught us mathematics in (and beyond) his seminar, and the theory of vector bundles was a reoccuring topic (among many others). In 1980, he asked one of us (I.P.) to report on A. Tyurin’s paper [T], and most importantly to try to understand this paper. The present note is a very preliminary progress report.

Acknowledgement. We acknowledge the support and hospitality of the Max Planck Institute for Mathematics in Bonn where the presented note was conceived. A. S. T. also acknowledges partial support from Jacobs University Bremen.

2. Notation and Conventions

The ground field is ℂ. Our notation is mostly standard: if X is an algebraic variety, (over ℂ), O_{X} denotes its structure sheaf, Ω_{X}^{1} (respectively T_{X}) denotes the cotangent (resp. tangent) sheaf on X under the assumption that X is smooth etc. If F is a sheaf on X, its cohomologies are denoted by H^{i}(F), h^{i}(F) := dim H^{i}(F), and χ(F) stands for the Euler characteristic of F. The Chern classes of F are denoted by c_{i}(F). If f : X → Y is a morphism, f^{∗} and f_{∗} denote respectively the inverse and direct image functors of O-modules. All vector bundles are assumed to have finite rank. The dual of a sheaf of O_{X}-modules F (as well as the the dual of a vector space) we denote by the superscript ^{∨}. Under an embedding of smooth varieties i : X → Y we always understand a closed immersion. Furthermore, in what follows for any ind-grassmannian X defined by (1), no embedding φ_{i} is an isomorphism.

We fix a finite dimensional space V and denote by X the grassmannian G(k; V) for k < dim V. In the sequel we write sometimes G(k; n) indicating simply the dimension of V. Below we will often consider (parts of) the following diagram of flag varieties: (2)

\xymatrix{ &  & Z := F l(k − 1, k, k + 1; V) \ar@{-->}[dl]_{π_{1}} \ar@{-->}[dr]^{π_{2}} \\  & Y := F l(k − 1, k + 1; V) \ar[dl]_{p_{1}} \ar@{-->}[dr]^{p_{2}} &  & X := G(k; V), \\ Y^{1} := G(k − 1; V) &  & Y^{2} := G(k + 1; V)}

under the assumption that k + 1 < dim V. Moreover we reserve the letters X, Y, Z for the varieties in the above diagram. By S_{k}, S_{k−1}, S_{k+1} we denote the tautological bundles on X,Y and Z, whenever they are defined (S_{k} is defined on X and Z, S_{k−1} is defined on Y^{1}, Y and Z, etc.). By O_{X}(i), i ∈ Z, we denote the isomorphism class (in the Picard group Pic X) of the line bundle (Λ^{k}(S_{k}^{∨}))^{⊗i}, where Λ^{k} stands for the k^{th} exterior power (in this case maximal exterior power as rkS_{k}^{∨} = k). The Picard group of Y is isomorphic to the direct product of the Picard groups of Y^{1} and Y^{2}, and by O_{Y} (i, j) we denote the isomorphism class of the line bundle p_{1}^{∗}(Λ^{k−1}(S_{k−1}^{∨}))^{⊗i} ⊗_{O_{Y}} p_{2}^{∗}(Λ^{k+1}(S_{k+1}^{∨}))^{⊗j}.

If φ : X = G(k; V) → X′ := G(k; V′) is an embedding, then φ^{∗}O_{X′}(1) ≃ O_{X}(d) for some d ∈ Z_{≥0}: by definition d is the degree deg φ of φ. We say that φ is linear if deg φ = 1. By a projective subspace (in particular a line, i.e. a 1-dimensional projective subspace) of X we mean a linearly embedded projective space into X. It is well known that all such are Schubert varieties of the form {V^{k} ∈ X|V^{k−1} ⊂ V^{k} ⊂ V^{t}} or {V ∈ X|V^{i} ⊂ V^{k} ⊂ V^{k+1}}, where V^{k} is a variable k-dimensional subspace of V, and V^{k−1}, V^{k+1}, V^{t}, V^{i} are fixed subspaces of V of respective dimensions k − 1, k + 1, t, i. (In what follows we will automatically assume that a given finite dimensional space written as V^{t} has dimension t). In other words, all projective subspaces of X are of the form G(1; V^{t}/V^{k−1}) or G(k − i, V^{k+1}/V^{i}). Note also that Y = F l(k − 1, k + 1; V) is the variety of lines in X = G(k; V).

3. The linear case

We consider the cases of linear and twisted ind-grassmannians separately. In the case of a linear ind-grassmannian, we show that Conjecture 1.1 is a straightforward corollary of existing results combined with the following proposition. We recall, [DP], that a standard extension of grassmannians is an embedding of the form

(3)  G(k; V) → G(k + a; V ⊕ Ŵ), {V^{k} ⊂ ℂ^{n}} ↦ {V^{k} ⊕ W ⊂ V ⊕ Ŵ },

where W is a fixed a-dimensional subspace of a finite dimensional vector space Ŵ.

Proposition 3.1. Let φ : X = G(k; V) → X′ := G(k′; V′) be an embedding of degree 1. Then φ is a standard extension, or φ factors through a standard extension ℙ^{r} → G(k′; V′) for some r.

Proof. We assume that k ≤ n − k, k ≤ n′ − k′, where n = dim V and n′ = dim V′, and use induction on k. For k = 1 the statement is obvious as the image of φ is a projective subspace of G(k′; V′) and hence φ is a standard extension. Assume that the statement is true for k − 1. Since deg φ = 1, φ induces an embedding φ_{Y} : Y → Y′, where Y = F l(k − 1, k + 1; V) is the variety of lines in X and Y = F l(k′ − 1, k′ + 1; V′) is the variety of lines in X′. Moreover, clearly we have a commutative diagram of natural projections and embeddings

[figure: Z; φ_{Z}; π_{1}; π_{2}; Z′; π′_{1}; π′_{2}; Y; X; Y′; X′,; φ_{Y}; φ]

where Z := F l(k − 1, k, k + 1; V) and Z′ := F l(k′ − 1, k′, k′ + 1; V′).

We claim that there is an isomorphism

(4)  φ_{Y}^{∗} O_{Y′}(1, 1) ≃ O_{Y} (1, 1).

Indeed, φ_{Y}^{∗} O_{Y′}(1, 1) is determined up to isomorphism by its restriction to the fibers of p_{1} and p_{2} (see diagram (2)), and therefore it is enough to check that

(5)  φ_{Y}^{∗} O_{Y′}(1, 1)_{|p_{1}^{−1}(V^{k−1})} ≃ O_{p_{1}^{−1}(V^{k−1})}(1),
(6)  φ_{Y}^{∗} O_{Y′}(1, 1)_{|p_{2}^{−1}(V^{k+1})} ≃ O_{p_{2}^{−1}(V^{k+1})}(1)

for some fixed subspaces V^{k−1} ⊂ V, V^{k+1} ⊂ V. Note that the restriction of φ to the projective subspace G(1; V/V^{k−1}) ⊂ X is simply an isomorphism of G(1; V/V^{k−1}) with a projective subspace of X′, hence the map induced by φ on the variety G(2; V/V^{k−1}) of projective lines in G(1; V/V^{k−1}) is an isomorphism with a grassmannian of 2-dimensional subspaces of an appropriate quotient space of V′. Note furthermore that p_{1}^{−1}(V^{k−1}) is nothing but the variety of lines G(2; V/V^{k−1}) in G(1; V/V^{k−1}), and that the image of G(2; V/V^{k−1}) under φ is nothing but φ_{Y} (p_{1}^{−1}(V^{k−1})). This shows that the restriction of φ_{Y}^{∗} O_{Y′}(1, 1) to G(2; V/V^{k−1}) is isomorphic to the restriction of O_{Y} (1, 1) to G(2; V/V^{k−1}), and we obtain (5). The isomorphism (6) follows from a very similar argument.

