Finite rank vector bundles on inductive limits of grassmannians

Joseph Donin^{∗} and Ivan Penkov^{†}

Abstract

If ℙ^{∞} is the projective ind-space, i.e. ℙ^{∞} is the inductive limit of linear embeddings of complex projective spaces, the Barth-Van de Ven-Tyurin (BVT) Theorem claims that every finite rank vector bundle on ℙ^{∞} is isomorphic to a direct sum of line bundles. We extend this theorem to general sequences of morphisms between projective spaces by proving that, if there are infinitely many morphisms of degree higher than one, every vector bundle of finite rank on the inductive limit is trivial. We then establish a relative version of these results, and apply it to the study of vector bundles on inductive limits of grassmannians. In particular we show that the BVT Theorem extends to the ind-grassmannian of subspaces commensurable with a fixed infinite dimensional and infinite codimensional subspace in ℂ^{∞}. We also show that for a class of twisted ind-grassmannians, every finite rank vector bundle is trivial.

2000 AMS Subject Classification: Primary 32L05, 14J60, Secondary 14M15.

To the memory of Andrey Tyurin

1 Introduction

About 30 years ago the study of “infinitely extendable” vector bundles of finite rank on projective spaces and grassmannians was initiated. In particular the following remarkable theorem was proved: any finite rank vector bundle on the infinite complex projective space ℙ^{∞} (or equivalently, any finite rank vector bundle on ℙ^{n} which admits an extension to ℙ^{m} for large enough m > n) is isomorphic to a direct sum of line bundles. For rank two bundles this was established by W. Barth and A. Van de Ven in [BV], and for arbitrary finite rank bundles the theorem was proved by A. Tyurin in [T]. In what follows we refer to this result as to the Barth-Van de Ven-Tyurin Theorem, or as to the BVT Theorem. Some first steps were made also towards understanding finite rank vector bundles on infinite grassmannians: R. Hartshorne conjectured that every finite rank bundle on an infinite grassmannian G(k, ∞) is homogeneous, see [BV]. For rank two bundles this conjecture is proved in [BV], and in the general case the conjecture is proved by E. Sato, [S2]. Sato established also a partial analog of Hartshorne’s conjecture for the infinite grassmanians of the classical simple groups, and reproved the BVT Theorem, see [S2] and [S1].

∗Work supported in part by Israel Academy of Sciences Grant no. 8007/99-03 and the Max Planck Institute for Mathematics, Bonn

†Work supported in part by an NSF grant and the Max Planck Institute for Mathematics, Bonn

The purpose of the present note is to revive this discussion and relate it with the more recent duscussion of homogeneous ind-spaces of locally linear ind-groups, see [DPW], [DP] and the references therein. Our starting point is an infinite sequence

G_{1} ⊂ G_{2} ⊂ ...

of complex linear algebraic groups and a subsequence

P_{1} ⊂ P_{2} ⊂ ...

of parabolic subgroups. This yields a sequence of morphisms

G_{1}/P_{1} ⟶ G_{2}/P_{2} ⟶ ... .  (1)

Let G/P denote the inductive limit of (1). We restrict ourselves here to the case when G_{N} ≃ GL(n_{N}) and P_{N} are maximal parabolic subgroups, i.e. G_{N}/P_{N} are grassmannians.

The study of line bundles on G/P gives some first hints on what general finite rank bundles on G/P might look like. An essential difference with the cases studied in [BV], [T], [S1], and [S2] is that the restriction maps

Pic(G_{N}/P_{N}) ⟶ Pic(G_{N−1}/P_{N−1})

on the Picard groups induced by inclusions in (1) are only injective and in general not surjective. Therefore Pic(G/P), the Picard group of the inductive limit, is isomorphic to ℤ or equals zero. In the first case the call the sequence (1) linear and in the second case we call it twisted.

Here is a brief description of our results. The simplest case for a sequence (1) is when all G_{N}/P_{N} are projective spaces. Consider more generally an arbitrary sequence of morphisms of projective spaces

ℙ^{i_{1}} ⟶^{φ_{1}} ℙ^{i_{2}} ⟶^{φ_{2}} ... .  (2)

If (2) is linear, i.e. all but finitely many Picard groups map isomorphically, the inductive limit is isomorphic to the projective ind-space ℙ^{∞}. Here the BVT Theorem claims that any finite rank bundle is isomorphic to a direct sum of line bundles. If (2) is twisted, i.e. the Picard group of the inductive limit ℙ_{tw}^{∞} of (2) equals zero, the problem of describing all finite rank vector bundles on ℙ_{tw}^{∞} was not posed in the 70’s.

Our first main result (Theorem 3.1) claims that every finite rank vector bundle on ℙ_{tw}^{∞} is trivial. This theorem, together with the BVT Theorem, can be extended to the relative case and yields a complete description of finite rank vector bundles on any inductive limit of relative projective spaces.

In Section 4 we apply the above result to the study of finite rank vector bundles on inductive limits of grassmannians G(k_{N}, n_{N}). We consider two types of morphisms of grassmannians in (1): standard inclusions and a certain class of twisted homogeneous morphisms which we call twisted extensions. For standard inclusions we show that, if lim_{N→∞} k_{N} = lim_{N→∞}(n_{N} − k_{N}) = ∞, any finite rank vector bundle on the inductive limit is isomorphic to a direct sum of line bundles. An interesting ind-variety which arises as the inductive limit of a sequence of standard inclusions satisfying the above condition is the ind-grassmanian G(V, ∞) of subspaces V′ ⊂ ℂ^{∞} commensurable with a fixed infinite dimensional and infinite codimensional subspace V ⊂ ℂ^{∞}. Therefore, any finite rank vector bundle on G(V, ∞) is isomorphic to a direct sum of line bundles. Finally, we prove that in the case of twisted extensions of grassmannians every finite rank vector bundle on the inductive limit is trivial.

