Line bundles for which a projectivized jet bundle is a product

Sandra Di Rocco and Andrew J. Sommese

June 25, 1998

Abstract We characterize the triples (X, L, H), consisting of line bundles L and H on a complex projective manifold X, such that for some positive integer k, the k-th holomorphic jet bundle of L, J_{k}(X, L), is isomorphic to a direct sum H ⊕ … ⊕ H. 1991 Mathematics Subject Classification. 14J40, 14M99.

Keywords and phrases. jet bundle, complex projective manifold, projective space, Abelian variety.

Introduction

Let X be a complex projective manifold. A large amount of information on the geometry of an embedding i : X ↪ ℙ^{N} is contained in the “bundles of jets” of the line bundles kL = i^{∗}O_{ℙ^{N}} (k) for k ≥ 1. The k-th jet bundle of a line bundle L (sometimes called the k-th principal part of L) is usually denoted by J_{k}(X, L), or by J_{k}(L) when the space X is clear from the context. The bundle J_{k}(kL) is spanned by the k-jets of global sections of kL. If α : ℙ(J_{1}(L)) → ℙ^{N} is the map given by the 1-jets of elements of H^{0}(X, L), then α(ℙ(J_{1}(L))_{x}) = 𝕋_{x}, where 𝕋_{x} is the embedded tangent space at x. Similarly ℙ(J_{k}(kL))_{x} is mapped to the “k-th embedded tangent space” at x (see [?] for more details). Given this interpretation of the projectivized jet bundle, it is natural to expect that a projectivized jet bundle is a product of the base and a projective space only under very rare circumstances.

In this paper we analyze the more general setting where L is a line bundle on X (with no hypothesis on its positivity) and J_{k}(L) = ⊕H, with H a line bundle on X. Some years ago the second author [?] analyzed the pairs (X, L) under the stronger hypothesis that J_{k}(L) is a trivial bundle. Though, the results in this paper do not follow from [?], the only possible projective manifolds with a projectivized jet bundle of some line bundle being a product of the base manifold and a projective space, turn out to be the same, i.e., Abelian varieties and projective space. We completely characterize the possible triples (X, L, H). In fact, on Abelian varieties we have the necessary and sufficient condition that L ∈ Pic_{0}(X), and if L = O_{ℙ^{n}}(a), then only the range a ≥ k or a ≤ −1 can occur.

A significant amount of the research on this paper was done during our stay at Max-Planck-Institut für Mathematik in Bonn, to which we would like to express our gratitude for the excellent working conditions. The second author would also like to thank the Alexander von Humboldt Foundation for its support.

1 Some preliminaries

We follow the usual notation of algebraic geometry. Often we denote the direct sum of m copies of a vector bundle E, for some integer m > 0, by ⨁_{m} E. We freely use the additive notation for line

bundles. We often use the same symbols for a vector bundle and its associated locally free sheaf of germs of holomorphic sections. We say a vector bundle is spanned if the global sections generate each fiber of the bundle. For general references we refer to [?] and [?].

By X we will always denote a projective manifold over the complex numbers ℂ and by L a holomorphic line bundle on X.

Let k be a nonnegative integer. The k-th jet bundle J_{k}(L) associated to L is defined as the vector bundle of rank \binom{k+n}{n} associated to the sheaf p^{∗}L/(p^{∗}L ⊗ J_{Δ}^{k+1}), where p : X × X → X is the projection on the first factor and J_{Δ} is the sheaf of ideals of the diagonal, Δ, of X × X. Let j_{k} : H^{0}(X, L) × X → J_{k}(L) be the map which associates to each section s ∈ H^{0}(X, L) its k-th jet. That means that locally, choosing coordinates (x_{1}, ..., x_{n}) and a trivialization of L in a neighborhood of x, j_{k}(s, x) = (a_{1}, ..., a\binom{n+k}{n}), where the a_{i}’s are the coefficients of the terms of degree up to k in the Taylor expansion of s around x. Notice that the map j_{k} is surjective if and only if H^{0}(X, L) generates all the k-jets at all points x ∈ X. For example j_{1} being surjective is equivalent to |L| defining an immersion of X in ℙ^{h^{0}(L)−1}.