The isomorphism (4) leaves us with two alternatives:

(7)  φ_{Y}^{∗} O_{Y′}(1, 0) ≃ O_{Y} or φ_{Y}^{∗} O_{Y′}(0, 1) ≃ O_{Y} ,

or

(8)  φ_{Y}^{∗} O_{Y′}(1, 0) ≃ O_{Y} (1, 0) or φ_{Y}^{∗} O_{Y′}(1, 0) ≃ O_{Y} (0, 1).

Let (7) hold, more precisely let φ_{Y}^{∗} O_{Y′}(1, 0) ≃ O_{Y}. Then φ_{Y} maps each fiber of p_{2} into a single point in Y′ (depending on the image in Y^{2} of this fiber), say ((V′)^{k′−1} ⊂ (V′)^{k′+1}), and moreover the space (V′)^{k′−1} is constant. Thus φ maps X into the projective subspace G(1; V′/(V′)^{k′−1}) of X′. If φ_{Y}^{∗} O_{Y′}(0, 1) ≃ O_{Y}, then φ maps X into the projective subspace G(1; (V′)^{k′+1}) of X′. Therefore, the Proposition is proved in the case (7) holds.

We assume now that (8) holds. It is easy to see that (8) implies that φ induces a linear embedding φ_{Y^{1}} of Y^{1} := G(k − 1; V) into G(k′ − 1; V′) or G(k′ + 1; V′). Assume that φ_{Y^{1}} : Y^{1} → (Y′)^{1} := G(k′ − 1; V′) (the other case is completely similar). Then, by the induction assumption, φ_{Y^{1}} is a standard extension or factors through a standard extension ℙ^{r} → (Y′)^{1}. If φ_{Y^{1}} is a standard extension corresponding to a fixed subspace W ⊂ Ŵ, then φ_{Y^{1}}^{∗}S_{k′−1} ≃ S_{k−1} ⊕ (W ⊗_{ℂ} O_{Y^{1}}) and we have a vector bundle monomorphism

(9)  0 → π_{1}^{∗}p_{1}^{∗}φ_{Y^{1}}^{∗}S_{k′−1} → π_{2}^{∗}φ^{∗}S_{k′}.

By restricting (9) to the fibers of π_{1} we see that the quotient line bundle π_{2}^{∗}φ^{∗}S_{k′}/π_{1}^{∗}p_{1}^{∗}φ_{Y^{1}}^{∗}S_{k′−1} is isomorphic to S_{k}/S_{k−1} ⊗ π_{1}^{∗}p_{1}^{∗}L, where L is a line bundle on Y^{1}. Applying π_{2∗} we obtain

(10)  0 → W ⊗_{ℂ} O_{X} → π_{2∗}(π_{2}^{∗}φ^{∗}S_{k′}) = φ^{∗}S_{k′} → π_{2∗}((S_{k}/S_{k−1}) ⊗ π_{1}^{∗}p_{1}^{∗}L) → 0.

Since rkφ^{∗}S_{k′} = k′ and dim W = k′ − k, rkπ_{2∗}((S_{k}/S_{k−1}) ⊗ π_{1}^{∗}p_{1}^{∗}L) = k, which implies immediately that L is trivial. Hence (10) reduces to 0 → W ⊗_{ℂ} O_{X} → φ^{∗}S_{k′} → S_{k} → 0, and thus

(11)  φ^{∗}S_{k′} ≃ S_{k} ⊕ (W ⊗_{ℂ} O_{X})

as there are no non-trivial extensions of S_{k} by a trivial bundle. Now (11) implies that φ is a standard extension.

It remains to consider the case when φ_{Y^{1}} maps Y^{1} into a projective subspace ℙ^{s} of (Y′)^{1}. Then ℙ^{s} is of the form G(1; V′/(V′)^{k′−2}) for some (V′)^{k′−2} ⊂ V′, or of the form G(k′ − 1; (V′)^{k′}) for some (V′)^{k′} ⊂ V′. The second case is clearly impossible because it would imply that φ maps X into the single point (V′)^{k′}. Hence ℙ^{s} = G(1; V′/(V′)^{k′−2}) and φ maps X into the grassmannian G(2; V′/(V′)^{k′−2}) in G(k′; V′). Let S′_{2} be the rank 2 tautological bundle on G(2; V′/(V′)^{k′−2}). Then its restriction S′′ := φ^{∗}S′_{2} to any line l in X is isomorphic to O_{l} ⊕ O_{l}(−1), and we claim that this implies one of the two alternatives:

(12)  S′′ ≃ O_{X} ⊕ O_{X}(−1)

or

(13)  S′′ ≃ S_{2} and k = 2, or S′′ ≃ (V ⊗_{ℂ} O_{X})/S_{2} and k = n − k = 2.

Let k ≥ 2. The evaluation map π_{1}^{∗}π_{1∗}π_{2}^{∗}S′′ → π_{2}^{∗}S′′ is a monomorphism of the line bundle π_{1}^{∗}L := π_{1}^{∗}π_{1∗}π_{2}^{∗}S′′ into π_{2}^{∗}S′′ (here L := π_{1∗}π_{2}^{∗}S′′). Restricting this monomorphism to the fibers of π_{2} we see immediately that π_{1}^{∗}L is trivial when restricted to those fibers and is hence trivial. Therefore L is trivial, i.e. π_{1}^{∗}L = O_{Z}. Push-down to X yields

(14)  0 → O_{X} → S′′ → O_{X}(−1) → 0,

and hence (14) splits as Ext^{1}(O_{X}(−1), O_{X}) = 0. Therefore (12) holds. For k = 2, there is an additional possibility for the above monomorphisms to be of the form π_{1}^{∗}O_{Y} (−1, 0) → π_{2}^{∗}S (or of the form π_{1}^{∗}O_{Y} (0, −1) → π_{2}^{∗}S if n − k = 2) which yields the option (13).

If (12) holds, φ maps X into an appropriate projective subspace of G(2; V′/(V′)^{k′−2}) which is then a projective subspace of X′, and if (13) holds, φ is a standard extension corresponding to a zero dimensional space W. The proof is now complete. □

We are ready now to prove the following theorem.

Theorem 3.2. Conjecture 1.1 holds for any linear ind-grassmannian X.

Proof. Assume that deg φ_{m} = 1 for all m, and apply Proposition 3.1. If infinitely many φ_{m}’s factor through respective projective subspaces, then X is isomorphic to P^{∞} and the BVT Theorem implies Conjecture 1.1. Otherwise, all φ_{m}’s are standard extensions of the form (3). There are two alternatives: lim_{m→∞} k_{m} = lim_{m→∞} n_{m} − k_{m} = ∞, or one of the limits lim_{m→∞} k_{m} or lim_{m→∞} n_{m} − k_{m} equals l for some l ∈ N. In the first case the claim of Conjecture 1.1 is proved in [DP]: Theorem 4.2. In the second case X is isomorphic to G(l; ∞), and therefore Conjecture 1.1 is proved in this case by E. Sato in [Sa2]. □

4. Auxilary results

In order to prove Conjecture 1.1 for rank 2 bundles E on a twisted ind-grassmannian X = lim_{→} G(k_{m}; V^{n_{m}}), we need to prove that the vector bundle E = lim_{←} E_{m} of rank 2 on X is trivial, i.e. that E_{m} is a trivial bundle on G(k_{m}; V^{n_{m}}) for each m. From this point on we assume that none of the grassmannians G(k_{m}; V^{n_{m}}) is a projective space, as for a twisted projective ind-space Conjecture 1.1 is proved in [DP] for bundles of arbitrary rank r.