Acknowledgements. We are grateful to A. Tyurin for a helpful discussion on the topic of this paper several weeks before he suddenly passed away in October 2002. We thank also Z. Ran for his comments and for making us aware of E. Sato’s work.

2 Preliminary results

2.1 Notation

The ground field is ℂ and we work in the category of complex analytic spaces. A vector bundle always means a vector bundle of finite rank. If E is a vector bundle, we denote by E^{∗} the dual vector bundle. For the tensor power of a line bundle E we write simply E^{k}. More generally, the convention E^{−k} = (E^{∗})^{k} enables us to assume that k ∈ ℤ. By H^{i}(E) we denote the i-th cohomology group of the sheaf of local sections of E, and put h^{i}(E) = dim H^{i}(E), and χ(E) := ∑_{i}(−1)^{i}h^{i}(E).

By G(k, n) we denote the grassmannian of k-dimensional subspaces in ℂ^{n}. For k = 1 we have the projective space ℙ^{n−1} := G(1, n). The grassmannian G(k, n) is isomorphic to G(n − k, n) via the isomorphism

{V ⊂ ℂ^{n}} ↦ {V^{⊥} ⊂ ℂ^{n}},

where ℂ^{n} is identified with its dual space. Under a projective subspace of G(k, n) we understand the set of k-dimentional subspaces V ⊂ ℂ^{n} such that U ⊂ V ⊂ W, where U ⊂ W are fixed subspaces of ℂ^{n} with dim U = k − 1, dim W > k, or dim W = k + 1, dim U < k. The projective subspace is a line if dim W − dim V = 2.

Let S_{k} be the vector bundle on G(k, n) with fiber V at the point V ∈ G(k, n). There is a canonical inclusion S_{k} ⊂ ℂ̃^{n}, where ℂ̃^{n} the trivial bundle on G(k, n) with fiber ℂ_{n}. We set S_{n−k} := (ℂ̃^{n}/S_{k})^{∗}. By definition, S_{k} and S_{n−k} are the tautological bundles on G(k, n). The Picard group Pic(G(k, n)) is isomorphic to ℤ, and both maximal exterior powers ∧^{k}S_{k} and ∧^{n−k}S_{n−k} are isomorphic generators of Pic(G(k, n)). We set O_{G(k,n)}(−1) := ∧^{k}S_{k} ≃ ∧^{n−k}S_{n−k} and O_{G(k,n)}(m) := O_{G(k,n)}(−1)^{−m} for m ∈ ℤ.

An ind-space is the union ∪_{N}X_{N} of analytic spaces X_{N} related by closed immersions

X_{1} ⟶^{φ_{1}} X_{2} ⟶^{φ_{2}} ... .  (3)

For instance, the projective ind-space ℙ^{∞} is the union ∪_{N}ℙ_{N} where ℙ^{N} ⊂ ℙ^{N+1} is the standard closed immersion. A morphism of ind-spaces φ : X → Y is a map whose restriction to each X_{N} is a morphism of X_{N} into Y_{j(N)} for some j(N). In this paper we will more generally consider sequences (3) of arbitrary morphisms. A vector bundle on the system (3) is a collection {E_{N}} of vector bundles E_{N} on X_{N} such that φ_{N}^{∗}E_{N+1} = E_{N}. If (3) determines an ind-space X, we speak of a vector bundle on X.

2.2 The Barth-Van de Ven-Tyurin Theorem

Theorem 2.1. If X ≃ ℙ^{1} or X ≃ ℙ^{∞}, every vector bundle E on X is isomorphic to a unique direct sum of line bundles, i.e. E ≃ ⊕_{j}O_{X}(d_{j}) for some unique integers d_{1} ≥ ... ≥ d_{rk E}.

For X ≃ ℙ^{1} this is a classical result due to A. Grothendieck. For X ≃ ℙ^{∞} the theorem has been proved by A. Tyurin in [T] (and earlier by W. Barth and A. Van de Ven, [BV], for vector bundles of rank two). E. Sato also presents a proof in [S1].

We will make extensive use of Theorem 2.1. In particular, if l ⊂ Y is a rational curve in a complex manifold or ind-space Y and E is a vector bundle on Y, we call the numbers d_{1}, ..., d_{rk E}, together with the multiplicities with which they occur in the isomorphism E|_{l} ≃ ⊕_{j}O_{Y} (d_{j}), the splitting data of E|_{l}. We always denote by d_{1} the maximal among all d_{j}, and by D – the difference d_{1} − d_{rk E}. When we need to emphasize the dependence of d_{j} and D on l, we write d_{j}(E|_{l}) and D(E|_{l}).

We will use the same convention when Y is an ind-space and l is replaced by an ind-subspace P ⊂ Y isomorphic to ℙ^{∞}. Note that, if E′ is a vector bundle on X ≃ ℙ^{1} or X ≃ ℙ^{∞} with splitting data d_{1} ≥ ... ≥ d_{rk E}, and ν_{1} is the multiplicity of d_{1}, there is a unique subbundle E′_{1} of E′ isomorphic to the direct sum of ν_{1} copies of O_{X}(d_{1}). Indeed, consider the twisted bundle E′ ⊗ O_{X}(−d_{1}) and its unique maximal subbundle E′′ generated by global sections. Then E′_{1} = E′′ ⊗ O_{X}(d_{1}).

We say that a vector bundle E on a grassmannian G(k, n) is uniform, if the splitting data d_{1}(E|_{l}), ..., d_{rk E}(E|_{l}) for a line l ⊂ G(k, n) does not depend on the choice of l. The bundle E is linearly trivial if its restriction E|_{l} is trivial for any line l ⊂ G(k, n). A linearly trivial bundle is necessarily trivial: for k = 1 this is a well-known result, and an induction argument on k yields the result also for G(k, n), see for instance [P].