We will often use the associated exact sequence:

0 → T_{X}^{∗(k)} ⊗ L → J_{k}(L) → J_{k−1}(L) → 0 (j_{k})

where T_{X}^{∗(k)} denotes the k-th symmetric power of the cotangent bundle of X. There is an injective bundle map [?, p. 52]

γ_{i,j} : J_{i+j}(L) → J_{i}(J_{j}(L)).

Using the sequence (j_{k}) it is easy to see that

det J_{k}(L) = \frac{1}{n + 1}\binom{n + k}{n}(kK_{X} + (n + 1)L)

Lemma 1.1 Let X be a compact Kähler variety and L a holomorphic line bundle on X. Then c_{1}(L) = 0 in H^{1}(T_{X}^{∗}) if and only if the bundle sequence (j_{1}) splits. If c_{1}(L) = 0 in H^{1}(T_{X}^{∗}), then J_{k}(L) ≅ J_{k}(O_{X}) ⊗ L.

Proof. A local computation, using Cech coverings, shows that the Atiyah class defined by the sequence (j_{1}) is the cocycle c_{1}(L) ∈ H^{1}(X, T_{X}^{∗}). Thus c_{1}(L) = 0 in H^{1}(T_{X}^{∗}) if and only if the bundle sequence (j_{1}) splits.

If c_{1}(L) = 0 in H^{1}(T_{X}^{∗}) then L has constant transition functions. From this, the isomorphism J_{k}(L) ≅ J_{k}(O_{X}) ⊗ L follows. Q.E.D.

Let A and B be vector bundles on X, we recall that if α ∈ H^{1}(A ⊗ B^{∗}) represents the vector bundle extension E, then any nonzero multiple λα gives an isomorphic extension. If λ = 0 this is of course false since we would get the trivial extension. It follows that:

Lemma 1.2 Let λ be a nonzero integer, then J_{1}(λL) ≅ J_{1}(L) ⊗ (λ − 1)L

Proof. Consider the extension :

0 → T_{X}^{∗} ⊗ λL → J_{1}(L) ⊗ (λ − 1)L → λL → 0 (j_{1}) ⊗ (λ − 1)L

represented by c_{1}(L) + c_{1}((λ − 1)L) = λc_{1}(L) = c_{1}(λL). Then, from what was observed above, J_{1}(λL), which is the vector bundle extension given by c_{1}(λL), is isomorphic to J_{1}(L) ⊗ (λ − 1)L.

Q.E.D.

2 The basic examples

In this section we will characterize line bundles with splitting k-th jet bundles on Abelian varieties and on ℙ^{n}. These turn out to be the only possible examples.

Proposition 2.1 Let L and H be line bundles on an Abelian variety X, then the following assertions are equivalent:

• J_{k}(L) ≅ ⊕H for some k;

• H ≅ L and L ∈ Pic_{0}(X).

In particular J_{k}(L) splits in the sum of spanned line bundles only when L = O_{X} and J_{k}(L) = ⊕O_{X}.

Proof. Let X be an Abelian variety and assume that J_{k}(L) = ⊕H for some line bundle H. Then the sequences (j_{m}) for m ≤ k imply that there is a surjection of the trivial bundle J_{k}(L) ⊗ (−H) onto L − H. Thus the line bundle L − H is a spanned line bundle with

\binom{n + k}{n}c_{1}(L − H) = 0,

which implies L = H. Using this we can find a direct summand L of J_{k}(L) whose image in J_{1}(L) maps onto L under the map J_{1}(L) → L → 0 of the sequence (j_{1}). Thus the sequence (j_{1}) splits and therefore c_{1}(L) = 0 in H^{1}(T_{X}^{∗}) by Lemma ??.

Conversely assume that we have a line bundle L ∈ Pic_{0}(X). Then by Lemma ??, we have that J_{k}(L) ≅ J_{k}(O_{X}) ⊗ L. Using the triviality of T_{X}^{(j)} for all j ≥ 0, it follows that J_{k}(O_{X}) is trivial.

Q.E.D.