The following known proposition gives a useful triviality criterion for vector bundles of arbitrary rank on grassmannians.

Proposition 4.1. A vector bundle E on X = G(k; n) is trivial iff its restriction E_{|l} is trivial for every line l in G(k; n), l ∈ Y = F l(k − 1, k + 1; n).

Proof. We recall the proof given in [P]. It uses the well known fact that the Proposition holds for any projective space, [OSS, Theorem 3.2.1]. Let first k = 2, n = 4, i.e. X = G(2; 4). Since E is linearly trivial, π_{2}^{∗}E is trivial along the fibers of π_{1} (we refer here to diagram (2)). Moreover, π_{1∗}π_{2}^{∗}E is trivial along the images of the fibers of π_{2} in Y. These images are of the form ℙ_{1}^{1} × ℙ_{2}^{1}, where ℙ_{1}^{1} (respectivelly ℙ_{2}^{1}) are lines in Y^{1} := G(1; 4) and Y^{2} := G(3; 4). The fiber of p_{1} is filled by lines of the form ℙ_{2}^{1}, and thus π_{1∗}π_{2}^{∗}E is linearly trivial, and hence trivial along the fibers of π_{1}. Finally the lines of the form ℙ_{1}^{1} fill Y^{1}, hence p_{1}_{∗}π_{1∗}π_{2}^{∗}E is also a trivial bundle. This implies that E = π_{2∗}π_{1}^{∗}p_{1}^{∗}(p_{1∗}π_{1∗}π_{2}^{∗}E) is also trivial.

The next case is the case when k = 2 and n is arbitrary, n ≥ 5. Then the above argument goes through by induction on n since the fiber of p_{1} is isomorphic to G(2; n − 1). The proof is completed by induction on k for k ≥ 3: the base of p_{1} is G(k − 1; n) and the fiber of p_{1} is G(2; n − 1). □

If C ⊂ N is a smooth rational curve in an algebraic variety N and E is a vector bundle on N, then by a classical theorem of Grothendieck, E_{|C} is isomorphic to ⨁_{i} O_{C}(d_{i}) for some d_{1} ≥ d_{2} ≥ ⋯ ≥ d_{rkE}. We call the ordered rkE-tuple (d_{1}, …, d_{rkE}) the splitting type of E_{|C} and denote it by d_{E}(C). If N = X = G(k; n), then the lines on N are parametrized by points l ∈ Y, and we obtain a map

Y → Z^{rkE} : l ↦ d_{E}(l).

By semicontinuity (cf. [OSS, Ch.I, Lemma 3.2.2]), there is a dense open set U_{E} ⊂ Y of lines with minimal splitting type with respect to the lexicographical ordering on Z^{rkE}. Denote this minimal splitting type by d_{E}. By definition, U_{E} = {l ∈ Y | d_{E}(l) = d_{E}} is the set of nonjumping lines of E, and its complement Y \ U_{E} is the proper closed set of jumping lines.

A coherent sheaf F over a smooth irreducible variety N is called normal if for every open set U ⊂ N and every closed algebraic subset A ⊂ U of codimension at least 2 the restriction map F (U) → F (U ∖ A) is surjective. It is well known that a rank-1 normal torsion-free sheaf F is reflexive, i.e. F^{∨∨} = F. Therefore, by [OSS, Ch.II, Theorem 2.1.4] F is necessarily a line bundle (see [OSS, Ch.II, 1.1.12 and 1.1.15]).

Theorem 4.2. Let E be a rank r vector bundle of splitting type d_{E} = (d_{1}, ..., d_{r}), d_{1} ≥ ... ≥ d_{r}, on X = G(k; n). If d_{s} − d_{s+1} ≥ 2 for some s < r, then there is a normal subsheaf F ⊂ E of rank s with the following properties: over the open set π_{2}(π_{1}^{−1}(U_{E})) ⊂ X the sheaf F is a subbundle of E, and for any l ∈ U_{E}

F_{|l} ≃ s ; ⨁_{i=1}O_{l}(d_{i}).

Proof. It is similar to the proof of Theorem 2.1.4 of [OSS, Ch.II]. Consider the vector bundle E′ = E ⨂ O_{X}(−d_{s}) and the evaluation map Φ : π_{1}^{∗}π_{1∗}π_{2}^{∗}E′ → π_{2}^{∗}E′. The definition of U_{E} implies that Φ_{|π_{1}^{−1}(U_{E})} is a morphism of constant rank s and that its image ImΦ ⊂ π_{2}^{∗}E′ is a subbundle of rank s over π_{1}^{−1}(U_{E}). Let M := π_{2}^{∗}E′/imΦ, let T (M) be the torsion subsheaf of M, and F′ := ker(π_{2}^{∗}E′ → M′ := M/T (M)). Consider the singular set Sing F′ of the sheaf F′ and set A := Z ∖ Sing F′. By the above, A is an open subset of Z containing π_{1}^{−1}(U_{E}) and f = π_{2}_{|A} : A → B := π_{2}(A) is a submersion with connected fibers.

Next, take any point l ∈ Y and put L := π_{1}^{−1}(l). By definition, L ≃ ℙ^{1}, and we have

(15)  T_{Z/X}_{|L} ≃ O_{L}(−1)^{⊕(n−2)},

where T_{Z/X} is the relative tangent bundle of Z over X. The construction of the sheaves F′ and M implies that for any l ∈ U_{E}: F′_{|L}^{∨} = ⊕_{i=1}^{s}O_{L}(−d_{i}+d_{s}), M′_{|L} = ⊕_{i=s+1}^{r}O_{L}(d_{i}−d_{s}). This, together with (15) and the condition d_{s} − d_{s+1} ≥ 2, immediately implies that H^{0}(Ω_{A/B}^{1} ⊗ F′^{∨} ⊗ M′_{|L}) = 0. Hence H^{0}(Ω_{A/B}^{1} ⊗ F′^{∨} ⊗ M′_{|π_{1}^{−1}(U_{E})}) = 0, and thus, since π_{1}^{−1}(U_{E}) is dense open in Z, Hom(T_{A/B}, Hom(F′, M′_{|A})) = H^{0}(Ω_{A/B}^{1} ⊗ F′^{∨} ⊗ M′_{|A}) = 0. Now we apply the Descent Lemma (see [OSS, Ch.II, Lemma 2.1.3]) to the data (f_{|π_{1}^{−1}(U_{E})} : π_{1}^{−1}(U_{E}) → V_{E}, F′_{|π_{1}^{−1}(U_{E})} ⊂ E′_{|π_{1}^{−1}(U_{E})}). Then F := (π_{2∗}F′) ⊗ O_{X}(−d_{s}) is the desired sheaf. □

5. The case rkE = 2

In what follows, when considering a twisted ind-grassmannian X = lim_{→} G(k_{m}; V^{n_{m}}) we set G(k_{m}; V^{n_{m}}) = X_{m}. Theorem 4.2 yields now the following corollary.