2.3 Local rigidity of direct sums of line bundles on ℙ^{N}

Let S be an analytic space and π_{1} and π_{2} denote respectively the projections of ℙ^{1} × S onto the first and the second factor. If E is a vector bundle on ℙ^{1} × S, E(x) denotes the restriction of E to π_{2}^{−1}(x) ≃ ℙ^{1}. In [D1] the following statement is proved.

Proposition 2.2. Let E be a vector bundle on ℙ^{1} × S, where S is a connected analytic space. Fix x_{0} ∈ S. Then

a) there exists an open neighborhood U ∋ x_{0} such that d_{1}(E(x)) ≤ d_{1}(E(x_{0})) for x ∈ U, i.e. d_{1}(E(x)) locally decreases;

b) ∑_{i}^{rk E(x)} ν_{i}(E(x))d_{i}(E(x)) = c_{1}(E(x)), where c_{1}(E(x)) is the first Chern class of E(x) and ν_{i}(E(x)) is the multiplicity of d_{i}(E(x)) (in particular ∑_{i}^{rk E(x)} ν_{i}(E(x))d_{i}(E(x)) does not depend on x ∈ S).

Proposition 2.2, applied to E^{∗} implies that d_{rk E}(E(x)) = −d_{1}(E^{∗}(x)) locally increases, hence D(E(x)) = d_{1} − d_{rk E} also locally decreases.

Proposition 2.3. Let E be a vector bundle on ℙ^{N}, N > 1, which is isomorphic to a direct sum of line bundles. Then E is locally rigid.

Proof. Note that H^{1}(End(E)) = 0. This follows from the fact that End(E) is isomorphic to a direct sum of line bundles, as H^{1}(L) = 0 for any line bundle L on ℙ^{N} for N > 1. Therefore the parameter space of the versal deformation of E consists of a single reduced point, see [D2].

2.4 Estimates related to vector bundles on ℙ^{1} × ℙ^{1}

In the case of ℙ^{1} × ℙ^{1} we call the fibers of π_{1} (respectively, π_{2}) vertical (resp., horizontal) sections of ℙ^{1} × ℙ^{1}, and for any vector bundle E on ℙ^{1} × ℙ^{1} we denote the twisted bundle E ⊗ π_{1}^{∗}(O(k)) ⊗ π_{2}^{∗}(O(l)) by E(k, l).

Lemma 2.4. Let E be a vector bundle on ℙ^{1} × ℙ^{1}. Let P be a vertical section and S be a horizontal section of ℙ^{1} × ℙ^{1}. Then d_{1}(E|_{P}), d_{1}(E|_{S}) ≥ −1 implies

h^{0}(E) ≤ (rk E)(d_{1}(E|_{P}) + 1)(d_{1}(E|_{S}) + 1).  (4)

If d_{1}(E|_{P}) < 0 or d_{1}(E|_{S}) < 0, we have h^{0}(E) = 0.

Proof. The second statement in the Lemma follows immediately from Proposition 2.2, since h^{0}(F) = 0 for any vector bundle F on ℙ^{1} when d_{1}(F) < 0. If d_{1}(E|_{P}) or d_{1}(E|_{S}) equals −1, then (4) is obvious, therefore we can assume that d_{1}(E|_{P}), d_{1}(E|_{S}) ≥ 0.

For any k there is the exact sequence

0 ⟶ E(0, −k − 1) ⟶ E(0, −k) ⟶ E|_{S} ⟶ 0.

The corresponding cohomology sequence gives

h^{0}(E(0, −k)) − h^{0}(E(0, −k − 1)) ≤ h^{0}(E|_{S}) ≤ (rk E)(d_{1}(E|_{S}) + 1).

Summation from k to zero yields

h^{0}(E) − h^{0}(E(0, −k − 1)) ≤ (rk E)(k + 1)(d_{1}(E|_{S}) + 1).  (5)

When k = d_{1}(E|_{P}) we have, h^{0}(E(0, −k − 1)) = 0, therefore (5) implies (4).

Lemma 2.5. Let E, P, and S be as in Lemma 2.4. Suppose d_{1}(E|_{P}), d_{1}(E|_{S}) ≥ 0 and d_{rk E}(E|_{P}), d_{rk E}(E|_{S}) ≤ 0. If P′ is another vertical section of ℙ^{1} × ℙ^{1}, then

D(E|_{P′}) ≤ 4(rk E)(D(E|_{P}) + 2)(D(E|_{S}) + 1) − χ(E) − χ(E^{∗}).  (6)

Proof. From the cohomology sequence of the exact sequence

0 ⟶ E ⟶ E(1, 0) ⟶ E|_{P′} ⟶ 0

we have

h^{0}(E) + h^{0}(E|_{P′}) ≤ h^{0}(E(1, 0)) + h^{1}(E).

This, together with the equality h^{2}(E) = h^{0}(E^{∗}(−2, −2)) (Serre duality), gives

d_{1}(E|_{P′}) ≤ h^{0}(E|_{P′}) ≤ h^{0}(E(1, 0)) − χ(E) + h^{0}(E^{∗}(−2, −2)).  (7)

Note that d_{1}((E^{∗}(−2, −2)|_{P}) = −d_{rk E}(E|_{P})−1 and d_{1}((E^{∗}(−2, −2)|_{S}) = −d_{rk E}(E|_{S})−1, so Lemma 2.4 yields

h^{0}(E^{∗}(−2, −2)) ≤ (rk E)(−d_{rk E}(E|_{P})(−d_{rk E}(E|_{S}).  (8)

Combining (7) and (8) we have

d_{1}(E|_{P′}) ≤ (rk E)(d_{1}(E|_{P}) + 2)(d_{1}(E|_{S}) + 1) − χ(E) + (rk E)(−d_{rk E}(E|_{P}))(−d_{rk E}(E|_{S})).