Proposition 2.2 J_{k}(O_{ℙ^{n}}(a)) is isomorphic to a direct sum \begin{matrix}⨁ \\ ( & {k+n \atop n} & ) \end{matrix} O_{ℙ^{n}}(q) for some k, a, q ∈ ℤ with

k > 0 if and only if q = a − k and either a ≥ k or a ≤ −1.

Proof. Let L = O_{ℙ^{n}}(a). First notice that if the k-th jet bundle splits as J_{k}(L) = \begin{matrix}⨁ \\ ( & {k+n \atop n} & ) \end{matrix} O_{ℙ^{n}}(q) for

some nonnegative k and some integer q, then

det J_{k}(L) = O_{ℙ^{n}} (\binom{n + k}{k}(a − k)) ,

which implies that q = a − k. Then the fact that the k-th jet bundle of O_{ℙ^{n}}(a) always has sections for a ≥ 0 rules out the cases a = 0, ..., k − 1.

Lemma ?? gives J_{1}(O_{ℙ^{n}}(a)) ≅ J_{1}(O_{ℙ^{n}}(1)) ⊗ O_{ℙ^{n}}(a − 1). Then J_{1}(O_{ℙ^{n}}(1)) = ⊕O_{ℙ} (see, e.g., [?]) implies

J_{1}(O_{ℙ^{n}}(a)) = ⨁_{n+1} O_{ℙ^{n}}(a − 1)

Dualizing γ_{1,1} and using

J_{1} (J_{1}(O_{ℙ^{n}}(a)) = J_{1} (⨁_{n+1} O_{ℙ^{n}}(a − 1)) = ⨁_{n+1} (⨁_{n+1} O_{ℙ^{n}}(a − 2))

we obtain the quotient

⨁_{n+1} (⨁_{n+1} O_{ℙ^{n}}(a − 2))^{∗} → J_{2}(O_{ℙ^{n}}(a))^{∗} → 0

and thus, tensoring by O_{ℙ^{n}}(a − 2)

⨁_{(n+1)^{2}} O_{ℙ^{n}} → J_{2}(O_{ℙ^{n}}(a))^{∗} ⊗ O_{ℙ^{n}}(a − 2) → 0

Then the vector bundle J_{2}(O_{ℙ^{n}}(a))^{∗} ⊗O_{ℙ^{n}}(a−2) is spanned with det(J_{2}(O_{ℙ^{n}}(a))^{∗} ⊗O_{ℙ^{n}}(a−2)) = O_{ℙ^{n}} which implies that J_{2}(O_{ℙ^{n}}(a))^{∗} ⊗ O_{ℙ^{n}}(a − 2) = ⊕O_{ℙ^{n}}. Iterating this argument one gets

J_{k}(O_{ℙ^{n}}(a))^{∗} ⊗ O_{ℙ^{n}}(a − k) = \begin{matrix}⨁ \\ ( & {k+n \atop n} & ) \end{matrix} O_{ℙ^{n}}. Q.E.D.

3 The main result

In this section we characterize complex projective manifolds having a line bundle whose projectivized k-jet bundle is a product of the base manifold and a projective space.

Theorem 3.1 Let L and H be holomorphic line bundles on X. Then J_{k}(L) = H ⊕ … ⊕ H if and only if the triple (X, L, H) is one of the two below:

1. (X, L, L) where X is Abelian and L ∈ Pic_{0}(X)

2. (ℙ^{n}, O_{ℙ^{n}}(a), O_{ℙ^{n}}(a − k)) and a ≥ k or a ≤ −1.

Proof. Assume J_{k}(L) = H ⊕ … ⊕ H. Tensoring the sequence (j_{k}) by H^{∗} gives the quotient \begin{matrix}⨁ \\ ( & {n+k \atop n} & ) \end{matrix} O_{X} → L ⊗ H^{∗} → 0. Tensoring the sequence (j_{k}) by H^{∗} and then dualizing it gives the

quotient \begin{matrix}⨁ \\ ( & {n+k \atop n} & ) \end{matrix} O_{X} → T_{X}^{(k)} ⊗ (H ⊗ L^{∗}) → 0. It follows that L ⊗ H^{∗}, T_{X}^{(k)} ⊗ (H ⊗ L^{∗}) and thus T_{X}^{(k)}

are spanned bundles over X.