Corollary 5.1. Let E = lim_{←} E_{m} be a rank 2 vector bundle on a twisted ind-grassmannian X = lim_{→} X_{m}. Then there exists m_{0} ≥ 1 such that d_{E_{m}} = (0, 0) for any m ≥ m_{0}.

Proof. Note first that the fact that X is twisted implies

(16)  c_{1}(E_{m}) = 0, m ≥ 1.

Indeed, c_{1}(E_{m}) is nothing but the integer corresponding to the line bundle Λ^{2}(E_{m}) in the identification of Pic X_{m} with Z. As X is twisted, c_{1}(E_{m}) = deg φ_{m} deg φ_{m+1} … deg φ_{m+k}c_{1}(E_{m+k+1}) for any k ≥ 1, in other words c_{1}(E_{m}) is divisible by larger and larger integers and hence c_{1}(E_{m}) = 0 (cf. [DP, Lemma 3.2]). Suppose that for any m_{0} ≥ 1 there exists m ≥ m_{0} such that d_{E_{m}} = (a_{m}, −a_{m}) with a_{m} > 0. Then Theorem 4.2 applies to E_{m} with s = 1, and hence E_{m} has a normal rank-1 subsheaf F_{m} such that

(17)  F_{m|l} ≃ O_{l}(a_{m})

for a certain line l in X_{m}. Since F_{m} is a torsion-free normal subsheaf of the vector bundle E, the sheaf F_{m} is a line bundle, i.e. F_{m} ≃ O_{X_{m}}(a_{m}). Therefore we have a monomorphism:

(18)  0 → O_{X_{m}}(a_{m}) → E_{m}, a_{m} ≥ 1.

This is clearly impossible. In fact, this monomorphism implies in view of (16) that any rational curve C ⊂ X_{m} of degree δ_{m} := deg φ_{1} · ... · deg φ_{m−1} has splitting type d_{E_{m}}(C) = (a′_{m}, −a′_{m}), where a′_{m} ≥ a_{m}δ_{m} ≥ δ_{m}. Hence, by semiconinuity, any line l ∈ X_{1} has splitting type d_{E_{1}}(l) = (b, −b), b ≥ δ_{m}. Since δ_{m} → ∞ as m_{0} → ∞, this is a contradiction. □

We now recall some standard facts about the Chow rings of X_{m} = G(k_{m}; V^{n_{m}}), (see, e.g., [F, 14.7]):

(i) A^{1}(X_{m}) = Pic(X_{m}) = ℤ[𝕍_{m}], A^{2}(X_{m}) = ℤ[𝕎_{1,m}] ⊕ ℤ[𝕎_{2,m}], where 𝕍_{m}, 𝕎_{1,m}, 𝕎_{2,m} are the following Schubert varieties: 𝕍_{m} := {V^{k_{m}} ∈ X_{m}| dim(V^{k_{m}} ∩ V_{0}^{n_{m}−k_{m}}) ≥ 1 for a fixed subspace V_{0}^{n_{m}−k_{m}−1} of V^{n_{m}}}, 𝕎_{1,m} := {V^{k_{m}} ∈ X_{m}| dim(V^{k_{m}} ∩ V_{0}^{n_{m}−k_{m}−1}) ≥ 1 for a fixed subspace V_{0}^{n_{m}−k_{m}−1} in V^{n_{m}}}, 𝕎_{2,m} := {V^{k_{m}} ∈ X_{m}| dim(V^{k_{m}} ∩ V_{0}^{n_{m}−k_{m}+1}) ≥ 2 for a fixed subspace V_{0}^{n_{m}−k_{m}+1} of V^{n_{m}}};

(ii) [𝕍_{m}]^{2} = [𝕎_{1,m}] + [𝕎_{2,m}] in A^{2}(X_{m}); (iii) A_{2}(X_{m}) = ℤ[ℙ_{1,m}^{2}]⊕ℤ[ℙ_{2,m}^{2}], where the projective planes ℙ_{1,m}^{2} (called α-planes) and ℙ_{2,m}^{2}

(called β-planes) are respectively the Schubert varieties ℙ_{1,m}^{2} := {V^{k_{m}} ∈ X_{m}| V_{0}^{k_{m}−1} ⊂ V^{k_{m}} ⊂ V_{0}^{k_{m}+2} for a fixed flag V_{0}^{k_{m}−1} ⊂ V_{0}^{k_{m}+2} in V^{n_{m}}}, ℙ_{2,m}^{2} := {V^{k_{m}} ∈ X_{m}| V_{0}^{k_{m}−2} ⊂ V^{k_{m}} ⊂ V_{0}^{k_{m}+1} for a fixed flag V_{0}^{k_{m}−2} ⊂ V_{0}^{k_{m}+1} in V^{n_{m}}};

(iv) the bases [𝕎_{i,m}] and [ℙ_{j,m}^{2}] are dual in the standard sense that [𝕎_{i,m}] · [ℙ_{j,m}^{2}] = δ_{i,j}.

Lemma 5.2. There exists m_{1} ∈ Z_{>0} such that for any m ≥ m_{1} one of the following holds: (1) c_{2}(E_{m}_{|ℙ_{1,m}^{2}}) > 0, c_{2}(E_{m}_{|ℙ_{2,m}^{2}}) ≤ 0,

(2) c_{2}(E_{m}_{|ℙ_{2,m}^{2}}) > 0, c_{2}(E_{m}_{|ℙ_{1,m}^{2}}) ≤ 0,

(3) c_{2}(E_{m}_{|ℙ_{1,m}^{2}}) = 0, c_{2}(E_{m}_{|ℙ_{2,m}^{2}}) = 0.

Proof. According to (i), for any m ≥ 1 there exist λ_{1m}, λ_{2m} ∈ Z such that

(19)  c_{2}(E_{m}) = λ_{1m}[𝕎_{1,m}] + λ_{2m}[𝕎_{2,m}].

Moreover, (iv) implies

(20)  λ_{jm} = c_{2}(E_{m}_{|ℙ_{j,m}^{2}}), j = 1, 2.

Next, (i) yields: (21) φ_{m}^{∗}[𝕎_{1,m+1}] = a_{11}(m)[𝕎_{1,m}] + a_{21}(m)[𝕎_{2,m}], φ_{m}^{∗}[𝕎_{2,m+1}] = a_{12}(m)[𝕎_{1,m}] + a_{22}(m)[𝕎_{2,m}],

where a_{ij}(m) ∈ ℤ. Consider the 2 × 2-matrix A(m) = (a_{ij}(m)) and the column vector Λ_{m} = (λ_{1m}, λ_{2m})^{t}. Then, in view of (iv), the relation (21) gives: Λ_{m} = A(m)Λ_{m+1}. Iterating this equation and denoting by A(m, i) the 2 × 2-matrix A(m) · A(m + 1) · ... · A(m + i), i ≥ 1, we obtain

(22)  Λ_{m} = A(m, i)Λ_{m+i+1}.