Since d_{1}(E|_{P}), −d_{rk E}(E|_{P}) ≤ D(E|_{P}) and d_{1}(E|_{S}), −d_{rk E}(E|_{S}) ≤ D(E|_{S}), we obtain

d_{1}(E|_{P′}) ≤ 2(rk E)(D(E|_{P}) + 2)(D(E|_{S}) + 1) − χ(E).  (9)

By adding to (9) the analogous inequality for E^{∗} we obtain (6).

3 The case of a twisted projective ind-space

Our first main result is the following theorem.

Theorem 3.1. Let

ℙ^{i_{1}} ⟶^{φ_{1}} ℙ^{i_{2}} ⟶^{φ_{2}} ...

be a twisted sequence of projective spaces, i.e. deg φ_{N} > 1 for infinitely many N, and let E = {E_{N}} be a vector bundle on this inductive system. Then E is trivial, i.e. all E_{N} are trivial.

Lemma 3.2. Let E be as in Theorem 3.1. Then all Chern classes c_{q}(E_{N}) equal zero.

Proof. By definition, c_{q}(E_{N}) ∈ H^{2q}(ℙ^{i_{N}} , ℤ). For each j > N, the homomorphism

H^{2q}(ℙ^{i_{j}}, ℤ) ⟶ H^{2q}(ℙ^{i_{N}} , ℤ)  (10)

induced by the composition φ_{j,N} := φ_{N} ∘ ... ∘ φ_{j} maps c_{q}(E_{j}) to c_{q}(E_{N}). On the other hand, we can identify H^{2q}(ℙ^{i_{N}} , ℤ) and H^{2q}(ℙ^{i_{j}}, ℤ) in a standard way with ℤ, and (10) considered as an endomorphism of ℤ is nothing but multiplication by (deg φ_{j})·...·(deg φ_{N}). Therefore c_{q}(E_{N}) ∈ ℤ is divisible by (deg φ_{j}) · ... · (deg φ_{N}) for all j > N. According to our assumption, lim_{j→∞}((deg φ_{j}) · ... · (deg φ_{N})) = ∞, consequently, c_{q}(E_{N}) = 0.

Proof of Theorem 3.1. Without restriction of generality we assume that i_{1} = 1. Fix N with i_{N} > 2 and set

D_{N} := max_{l′} D(E_{N}|_{l′}),

where l′ runs over all lines in ℙ^{i_{N}}. Let l be a line in ℙ^{i_{N}} such that D(E|_{l}) = D_{N}, and Q be a projective subspace in ℙ^{i_{N}} of dimension i_{N} − 2 not intersecting l. We can choose homogeneous coordinates z_{0}, ..., z_{i_{N}} in ℙ^{i_{N}} such that l is defined by z_{i} = 0, i ≥ 2, and Q is defined by z_{0} = z_{1} = 0. Furthermore, we fix a morphism f : ℙ^{1} → ℙ^{i_{N}} of degree deg φ_{1,n} such that

f(ℙ^{1}) ∩ l = ∅, f(ℙ^{1}) ∩ Q = ∅,  (11)

and

D(f^{∗}E_{N}) ≤ D(E_{1}).  (12)

To see that f exists, note that by Proposition 2.2 there is a neighborhood of unity U in the group of linear transformation of ℙ^{i_{N}} such that for any g ∈ U the morphism g ∘ φ_{1,N} : ℙ^{1} → ℙ^{i_{N}} (where φ_{1,N} = φ_{N} ∘ φ_{N−1} ∘ ... ∘ φ_{1}) satisfies (12). Moreover, it is obvious that one can choose g in such a way that (11) holds.

We now extend the morphism f to a morphism f̃ : ℙ^{1} × ℙ^{1} → ℙ^{i_{N}} in the following way. Let f be given in coordinates as z_{i} = f_{i}(x_{0}, x_{1}), where x_{0}, x_{1} are homogeneous coordinates on the space ℙ^{1} which we identify with the second factor of ℙ^{1} × ℙ^{1}. Let t_{0}, t_{1} be homogeneous coordinates on the first factor of ℙ^{1} × ℙ^{1}. We define f̃ by putting z_{i} = t_{0}f_{i}(x_{0}, x_{1}) for i = 0, 1 and z_{i} = t_{1}f_{i}(x_{0}, x_{1}) for i ≥ 2. Condition (11) ensures that f̃ is well-defined. Set Ẽ_{N} := f̃_{N}^{∗}E_{N}.

The morphism f̃ has the following properties.

a) The restriction of f̃ to the vertical fiber P over the point (t_{0}, t_{1}) = (1, 1) coincides with f. Therefore (12) implies

D(Ẽ_{N}|_{P}) ≤ D(E_{1}).  (13)

b) The restriction of f̃ to the vertical fiber P′ over the point (t_{0}, t_{1}) = (1, 0) is a morphism of P′ onto l of degree deg φ_{1,N}. Thus

D(Ẽ_{N}|_{P′}) = (deg φ_{1,N})D_{N}.  (14)

c) The restriction of f̃ to any horizontal fiber S is a linear embedding. Therefore

D(Ẽ_{N}|_{S}) ≤ D_{N}.  (15)

The hypotheses of Lemma 2.5 hold for Ẽ_{N}, since c_{1}(Ẽ_{N}|_{P}) = c_{1}(Ẽ_{N}|_{S}) = 0 by Lemma 3.2. Therefore Lemma 2.5, together with (13), (14), and (15), yields

(deg φ_{1,N})D_{N} ≤ 4(rk E_{N})(D(E_{1}) + 2)(D_{N} + 1) − χ(E_{N}) − χ(E_{N}^{∗}).  (16)

By the Riemann-Roch Theorem and the vanishing of all Chern classes of E_{N} (Lemma 3.2), we have

χ(E_{N}) = χ(E_{N}^{∗}) = nχ(O_{P^{N}}) = rk E_{N}.