First assume that the canonical bundle K_{X} is nef. Since T_{X}^{(k)} is spanned, we conclude that det T_{X}^{(k)}, which is a spanned negative multiple of K_{X} is trivial. Thus T_{X}^{(k)} is a trivial bundle. From this we conclude that L − H is trivial. It follows that the trivial bundle J_{k}(L)^{∗} ⊗ L has a filtration with quotient bundles T_{X}^{(j)} for 0 ≤ j ≤ k. This shows that T_{X}^{(j)} is trivial for all j > 0. Thus under the assumption that K_{X} is nef, we conclude that X would be an Abelian variety. Proposition ?? implies also that L ∈ Pic_{0}(X). This gives the first case of the theorem.

We are thus left with the case when K_{X} is not nef. Since T_{X}^{(k)} is spanned −K_{X} is nef. The cone theorem then yields the existence of an extremal ray ℝ_{+}[γ], with 1 ≤ −K_{X} · γ ≤ n + 1. Let l be the normalization of γ and let

T_{X|l} = ⨁ O_{l}(a_{i}), (L ⊗ H^{∗})_{l} = O_{l}(b),

where by abuse of notation we denote the pullback of a bundle E on γ to l by E_{l}. Writing for simplicity T_{X|l}^{(k)} = (⊕_{i}O_{l}(ka_{i})) ⊕ P we get

T_{X|l}^{(k)} ⊗ (L ⊗ H^{∗})_{l}^{∗} = (⊕_{i}O(ka_{i} − b)) ⊕ (P ⊗ O_{l}(−b))

Since T_{X|l}^{(k)} ⊗ (L ⊗ H^{∗})_{l}^{∗} is spanned we conclude that ka_{i} ≥ b ≥ 0. Note that b > 0. Indeed if b = 0, then

0 = deg det J_{k}(L)_{l} ⊗ (−H_{l}) = \frac{1}{n + 1}\binom{n + k}{n}kK_{X} · l ≠ 0.

Thus a_{i} > 0 for all i’s. Moreover from sheaf injection 0 → T_{l} → T_{X|l} we see that T_{X|l} must contain

a factor O_{l}(a_{i}) with a_{i} ≥ 2. Then −K_{X} · γ = ∑_{i=1}^{n+1} a_{i} ≤ n + 1 implies that a_{j} = 2 for one j and

a_{i} = 1 for i ≠ j, i.e., −K_{X} ·γ = n+1. Now from Mori’s proof of Hartshorne conjecture (see [?, §4]) we deduce that X must be ℙ^{n}. At this point we have recovered case (2) by applying Proposition ??.

Propositions ?? and ?? show that if we are in cases (1) and (2) respectively, then J_{k}(L) = H ⊕ … ⊕ H satisfied. Q.E.D.

References

[BSo] M.C. Beltrametti and A.J. Sommese, The Adjunction Theory of Complex Projective Varieties, Expositions in Math. 16 (1995), W. de Gruyter, Berlin.

[GH] P. Griffiths and J. Harris, Algebraic geometry and local differential geometry, Ann. Sci. Ècole Norm. Sup. (4) 12 (1979), 355-432.

[KSp] A. Kumpera and D. Spencer, Lie Equations; Vol. I: General Theory, Ann. of Math. Stud. 73 (1972), Princeton University Press, Princeton, N.J.

[L] R. Lazarsfeld, Some applications of the theory of positive vector bundles, Complete Intersections, CIME course 1983, Acireale (Catania) ed. by S. Greco and R. Strano, Lecture Notes in Math. 1092 (1984), 29–61, Springer-Verlag, New York.

[So] A.J. Sommese, Compact Complex Manifolds possessing a line bundle with a trivial jet bundle, Abh. Math. Sem. Univ. Hamburg, 47 (1978), 79-91.

Sandra Di Rocco  Andrew J. Sommese
Department of Mathematics  Department of Mathematics
KTH, Royal Institute of Technology University of Notre Dame
S-100 44 Stockholm, Sweden  Notre Dame, Indiana 46556, U.S.A.
e-mail: sandra@math.kth.se  e-mail: sommese@nd.edu
http://www.math.kth.se/∼sandra  http://www.nd.edu/∼sommese