The twisting condition φ_{m}^{∗}[𝕍_{m+1}] = deg φ_{m}[𝕍_{m}] together with (ii) implies: φ_{m}^{∗}([𝕎_{1,m+1}] + [𝕎_{2,m+1}]) = (deg φ_{m})^{2}([𝕎_{1,m}] + [𝕎_{2,m}]). Substituting (21) into the last equality, we have: a_{11}(m) + a_{12}(m) = a_{21}(m) + a_{22}(m) = (deg φ_{m})^{2}, m ≥ 1. This means that the column vector v = (1, 1)^{t} is the eigenvector of the matrix A(m) with the eigenvalue (deg φ_{m})^{2}. Hence, it is the eigenvector of A(m, i) with the eigenvalue d_{m,i} = (deg φ_{m})^{2}(deg φ_{m+1})^{2}...(deg φ_{m+i})^{2} :

(23)  A(m, i)v = d_{m,i}v.

Notice that the entries of A(m), m ≥ 1, are nonnegative integers (in fact, from the definition of Schubert varieties 𝕎_{j,m+1} it follows quickly that φ_{m}^{∗}[𝕎_{j,m+1}] is an effective cycle on X_{m}, so that (21) and (iv) give 0 ≤ φ_{m}^{∗}[𝕎_{i,m+1}] · [ℙ_{j,m}^{2}] = a_{ij}(m)); hence also the entries of A(m, i), m, i ≥ 1, are nonnegative integers). Besides, clearly d_{m,i} → ∞ as i → ∞ for any m ≥ 1. This, together with (22) and (23), implies that, for m ≫ 1, λ_{1m} and λ_{2m} cannot both be nonzero and have the same sign. This together with (20) is equivalent to the statement of the Lemma. □

In what follows we denote the α-planes and the β-planes on X = G(2; 4) respectively by ℙ_{α}^{2} and ℙ_{β}^{2}.

Proposition 5.3. There exists no rank 2 vector bundle E on the grassmannian X = G(2; 4) such that:

(a) c_{2}(E) = a[ℙ_{α}^{2}], a > 0,

(b) E_{|ℙ_{β}^{2}} is trivial for a generic β-plane ℙ_{β}^{2} on X.

Proof. Now assume that there exists a vector bundle E on X satisfying the conditions (a) and (b) of the Proposition. Fix a β-plane P ⊂ X such that

(24)  E_{|P} ≃ O_{P}^{⊕2}.

As X is the grassmannian of lines in ℙ^{3}, the plane P is the dual plane of a certain plane P̃ in ℙ^{3}. Next, fix a point x_{0} ∈ ℙ^{3} ∖ P̃ and denote by S the variety of lines in ℙ^{3} which contain x_{0}. Consider the variety Q = {(x, l) ∈ ℙ^{3} × X | x ∈ l ∩ P̃} with natural projections p : Q → S : (x, l) ↦ Span(x, x_{0}) and σ : Q → X : (x, l) ↦ l. Clearly, σ is the blowing up of X at the plane P, and the exceptional divisor D_{P} = σ^{−1}(P) is isomorphic to the incidence subvariety of P × P̃. Moreover, one easily checks that Q ≃ ℙ(O_{S}(1) ⊕ T_{S}(−1)), so that the projection p : Q → S coincides with the structure morphism ℙ(O_{S}(1) ⊕ T_{S}(−1)) → S. Let O_{Q}(1) be the Grothendieck line bundle on Q such that p_{∗}O_{Q}(1) = O_{S}(1) ⊕ T_{S}(−1). Using the Euler exact triple on Q

(25)  0 → Ω_{Q/S}^{1} → p^{∗}(O_{S}(1) ⊕ T_{S}(−1)) ⊗ O_{Q}(−1) → O_{Q} → 0,

we find the p-relative dualizing sheaf ω_{Q/S} := det(Ω_{Q/S}^{1}):

(26)  ω_{Q/S} ≃ O_{Q}(−3) ⊗ p^{∗}O_{S}(2).

Set E := σ^{∗}E. By construction, for each y ∈ S the fiber S_{y} = p^{−1}(y) is a plane such that l_{y} = S_{y} ∩ D_{Y} is a line, and, by (24),

(27)  E_{|l_{y}} ≃ O_{l_{y}}^{⊕2}.

Furthermore, σ(S_{y}) is an α-plane in X, hence by condition (a) of the Proposition we obtain that E_{|S_{y}} is a stable vector bundle for a > 1, and that E_{|S_{y}} fits into an exact triple 0 → O_{S_{y}} → E_{|S_{y}} → I_{z,S_{y}} → 0 for a certain point z ∈ S_{y} for a = 1. In both cases it is well known (and immediately verified) that (27) implies

(28)  h^{1}(E_{|S_{y}}(−1)) = h^{1}(E_{|S_{y}}(−2)) = a, h^{1}(E_{|S_{y}} ⊗ Ω_{S_{y}}^{1}) = 2a + 2,
h^{i}(E_{|S_{y}}(−1)) = h^{i}(E_{|S_{y}}(−2)) = h^{i}(E_{|S_{y}} ⊗ Ω_{S_{y}}^{1}) = 0, i ≠ 1

(see [OSS, p.285]). Consider the sheaves of O_{S}-modules

(29) E_{−1} := R^{1}p_{∗}(E ⊗O_{Q}(−2)⊗p^{∗}O_{S}(2)), E_{0} := R^{1}p_{∗}(E ⊗Ω_{Q/S}^{1}), E_{1} := R^{1}p_{∗}(E ⊗O_{Q}(−1)).

The equalities (28) together with semicontinuity imply that E_{−1}, E_{1} and E_{0} are locally free O_{S}-modules, and rk(E_{−1}) = rk(E_{1}) = a, and rk(E_{0}) = 2a + 2. Moreover,

(30)  R^{i}p_{∗}(E ⊗ O_{Q}(−2)) = R^{i}p_{∗}(E ⊗ Ω_{Q/S}^{1}) = R^{i}p_{∗}(E ⊗ O_{Q}(−1)) = 0

for i ≠ 1. Note that E^{∨} ≃ E as c_{1}(E) = 0 and rkE = 2. Furthermore, (26) implies that the nondegenerate pairing (p-relative Serre duality) R^{1}p_{∗}(E⊗O_{Q}(−1))⊗R^{1}p_{∗}(E^{∨}⊗O_{Q}(1)⊗ω_{Q/S}) → R^{2}p_{∗}ω_{Q/S} = O_{S} can be rewritten as E_{1} ⊗ E_{−1} → O_{S}, thus giving an isomorphism

(31)  E_{−1} ≃ E_{1}^{∨}.

Similarly, since E^{∨} ≃ E and Ω_{Q/S}^{1} ≃ T_{Q/S} ⊗ω_{Q/S}, p-relative Serre duality yields a nondegenerate pairing E_{0}⊗E_{0} = R^{1}p_{∗}(E⊗Ω_{Q/S}^{1})⊗R^{1}p_{∗}(E⊗Ω_{Q/S}^{1}) = R^{1}p_{∗}(E⊗Ω_{Q/S}^{1})⊗R^{1}p_{∗}(E^{∨}⊗T_{Q/S}⊗ω_{Q/S}) → R^{2}p_{∗}ω_{Q/S} = O_{S}. Therefore E_{0} is self-dual, i.e. E_{0} ≃ E_{0}^{∨}, and in particular c_{1}(E_{0}) = 0.

Now, let J denote the fiber product Q ×_{S} Q with projections Q ←^{pr_{1}} J →^{pr_{2}} Q such that p ∘ pr_{1} = p ∘ pr_{2}. Put F_{1} ⊠ F_{2} := pr_{1}^{∗}F_{1} ⊗ pr_{2}^{∗}F_{2} for sheaves F_{1} and F_{2} on Q, and consider the standard O_{J}-resolution of the structure sheaf O_{Δ} of the diagonal Δ ↪ J:

(32) 0 → O_{Q}(−1) ⊗ p^{∗}O_{S}(2) ⊠ O_{Q}(−2) → Ω^{1}_{Q/S}(1) ⊠ O_{Q}(−1) → O_{J} → O_{Δ} → 0.