Therefore (16) turns into

(deg φ_{1,N})D_{N} ≤ 4(rk E_{N})(D(E_{1}) + 2)(D_{N} + 1) − 2 rk E_{N}.  (17)

As rk E_{N} = rk E and D(E_{1}) do not depend on N, we see that the right hand side of (17) is a linear function of D_{N}, while the left hand side grows with N faster than a linear function of D_{N}, as lim_{N→∞} deg φ_{1,N} = ∞ by our hypothesis. Therefore, inequality (17) can hold only if D_{N} equals zero for large enough N and hence for all N. This means that all vector bundles E_{N} are linearly trivial, and thus trivial.

Theorem 3.1 together with Theorem 2.1 leads to the following general description of a vector bundle on an inductive limit of relative projective spaces. Let p_{N} : M_{N} → S_{N} be a relative projective space, i.e. a locally trivial fibration with base a connected complex analytic space S_{N} and with fiber the projective space ℙ^{i_{N}}. Let furthermore

\begin{matrix}M_{1} & ⟶^{φ_{1}} & M_{2} & ⟶^{φ_{2}} & ... \\ ^{p_{1}}↓ & & ^{p_{2}}↓ \\ S_{1} & ⟶ & S_{2} & ⟶ & ... \end{matrix}  (18)

be a commutative diagram. Note that if the restriction of φ_{N} to a fiber P_{N} of p_{N} has degree d, then the restriction of φ_{N} to any fiber of p_{N} has also degree d. Therefore it makes sense to call (18) linear if the degree of the restriction of φ_{N} on the fibers of p_{N} equals one for almost all N, and twisted otherwise.

Theorem 3.3. Let E = {E_{N}} be a vector bundle on the upper row of (18). Then there exist integers d_{1} ≥ ... ≥ d_{rk E} such that, for large enough N and any fiber P_{N} of p_{N}, E_{N}|_{P_{N}} ≃ ⊕_{r}O_{P_{N}} (d_{r}). Furthermore, if (18) is twisted, d_{1} = ... = d_{rk E} = 0.

Proof. The fact that E_{N}|_{P_{N}} is isomorphic to a direct sum of line bundles follows from Theorems 2.1 and 3.1. The fact that the splitting data of E_{N}|_{P_{N}} does not depend on P_{N} follows from Proposition 2.3. If (18) is linear, the degree of P_{N} equals one for large enough N, thus the splitting data E_{N}|_{P_{N}} does also not depend on N for large enough N. Finally, if (18) is twisted, each d_{r}(E_{N}|_{P_{N}}) is divisible by (deg φ_{N}) · ... · (deg φ_{j}) for all j > N, and therefore d_{r}(E_{N}|_{P_{N}}) = 0.

In general it is not true that E is isomorphic to a direct sum of line bundles, or that E admits a filtration whose associated quotient are line bundles. The reader will easily prove that the latter holds only under the additional assumption that all d_{r} are distinct. Finally, if (18) is twisted, Theorem 3.3 implies that each E_{N} is the pull-back p_{N}^{∗}E′_{N} for some vector bundle E′ = {E′_{N}} on the lower row of (18).

4 The case of ind-grassmannians

In this section we consider two different types of closed immersions

G(k_{1}, n_{1}) ⟶^{f_{1}} G(k_{2}, n_{2}) ⟶^{f_{2}} ...  (19)

and characterize vector bundles on the corresponding ind-spaces.

4.1 Standard extensions of grassmannians

We define a standard extension of grassmannians as a closed immersion of the form

λ_{r,m} : G(k, n) → G(k + r, n + m), {V ⊂ ℂ^{n}} ↦ {V ⊕ W ⊂ ℂ^{n} ⊕ ℂ^{m}},

where W ⊂ ℂ^{m} is a fixed subspace of dimension r ≥ 0.

Proposition 4.1. Assume that a sequence (19) is given, where f_{N} are standard extensions. Let E = {E_{N}} be a vector bundle on (19). Then for each N, E_{N} is a uniform bundle on G(k_{N}, n_{N}).

Proof. For each N consider the natural diagram

\begin{matrix}F (k_{N} − 1, k_{N}, n_{N}) & ⟶^{π_{N}} & G(k_{N}, n_{N}) \\ p^{N} & ↓ \\ G(k_{N} − 1, n_{N}), \end{matrix}  (20)

where F (k_{N} −1, k_{N}, n_{N}) stands for the space of all flags of type (k_{N} −1, k_{N}) in ℂ^{n_{N}}. Note that f_{N} induce morphisms between the diagrams (20) for N and N + 1. Furthermore, the embeddings F (k_{N} − 1, k_{N}, n_{N}) → F (k_{N+1} − 1, k_{N+1}, n_{N+1}) and the morphisms p_{N} define a relative projective ind-space. The vector bundles π_{N}^{∗}E_{N} define a vector bundle on this relative projective ind-space. By Theorem 3.3 the splitting data of the restriction π_{N}^{∗}E_{N}|_{P_{N}} on each fiber P_{N} of p_{N} does not depend on N and P_{N}. This implies the result, as any projective line l ⊂ G(k_{N}, n_{N}) is a line in some fiber P_{N}, and the splitting data of E|_{l} and π_{N}^{∗}E_{N}|_{P_{N}} are equal.

Theorem 4.2. Assume that a sequence (19) is given, where f_{N} are standard extensions and lim_{N→∞} k_{N} = lim_{N→∞}(n_{N} − k_{N}) = ∞. Then any vector bundle E = {E_{N}} on the inductive limit of (19) is isomorphic to a direct sum of line bundles.

Proof. By Proposition 4.1, each E_{N} is a uniform bundle. We will prove the Theorem by induction on rk E = rk E_{N}. Fix N and let d_{1}, ..., d_{rk E_{N}} be the splitting data for the restriction E_{N}|_{l}, where l ⊂ G(k_{N}, n_{N}) is any line. Denote by ν_{1} the multiplicity of d_{1}. As each f_{j} induces an isomorphisms of Picard groups, the splitting data d_{1}, ..., d_{rk E_{N}} does not depend on N. Fix x ∈ G(k_{N}, n_{N}). Any line l ⊂ G(k_{N}, n_{N}) passing through x determines a subspace E′_{l,x} in the fiber (E_{N})_{x}: this is the fiber of the unique subbundle of E_{N}|_{l} isomorphic to the direct sum of ν_{1} copies of O_{l}(d_{1}), see Subsection 2.2. The variety of all lines l passing through x is isomorphic to the direct product ℙ^{k_{N} −1} × ℙ^{n_{N} −k_{N} −1}, hence we have a morphism

ψ_{x} : ℙ^{k_{N} −1} × ℙ^{n_{N} −k_{N} −1} ⟶ G(ν_{1}, rk E_{N}), l ↦ E′_{l,x}.

By our assumption, lim_{N→∞} k_{N} = lim_{N→∞}(n_{N} − k_{N}) = ∞. Therefore, for large N, the morphism ψ_{x} is trivial, i.e. it determines a fixed subspace (E′_{N})_{x} in (E_{N})_{x}. In this way we obtain a subbundle E′_{N} ⊂ E_{N}. Clearly f_{N}^{∗}(E′_{N+1}) = E′_{N}, therefore E′ := {E′_{N}} is a well-defined subbundle of E.

If rk E′ = rk E, twisting by the line bundle {O_{G(k_{N} ,n_{N})}(−d_{1})} yields a bundle whose restriction to any line in G(k_{N}, n_{N}) is trivial for all N. Therefore the twisted bundle is trivial on each G(k_{N}, n_{N}), and hence trivial. Thus E is isomorphic to a direct sum of ν_{1} copies of {O_{G(k_{N} ,n_{N})}(d_{1})}. If rk E′ < rk E, the induction assumption implies that both E′ and E/E′ are isomorphic to direct sums of line bundles. Finally, the observation that there are no non-trivial extensions of line bundles on G(k_{N}, n_{N}) for k_{N} > 1 (by the Bott-Borel-Weil Theorem, H^{1}(L) = 0 for any line bundle L on G(k_{N}, n_{N}) unless k_{N} = n_{N} − k_{N} = 1) implies that E is isomorphic to a direct sum of line bundles.

Here is an important special case of Theorem 4.2. If V ⊂ ℂ^{∞} is an arbitrary fixed subspace, a subspace V′ ⊂ ℂ^{∞} is commensurable with V, if there exists a finite dimensional subspace U ⊂ ℂ^{∞} such that V ⊂ V′ + U, V′ ⊂ V + U, and dim V ∩ U = dim V′ ∩ U. The set of all subspaces V′ comensurable with V is, by definition, the ind-grassmannian G(V, ∞). If V is finite dimensional and dim V = k, then G(V, ∞) = G(k, ∞). In [DP] an explicit construction of G(V, ∞) as an ind-space is given. Moreover, G(V, ∞) is the inductive limit of standard extensions, and the conditions of Theorem 4.2 are satisfied for G(V, ∞) if and only if dim V = codim_{ℂ^{∞}}V = ∞. Therefore, by Theorem 4.2, we conclude that in the latter case every vector bundle on G(k, ∞) is isomorphic to a direct sum of line bundles.

More generally, if V_{1} ⊂ ... ⊂ V_{r} is any fixed flag in ℂ^{∞}, the ind-variety F (V_{1}, ..., V_{r}, ∞) of flags V′_{1} ⊂ ... ⊂ V′_{r} comensurable with V_{1} ⊂ ... ⊂ V_{r}, is constructed in [DP]. We leave it to the reader to prove the following corollary of Theorem 4.2 by double induction on rk E and r.

Corollary 4.3. Let dim V_{1} = dim(V_{2}/V_{1}) = ... = dim(ℂ^{∞}/V_{r}) = ∞. Then every vector bundle E on F (V_{1}, ..., V_{r}, ∞) is isomorphic to a direct sum of line bundles.

4.2 Twisted extensions of grassmannians

An alternative definition of a grassmannian is as a homogeneous space GL(n)/P for a maximal parabolic subgroup P ⊂ GL(n). We call a morphism f : G(k_{1}, n_{1}) → G(k_{2}, n_{2}) homogeneous if it is induced by a group homomorphism f̃ : GL(n_{1}) → GL(n_{2}). A homogeneous morphism is a closed immersion or its image is a point.

Consider a homogeneous morphism f : G(k_{1}, n_{1}) → G(k_{2}, n_{2}), where G(k_{1}, n_{1}) = GL(n_{1})/P_{1}, G(k_{2}, n_{2}) = GL(n_{2})/P_{2}, and P_{1}, P_{2} are maximal parabolic subgroups such what f̃(P_{1}) ⊂ P_{2}. The reductive part of P_{1} is isomorphic to GL(k_{1}) × GL(n_{1} − k_{1}) and the reductive part of P_{2} is isomorphic to GL(k_{2}) × GL(n_{2} − k_{2}). Note that f̃(GL(k_{1}) × GL(n_{1} − k_{1})) is contained in one component of the direct product GL(k_{2}) × GL(n_{2} − k_{2}) if and only if the induced morphism f : G(k_{1}, n_{1}) → G(k_{2}, n_{2}) is a morphism into a point. In the sequel we adopt a convention: whenever we write a homogeneous morphism as f : G(k_{1}, n_{1}) → G(k_{2}, n_{2}) and f is a closed immersion, we automatically assume that f̃(GL(k_{1})) ⊂ GL(k_{2}) and f̃(GL(n_{1} − k_{1})) ⊂ GL(n_{2} − k_{2}). Furthermore, we call a morphism φ : ℙ^{N} → G(k, n) k-split if φ^{∗}S_{k} is isomorphic to a direct sum of line bundles on ℙ^{N}. For example, Theorem 2.1 implies that any morphism ℙ^{1} → G(k, n) is k-split.