Twist this sequence by the sheaf (E ⊗ O_{Q}(−1)) ⊠ O_{Q}(1) and apply the functor R^{i}pr_{2∗} to the resulting sequence. In view of (29) and (30) we obtain the following monad for E:

(33)  0 → p^{∗}E_{−1} ⊗ O_{Q}(−1) →^{λ} p^{∗}E_{0} →^{μ} p^{∗}E_{1} ⊗ O_{Q}(1) → 0, ker(μ)/im(λ) = E.

Put R := p^{∗}h, where h is the class of a line in S. Furthermore, set H := σ^{∗}H_{X}, [ℙ_{α}] := σ^{∗}[ℙ_{α}^{2}], [ℙ_{β}] := σ^{∗}[ℙ_{β}^{2}], where H_{X} is the class of a hyperplane section of X (via the Plücker embedding), and respectively, [ℙ_{α}^{2}] and [ℙ_{β}^{2}] are the classes of an α- and β-plane. Note that, clearly, O_{Q}(H) ≃ O_{Q}(1). Thus, taking into account the duality (31), we rewrite the monad (33) as:

(34) 0 → p^{∗}E_{1}^{∨} ⊗ O_{Q}(−H) →^{λ} p^{∗}E_{0} →^{μ} p^{∗}E_{1} ⊗ O_{Q}(H) → 0,  ker(μ)/im(λ) ≃ E.

As a next step, we are going to express all Chern classes of the sheaves in the monad (34) in terms of a. We start by writing down the Chern polynomials of the bundles p^{∗}E_{1} ⊗ O_{Q}(H) and p^{∗}E_{1}^{∨} ⊗ O_{Q}(−H) in the form:

(35) c_{t}(p^{∗}E_{1} ⊗ O_{Q}(H)) = ∏_{i=1}^{a}(1 + (δ_{i} + H)t), c_{t}(p^{∗}E_{1}^{∨} ⊗ O_{Q}(−H)) = ∏_{i=1}^{a}(1 − (δ_{i} + H)t),

where δ_{i} are the Chern roots of the bundle p^{∗}E_{1}. Thus

(36)  cR^{2} = ∑_{i=1}^{a} δ_{i}^{2}, dR = ∑_{i=1}^{a} δ_{i}.

for some c, d ∈ ℤ. Next we invoke the following easily verified relations in A^{·}(Q):

(37) H^{4} = RH^{3} = 2[pt], R^{2}H^{2} = R^{2}[ℙ_{α}] = RH[ℙ_{α}] = H^{2}[ℙ_{α}] = RH[ℙ_{β}] = H^{2}[ℙ_{β}] = [pt],
[ℙ_{α}][ℙ_{β}] = R^{2}[ℙ_{β}] = R^{4} = R^{3}H = 0,

where [pt] is the class of a point. This, together with (36), gives: (38)

∑_{1≤i<j≤a} δ_{i}^{2}δ_{j}^{2} = ∑_{1≤i<j≤a}(δ_{i}^{2}δ_{j}+δ_{i}δ_{j}^{2})H = 0, ∑_{1≤i<j≤a} δ_{i}δ_{j}H^{2} = \frac{1}{2}(d^{2}−c)[pt], ∑_{1≤i≤a}(δ_{i}+δ_{j})H^{3} = 2(a−1)d[pt].

Note that, since c_{1}(E_{0}) = 0,

(39)  c_{t}(p^{∗}E_{0}) = 1 + bR^{2}t^{2}

for some b ∈ ℤ. Furthermore,

(40)  c_{t}(E) = 1 + a[ℙ_{α}]t^{2}

by the condition of the Proposition. Substituting (39) and (40) into the polynomial f(t) := c_{t}(E)c_{t}(p^{∗}E_{1} ⊗ O_{Q}(H))c_{t}(p^{∗}E_{1}^{∨} ⊗ O_{Q}(−H)), we have f(t) = (1 + a[ℙ_{α}]t^{2}) ∏_{i=1}^{a}(1 − (δ_{i} + H)^{2}t^{2}). Expanding f(t) in the variable t and using (36)-(38), we obtain:

(41)  f(t) = 1 + (a[ℙ_{α}] − cR^{2} − 2dRH − aH^{2})t^{2} + e[pt]t^{4},

where

(42)  e = −3c − a(2d + a) + (a − 1)(a + 4d) + 2d^{2}.

Next, the monad (34) implies f(t) = c_{t}(p^{∗}E_{0}). A comparison of (41) with (39) yields

(43)  c_{2}(E) = a[ℙ_{α}] = (b + c)R^{2} + 2dRH + aH^{2},
(44)  e = c_{4}(p^{∗}E_{0}) = 0.

The relation (44) is the crutial relation which enables us to express the Chern classes of all sheaves in the monad (34) just in terms of a. More precisely, (43) and (37) give 0 = c_{2}(E)[ℙ_{β}] = 2d+a, hence a = −2d. Substituting these latter equalities into (42) we get e = −a(a−2)/2−3c. Hence c = −a(a − 2)/6 by (44). Since a = −2d, (36) and the equality c = −a(a − 2)/6 give c_{1}(E_{1}) = −a/2, c_{2}(E_{1}) = (d^{2} − c)/2 = a(5a − 4)/24. Substituting this into the standard formulas e_{k} := c_{k}(p^{∗}E_{1} ⊗ O_{Q}(H)) = ∑_{i=0}^{2} \binom{a−i}{k−i}R^{i}H^{k−i}c_{i}(E_{1}), 1 ≤ k ≤ 4, we obtain

(45)  e_{1} = −aR/2 + aH, e_{2} = (5a^{2}/24 − a/6)R^{2} + (a^{2} − a)(−RH + H^{2})/2,

e_{3} = (5a^{3}/24 − 7a^{2}/12 + a/3)R^{2}H + (−a^{3}/4 + 3a^{2}/4 − a/2)RH^{2} + (a^{3}/6 − a^{2}/2 + a/3)H^{3},

e_{4} = (−7a^{4}/144 + 43a^{3}/144 − 41a^{2}/72 + a/3)[pt].

It remains to write down explicitely c_{2}(p^{∗}E_{0}): (37), (43) and the relations a = −2d, c = −a(a − 2)/6 give a = c_{2}(E)[ℙ_{α}] = b + c, hence

(46)  c_{2}(E_{0}) = b = (a^{2} + 4a)/6

by (39).

Our next and final step will be to obtain a contradiction by computing the Euler characteristic of the sheaf E and two different ways. We first compute the Todd class td(T_{Q}) of the bundle T_{Q}. From the exact triple dual to (25) we find c_{t}(T_{Q/S}) = 1 + (−2R + 3H)t + (2R^{2} − 4RH + 3H^{2})t^{2}. Next, c_{t}(Q) = c_{t}(T_{Q/S})c_{t}(p^{∗}T_{Q}). Hence c_{1}(T_{Q}) = R+3H, c_{2}(T_{Q}) = −R^{2}+5RH+3H^{2}, c_{3}(T_{Q}) = −3R^{2}H + 9H^{2}R, c_{4}(T_{Q}) = 9[pt]. Substituting into the formula for the Todd class of T_{Q}, td(T_{Q}) = 1 + \frac{1}{2}c_{1} + \frac{1}{12}(c_{1}^{2} + c_{2}) + \frac{1}{24}c_{1}c_{2} − \frac{1}{720}(c_{1}^{4} − 4c_{1}^{2}c_{2} − 3c_{2}^{2} − c_{1}c_{3} + c_{4}), where c_{i} := c_{i}(T_{Q}) (see, e.g., [H, p.432]), we get:

(47)  td(T_{Q}) = 1 + \frac{1}{2}R + \frac{3}{2}H + \frac{11}{12}RH + H^{2} + \frac{1}{12}HR^{2} + \frac{3}{4}H^{2}R + \frac{3}{8}H^{3} + [pt].

Next, by the condition of Proposition c_{1}(E) = 0, c_{2}(E) = a[ℙ_{α}], c_{3}(E) = c_{4}(E) = 0. Substituting this into the general formula for the Chern character of a vector bundle F,

ch(F) = rk(F) + c_{1} + (c_{1}^{2} − 2c_{2})/2 + (c_{1}^{3} − 3c_{1}c_{2} − 3c_{3})/6 + (c_{1}^{4} − 4c_{1}^{2}c_{2} + 4c_{1}c_{3} + 2c_{2}^{2} − 4c_{4})/24,

c_{i} := c_{i}(F) (see, e.g., [H, p.432]), and using (47), we obtain by the Riemann-Roch Theorem for F = E

(48)  χ(E) = \frac{1}{12}a^{2} − \frac{23}{12}a + 2.

In a similar way, using (45), we obtain:

(49)  χ(p^{∗}E_{1} ⊗ O_{Q}(H)) + χ(p^{∗}E_{1}^{∨} ⊗ O_{Q}(−H)) = \frac{5}{216}a^{4} − \frac{29}{216}a^{3} − \frac{1}{54}a^{2} + \frac{113}{36}a.

Next, in view of (46) and the equality c_{1}(E_{0}) = 0 the Riemann-Roch Theorem for E_{0} easily gives

(50)  χ(p^{∗}E_{0}) = χ(E_{0}) = −\frac{1}{6}a^{2} + \frac{4}{3}a + 2.

Together with (48) and (49) this yields

Φ(a) := χ(p^{∗}E_{0})−(χ(E)+χ(p^{∗}E_{1}⊗O_{Q}(H))+χ(p^{∗}E_{1}^{∨}⊗O_{Q}(−H))) = −\frac{5}{216}a(a−2)(a−3)(a− \frac{4}{5}).

The monad (34) implies now Φ(a) = 0. The only positive integer roots of the polynomial Φ(a) are a = 2 and a = 3. However, (48) implies χ(E) = −\frac{3}{2} for a = 2, and (50) implies χ(p^{∗}E_{0}) = \frac{9}{2} for a = 3. This is a contradiction as the values of χ(E) and χ(p^{∗}E_{0}) are integers by definition. □

We need a last piece of notation. Consider the flag variety F l(k_{m} − 2, k_{m} + 2; V). Any point u = (V^{k_{m}−2}, V^{k_{m}+2}) ∈ F l(k_{m} − 2, k_{m} + 2; V) determines a standard extension

(51)  i_{u} : X = G(2; 4) ↪ X_{m},
(52)  W^{2} ↦ V^{k_{m}−2} ⊕ W^{2} ⊂ V^{k_{m}+2} = V^{k_{m}−2} ⊕ W^{4},

where W^{2} ∈ X = G(2; W^{4}) and an isomorphism V^{k_{m}−2} ⊕ W^{4} ≃ V^{k_{m}+2} is fixed (clearly i_{u} does not depend on the choice of this isomorphism). We clearly have isomorphisms of Chow groups

(53)  i_{u}^{∗} : A^{2}(X_{m}) →^{∼} A^{2}(X), i_{u∗} : A_{2}(X) →^{∼} A_{2}(X_{m}),

where the flag variety Y_{m} := F l(k_{m} − 1, k_{m} + 1; V^{n_{m}}) (respectively, Y := F l(1, 3; 4)) is the set of lines in X_{m} (respectively, in X).

Theorem 5.4. Let X = lim_{→} X_{m} be a twisted ind-grassmannian. Then any vector bundle E = lim_{←} E_{m} on X of rank 2 is trivial, and hence Conjecture 1.1(iv) holds for vector bundles of rank 2.

Proof. Fix m ≥ max{m_{0}, m_{1}}, where m_{0} and m_{1} are as in Corollary 5.1 and Lemma 5.2. For j = 1, 2, let E^{(j)} denote the restriction of E_{m} to a projective plane of type ℙ_{j,m}^{2}, T^{j} ≃ F l(k_{m} − j, k_{m}+3−j, V^{n_{m}}) be the variety of planes of the form ℙ_{j,m}^{2} in X_{m}, and Π^{j} := {ℙ_{j,m}^{2} ∈ T^{j}| E_{m}_{|ℙ_{j,m}^{2}} is properly unstable (i.e. not semistable)}. As semistability is an open condition, Π^{j} is a closed subset of T^{j}.

(i) Assume that c_{2}(E^{(1)}) > 0. Then, since m ≥ m_{1}, Lemma 5.2 implies c_{2}(E^{(2)}) ≤ 0.

(i.1) Suppose that c_{2}(E^{(2)}) = 0. If Π^{2} ≠ T^{2}, then for any ℙ_{2,m}^{2} ∈ T^{2} ∖ Π^{2} the corresponding bundle E^{(2)} is semistable, hence E^{(2)} is trivial as c_{2}(E^{(2)}) = 0, see [DL, Prop. 2.3,(4)]. Thus, for a generic point u ∈ F l(k_{m} − 2, k_{m} + 2; V^{n_{m}}), the bundle E = i_{u}^{∗}E_{m} on X = G(2; 4) satisfies the conditions of Proposition 5.3, which is a contradiction.

We therefore assume Π^{2} = T^{2}. Then for any ℙ_{2,m}^{2} ∈ T^{2} the corresponding bundle E^{(2)} has a maximal destabilizing subsheaf 0 → O_{ℙ_{2,m}^{2}}(a) → E^{(2)}. Moreover a > 0. In fact, otherwise the condition c_{2}(E^{(2)}) = 0 would imply that a = 0 and E^{(2)}/O_{ℙ_{2,m}^{2}} = O_{ℙ_{2,m}^{2}}, i.e. E^{(2)} would be trivial, in particular semistable. Hence

(54)  d_{E^{(2)}} = (a, −a).

Since any line in X_{m} is contained in a plane ℙ_{2,m}^{2} ∈ T^{2}, (54) implies d_{E_{m}} = (a, −a) with a > 0 for m > m_{0}, contrary to Corollary 5.1.

(i.2) Assume c_{2}(E^{(2)}) < 0. Since E^{(2)} is not stable for any ℙ_{2,m}^{2} ∈ T^{2}, its maximal destabilizing subsheaf 0 → O_{ℙ_{2,m}^{2}}(a) → E^{(2)} clearly satisfies the condition a > 0, i.e. E^{(2)} is properly unstable, hence Π^{2} = T^{2}. Then we again obtain a contradiction as above.

(ii) Now we assume that c_{2}(E^{(2)}) > 0. Then, replacing E^{(2)} by E^{(1)} and vice versa, we arrive to a contradiction by the same argument as in case (i).