Proposition 4.4. a) Any homogeneous morphism φ : ℙ^{N} = G(1, N) → G(k, n) is k-split.

b) Let f : G(k_{1}, n_{1}) → G(k_{2}, n_{2}) be a homogeneous embedding of grassmannians and φ : ℙ^{N} → G(k_{1}, n_{1}) be a k_{1}-split morphism. Then the composition f ∘ φ : ℙ^{N} → G(k_{2}, n_{2}) is k_{2}-split.

Proof. a) Let ℙ^{N} = GL(N + 1)/P_{1}, and G(k, N) = GL(n)/P_{2}. The reductive parts of P_{1} and P_{2} are isomorphic respectively to GL(1)×GL(N) and GL(k)×GL(n−k). According to our convention, the group homomorphism φ̃ : GL(N + 1) → GL(n) which induces φ maps GL(1) into GL(k), and GL(N) into GL(n − k). Therefore the structure group of φ^{∗}S_{k} is reduced to GL(1), which implies that φ^{∗}S_{k} is a direct sum of line bundles.

b) Let f̃: GL(n_{1}) → GL(n_{2}) be a group homomorphism which induces the morphism f. Since f̃(GL(k_{1})) ⊂ GL(k_{2}), the GL(n_{1})-homogeneous bundle f^{∗}S_{k_{2}} is a direct summand in the tensor product of several copies of S_{k_{1}} and S_{k_{1}}^{∗}. This, together with the fact, that the morphism φ is k_{1}-split, implies that (f ∘ φ)^{∗}S_{k_{2}} is a direct summand of a bundle isomorphic to a direct sum of line bundles on ℙ^{N}. Hence (f ∘ φ)^{∗}S_{k_{2}} is isomorphic itself to a direct sum of line bundles, i.e. the morphism f ∘ φ is k_{2}-split.

Here is a coordinate form of a k-split map. Let t = (t_{0}, ..., t_{N}) be homogeneous coordinates on ℙ^{N}, and v_{1}, ..., v_{n} be a basis in ℂ^{n}. Then the reader will check that a morphism φ : ℙ^{N} → G(k, n) is k-split if and only if φ can be presented in the form

φ(t) = span{ ∑_{j=1}^{n} φ_{i,j}(t)v_{j}, i = 1, ..., k}  (21)

for some homogeneous polynomials φ_{i,j}(t) in t. It is clear that if φ : ℙ^{N} → G(k, n) is represented in the form (21), then deg φ_{i,j} does not depend on j and deg φ = ∑_{i=1}^{k} deg φ_{i,j}.

Lemma 4.5. Let φ : ℙ^{N} → G(k, n) be a k-split morphism and λ_{0,k} : G(k, n) → G(k, n+k) be a standard extension (see Subsection 4.1). Then the composition λ_{0,k} ∘ φ : ℙ^{N} → G(k, n + k) factors through a standard extension ℙ^{N} → ℙ^{N+1}, via a k-split morphism φ : ℙ^{N+1} → G(k, n + k).

Proof. Let φ be presented in the form (21), and let the vectors v_{n+1}, ...v_{n+k} extend the set v_{1}, ..., v_{n} to a basis in ℂ^{n+k}. Fix homogeneous coordinates t′ = (t_{0}, ..., t_{N+1}) in ℙ^{N+1} and identify ℙ^{N} with the hyperplane t_{N+1} = 0. Then φ̃ : ℙ^{N+1} → G(k, n + k) is given by the formula

φ̃(t′) = span{ ∑_{j=1}^{n} φ_{i,j}(t)v_{j} + t_{N+1}^{δ_{i}}v_{n+i}, i = 1, ..., k},

where δ_{i} := deg φ_{i,j}.

We need one last definition. We call a homogeneous morphism f : G(k_{1}, n_{1}) → G(k_{2}, n_{2}) a twisted extension if f decomposes as λ_{0,n_{2}−n}∘h, where h : G(k_{1}, n_{1}) → G(k_{2}, n) is a homogeneous morphism for n_{2} − n ≥ k_{2}, and λ_{0,n_{2}−n} : G(k_{2}, n) → G(k_{2}, n_{2}) is a standard extension.

Proposition 4.6. Assume that a sequence (19) is given, where f_{N} are twisted extensions, infinitely many of which have degree greater than one. Suppose in addition that lim_{N→∞} k_{N} = lim_{N→∞}(n_{N} − k_{N}) = ∞. Then any vector bundle E = {E_{N}} on the inductive limit of (19) is trivial.

Proof. Let E = {E_{N}} be a vector bundle on the inductive limit of (19). Fix N and a (k_{N} − 1)-dimensional subspace of ℂ^{n_{N}}. This determines a projective subspace ℙ^{n_{N} −k_{N}} of G(k_{N}, n_{N}), and the closed immersion φ_{N} : ℙ^{n_{N} −k_{N}} → G(k_{N}, n_{N}) is k_{N}-split as φ^{∗}S_{k_{N}} ≃ (k_{N} − 1)O_{ℙ^{n_{N} −k_{N}}} ⊕ O_{ℙ^{n_{N} −k_{N}}}(−1). Furthermore, let f_{N} = λ_{0,m_{N}} ∘ h_{N}, where h_{N} : G(k_{N}, n_{N}) → G(k_{N+1}, n_{N+1} − m_{N}) is a homogeneous closed immersion and m_{N} ≥ k_{N+1}. Proposition 4.4 b) implies that the composition h_{N} ∘ φ_{N} is k_{N+1}-split. Then, by Lemma 4.5, the morphism f_{N} ∘ φ_{N} = λ_{0,m_{N}} ∘ (h_{N} ∘ φ_{N}) factors through a standard extension ℙ^{n_{N} −k_{N}} → ℙ^{n_{N} −k_{N} +1} via a homogeneous immersion φ_{N+1} : ℙ^{n_{N} −k_{N} +1} → G(k_{n+1}, n_{N+1}). Proceeding by induction on j, we build homogeneous immersions φ_{N+j} : ℙ^{n_{N} −k_{N} +j} → G(k_{N+j}, n_{N+j}) and standard extensions ℙ^{n_{N} −k_{N} +j} → ℙ^{n_{N} −k_{N} +j+1} which form the commutative diagram

\begin{matrix}ℙ^{n_{N} −k_{N}} & ⟶ & ℙ^{n_{N} −k_{N} +1} & ⟶ & ... \\ ^{φ_{N}} ↓ & & ^{φ_{N+1}}↓ \\ G(k_{N}, n_{N}) & ⟶^{f_{N}} & G(k_{N+1}, n_{N+1}) & ⟶^{f_{N+1}} & ... . \end{matrix}  (22)

Applying the BVT theorem to the inductive limit of the upper row of (22), we conclude that E_{N}|_{ℙ^{n_{N} −k_{N}}} is isomorphic to a direct sum of line bundles. Furthermore, one sees exactly as in Proposition 4.1 that E_{N} is a uniform bundle. Now the argument in the proof of Theorem 4.2 shows that, since lim_{N→∞} k_{N} = lim_{N→∞}(n_{N} − k_{N}) = ∞, E_{N} is isomorphic to a direct sum of line bundles. Therefore E is also isomorphic to a direct sum of line bundles. Finally, as deg f_{N} > 1 for infinitely many N, a line bundle on this inductive limit is necessarily trivial, hence E is trivial.

Here are two examples to Proposition 4.6.

Let ℂ^{w} be a proper subspace of ℂ^{m}. Consider the closed immersion

G(k, n) ⟶ G(kw, nm),  {V ⊂ ℂ^{n}} ↦ {V ⊗ ℂ^{w} ⊂ ℂ^{n} ⊗ ℂ^{m}}.  (23)

It factors, via a standard extension, through the closed immersion

G(k, n) ⟶ G(kw, nw), {V ⊂ ℂ^{n}} ↦ {V ⊗ ℂ^{w} ⊂ ℂ^{n} ⊗ ℂ^{w}}.

Fixing a sequence of pairs ℂ^{w_{N}} ⊂ ℂ^{m_{N}}, determines a sequence of immersions (23). If the set {w_{N}} is bounded, for sufficiently large N all embeddings in this sequence are twisted extensions as

nm_{1}...m_{N+1} − nw_{1}...w_{N+1} ≥ kw_{1}...w_{N+1}

for large enough N. Therefore Proposition 4.6 implies that any vector bundle on the inductive limit is trivial.

Finally consider a closed immersion of the form

G(k, n) ⟶ G(k(k + 1)/2, n^{2}),  {V ⊂ ℂ^{n}} ↦ {S^{2}(V) ⊂ ℂ^{n} ⊗ ℂ^{n}},  (24)

where S^{2} stands for symmetric square. By iteration of (24) we obtain another example of a sequence of twisted extensions. Indeed, (24) factors, via a standard extension, through the closed immersion

G(k, n) ⟶ G(k(k + 1)/2, n(n + 1/2), {V ⊂ ℂ^{n}} ↦ {S^{2}(V) ⊂ S^{2}(ℂ^{n})},

and the inequality

k(k + 1)/2 ≤ n^{2} − n(n + 1)/2

needed in the definition of a twisted extension, is trivially satisfied. Proposition 4.6 implies that any vector bundle on the corresponding inductive limit is also trivial.

5 Conclusion

The results of this paper lead naturally to the following question. Consider an arbitrary sequence of morphisms

G(k_{1}, n_{1}) ⟶^{f_{1}} G(k_{2}, n_{2}) ⟶^{f_{2}} ... .  (25)

Is it true that, if E = {E_{N}} is an arbitrary vector bundle on (25) then each E_{N} is a homogeneous bundle on G(k_{N}, n_{N})? In fact, the BVT Theorem, both main theorems of Sato (see [S2]), Theorems 3.1 and 4.2, as well as Proposition 4.6, are all equivalent to the affirmative answer to the above question for the cases they apply to. More precisely, the description of vector bundles on sequences (25) in each specific case considered in the above statements, is equivalent to the descriptions of systems of homogeneous bundles {E_{N}} with f_{N+1}^{∗}E_{N+1} = E_{N}. Note that Theorem 3.1 applies to not necessarily homogeneous morphisms, while in all other statements homogeneous morphisms are considered. It would be very interesting to give an answer to the above general question even under the assumption that all morphisms f_{N} are homogeneous.

References

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

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

[DP] I. Dimitrov and I. Penkov, Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups, preprint 2001.

[D1] J. Donin, Deformations of holomorphic vector bundles on the Riemann sphere, Mat. Sb. (N.S.) 71(113) (1966), 495–502 (Russian).

[D2] J. Donin, Construction of a versal family of deformations for holomorphic principal bundles over a compact complex space, Mat. Sb. (N.S.) 94(136) (1974), 430–443 (Russian).

[P] I. Penkov, The Penrose transform on general grassmannians, C. R. Acad. Bulgare Sci. 33 (1980), 1439–1442.

[S1] E. Sato, On the decomposability of infinitely extendable vector bundles on projective spaces and Grassmann varieties, J. Math. Kyoto Univ. 17 (1977), 127–150.

[S2] 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.

J. Donin Department of Mathematics Bar-Ilan University 52900 Ramat Gan, Israel donin@macs.biu.ac.il

I. Penkov Department of Mathematics University of California Riverside Riverside, CA 92521, USA penkov@math.ucr.edu