(iii) We must therefore assume c_{2}(E^{(1)}) = c_{2}(E^{(2)}) = 0. Set D(E_{m}) := {l ∈ Y_{m}| d_{E_{m}}(l) ≠ (0, 0)} and D(E) := {l ∈ Y | d_{E}(l) ≠ (0, 0)}. By Corollary 5.1, d_{E_{m}} = (0, 0), respectively, d_{E}(l) = (0, 0) for a generic line l ∈ Y. Then by deformation theory [B], D(E_{m}) (respectively, D(E)) is an effective divisor on Y_{m} (respectively, on Y). Hence, O_{Y} (D(E)) = p_{1}^{∗}O_{Y^{1}}(a) ⊗ p_{2}^{∗}O_{Y^{2}}(b) for some a, b ≥ 0, where p_{1}, p_{2} are as in diagram (2). Note that each fiber of p_{1} (resp., of p_{2}) is a plane ℙ̃_{α}^{2} dual to some α-plane ℙ_{α}^{2} (respectively, a plane ℙ̃_{β}^{2} dual to some β-plane ℙ_{β}^{2}). Thus, setting D(E_{|ℙ_{α}^{2}}) := {l ∈ ℙ̃_{α}^{2}| d_{E}(l) ≠ (0, 0)}, D(E_{|ℙ_{β}^{2}}) := {l ∈ ℙ̃_{β}^{2}| d_{E}(l) ≠ (0, 0)}, we obtain Oℙ̃_{α}^{2}(D(E_{|ℙ_{α}^{2}})) = O_{Y} (D(E))_{|ℙ̃_{α}^{2}} = Oℙ̃_{α}^{2}(b), Oℙ̃_{β}^{2}(D(E_{|ℙ_{β}^{2}})) = O_{Y} (D(E))_{|ℙ̃_{β}^{2}} = Oℙ̃_{β}^{2}(a). Now if E_{|ℙ_{α}^{2}} is semistable, a theorem of Barth [OSS, Ch. II, Theorem 2.2.3] implies that D(E_{|ℙ_{α}^{2}}) is a divisor of degree c_{2}(E_{|ℙ_{α}^{2}}) = a on ℙ_{α}^{2}. Hence a = c_{2}(E^{(1)}) = 0 for a semistable E_{|ℙ_{α}^{2}}. If E_{|ℙ_{α}^{2}} is not semistable, it is unstable and the equality d_{E}(l) = (0, 0) yields d_{E_{|ℙ^{2}_{α}}} = (0, 0). Then the maximal destabilizing subsheaf of E_{|ℙ_{α}^{2}} is isomorphic to O_{ℙ_{α}^{2}} and, since c_{2}(E_{|ℙ_{α}^{2}}) = 0, we obtain an exact triple 0 → O_{ℙ_{α}^{2}} → E_{|ℙ_{α}^{2}} → O_{ℙ_{α}^{2}} → 0, such that E_{|ℙ_{α}^{2}} ≃ O_{ℙ_{α}^{2}}^{⊕2} is semistable, a contradiction. This shows that a = 0 whenever c_{2}(E^{(1)}) = c_{2}(E^{(2)}) = 0. Similarly, b = 0. Therefore D(E_{m}) = ∅, and Proposition 4.1 implies that E_{m} is trivial. Therefore E is trivial as well. □

In [DP] Conjecture 1.1 (iv) was proved not oly when X is a twisted projective ind-space, but also for finite rank bundles on special twisted ind-grassmannians defined through certain homogeneous embeddings φ_{m}. These include embeddings of the form

G(k; n) → G(kw; na)
V^{k} ⊂ V ↦ V^{k} ⊗ W ⊂ V ⊗ W,

where W is a fixed a-dimensional subspace, or of the form

G(k; n) → G (\frac{k(k + 1)}{2}; n^{2})

V^{k} ⊂ V ↦ S^{2}(V^{k}) ⊂ V ⊗ V.

More precisely, Conjecture 1.1 (iv) was proved in [DP] for twisted ind-grassmannians whose defining embeddings are homogeneous embeddings satisfying some specific numerical conditions relating the degrees deg φ_{m} with the pairs of integers (k_{m}, n_{m}). There are many twisted indgrassmannians for which those conditions are not satisfied. For instance, this applies to the ind-grassmannians defined by iterating each of the following embeddings:

G(k; n) → G (\frac{k(k + 1)}{2}; \frac{n(n + 1)}{2})

V^{k} ⊂ V ↦ S^{2}(V^{k}) ⊂ S^{2}(V),

G(k; n) → G (\frac{k(k − 1)}{2}; \frac{n(n − 1)}{2})

V^{k} ⊂ V ↦ Λ^{2}(V^{k}) ⊂ Λ^{2}(V).

Therefore the resulting ind-grassmannians G(k, n, S^{2}) and X(k, n, Λ^{2}) are examples of twisted ind-grassmannians for which Theorem 5.4 is new.

References

[BV] W. Barth, A. Van de Ven, On the geometry in codimension 2 in Grassmann manifolds, Lecture Notes in Math. 412 (1974), 1-35.

[B] E. Brieskorn, Über holomorphe ℙ_{n}-bündel uber ℙ_{1}, Math. Ann. 157 (1967), 351-382.

[CT] I. Coanda, G. Trautmann, The splitting criterion of Kempf and the Babylonian tower theorem, arXiv:math.AG/0411636 v1 29 Nov 2004.

[DiP] I. Dimitrov, I. Penkov, Ind-varieties of generalized flags as homogeneous spaces for classical indgroups, IMRN 2004, no 55, 2935-2953

[DL] J. M. Drezet, J. Le Potier, Fibres stables et fibres exceptionnels sur ℙ_{2}, Ann. Sci. Ec. Norm. Super. IV. Ser. 18 (1985), 193-243.

[DP] J. Donin, I. Penkov, Finite rank vector bundles on inductive limits of Grassmannians, IMRN 2003, no 34, 1871-1887.

[DPW] I. Dimitrov, I. Penkov, J. A. Wolf, A Bott-Borel-Weil theorem for direct limits of algebraic groups. Amer. J. of Math. 124 (2002), 955-998.

[F] W. Fulton, Intersection Theory, Springer, Berlin, 1998.

[H] R. Hartshorne, Algebraic Geometry, Springer, New York, 1977.

[OSS] C. Okonek, M. Schneider, H. Spindler, Vector Bundles on Complex Projective Spaces, Birkhäuser, [P] I. Penkov, The Penrose transform on general grassmannians, C. R. Acad. Bulg. des Sci. 22 (1980),

1439-1442.

[Sa1] E. Sato, On the decomposability of infinitely extendable vector bundles on projective spaces and Grassmann varieties, J. Math. Kyoto Univ. 17 (1977), 127-150.

[Sa2] E. Sato, On infinitely extendable vector bundles on G/P, J. Math. Kyoto Univ. 19 (1979), 171-189.

[T] A. N. Tyurin, Vector bundles of finite rank over infinite varieties, Math. USSR Izvestija 10 (1976), 1187-1204.

Jacobs University Bremen^{1}, School of Engineering and Science, Campus Ring 1, 28759 Bremen, Germany

  E-mail address: i.penkov@iu-bremen.de

Department of Mathematics, State Pedagogical University, Respublikanskaya Str. 108 150 000 Yaroslavl, Russia

  E-mail address: astikhomirov@mail.ru

1International University Bremen prior to Spring 2007