Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 37 (1996), No. 1, 87-101.

On some Special Fano Manifolds

Antonio Lanteri and Cristiana S. Sacchi *

Dipartimento di Matematica “F. Enriques” Università degli Studi di Milano, via C. Saldini, 50, 20133 Milano, Italy E-mail address: lanteri@vmimat.mat.unimi.it

Abstract. Some properties of Fano manifolds of product type involving the coindex and the number of factors are discussed and a contribution to the study of special Fano P-bundles of low corank is given; in particular standard special Fano P-bundles of corank 3 are classified. A question related to a vector bundle adjunction conjecture is also discussed. MSC 1991: 14J45, 14J40

Introduction

In this paper we deal with Fano manifolds of special kind, such as products and P-bundles with some extra conditions.

For a Fano manifold of product type all factors are Fano. So we simply call such a manifold a Fano product. Let X = Y_{1} × … × Y_{s} be a Fano product. By investigating some inequalities involving the coindex c of X, the coindexes of the factors and their number s, we prove that s ≤ c + 1 and characterize the X for which c − 2 ≤ s ≤ c + 1. In some sense our results are best possible. The case s = c − 2 involves at worst Mukai manifolds as factors; but in some cases with s = c − 3 one of the Y_{i} belongs to the still unexplored class of Fano manifolds of coindex 4.

Besides products, some P-bundles are Fano manifolds. Recently, several papers appeared on the classification of Fano P-bundles of low dimensions [11], [12], while other recent papers related to a conjecture of Mukai drew attention to ample vector bundles E on a projective manifold B such that K_{B} + det E is not nef (or not ample, or trivial) [3], [8], [10], [14]. The obvious link between these subjects is that if −(K_{B} + det E) is nef, then P = P_{B}(E) is a Fano manifold. In this paper we also investigate some properties of Fano P-bundles of the above type P_{B}(E) satisfying the condition det E = −K_{B}. We

* Research done within the framework of the 40% project “Algebraic Geometry” of the Italian M.U.R.S.T. The first author is a member of G.N.S.A.G.A. of the Italian C.N.R.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1996 Heldermann Verlag, Berlin define these to be special (see section 2). Up to replacing E with E ⊗ L, for a suitable line bundle L on B, they simply coincide with Fano P-bundles of maximal index. If P is a special Fano P-bundle as above, letting r = rk E and b = dim B, it follows from [3] that cork P ≥ 0, where, by definition, cork P = b + 1 − r, and several results scattered in the literature provide the classification of special Fano P-bundles with cork P ≤ 2, up to a still undecided case when cork P = 2. In this paper, after providing several examples in the range cork P ≥ 3 we classify a special class of Fano P-bundles of corank 3 on product manifolds, which we call standard. Roughly speaking, this means that E comes from vector bundles on the factors (for the precise definition see section 3). There are several such P-bundles of corank 3, while they are very few for cork P = 2 and do not appear at all when cork P ≤ 1. Although this classification relies on the known results for Fano P-bundles with cork P ≤ 2, we would like to stress that our result is effective: actually the undecided case mentioned above in this context is irrelevant.

The paper is organized as follows. In section 1 we study Fano products. In section 2, after discussing general properties of special Fano P-bundles, we review the results which are known for corank ≤ 2 and produce some new examples of these bundles with corank ≥ 3. In section 3 we classify standard special Fano P-bundles of corank 3. Finally in section 4 we relate the above setting to a conjecture on adjunction for vector bundles [9].

1. Fano products

(1.0) Let X = Y_{1} × … × Y_{s} (s ≥ 2) be a Fano manifold. Put N = dim X, r = index(X), c = coindex(X) and n_{i} = dim Y_{i}, i = 1, …, s. So

(1.0.1)  ∑_{i=1}^{s} n_{i} = N = r + c − 1.

We always assume that no Y_{i} is a product. The aim of this section is to provide an upper bound for s in terms of c and to classify the X as above near the upper bound.

(1.1) Remark. Let X be as in (1.0) and let p_{i} denote the projection onto the i-th factor. Then

(1) Y_{i} is a Fano manifold for all i;

(2) Pic(X) ≅ ∏_{i=1}^{s} p_{i}^{∗} Pic(Y_{i});

(3) r = g.c.d.{r_{i}}, where r_{i} = index(Y_{i}).

Proof. 1) Since the normal bundle N_{Y_{i}|X} is trivial, Y_{i} being a factor of X, by adjunction we get K_{Y_{i}} = (K_{X})_{Y_{i}}, hence −K_{Y_{i}} is ample since −K_{X} is.

2) Note that Pic(X) ≅ H^{2}(X, Z). Similarly Pic(Y_{i}) ≅ H^{2}(Y_{i}, Z), by (1). So the assertion follows from the Künneth decomposition.

3) Write −K_{X} = rH, −K_{Y_{i}} = r_{i}H_{i}, with H ∈ Pic(X) and H_{i} ∈ Pic(Y_{i}) ample, with no proper submultiples. We thus get

rH = −K_{X} = − ∑_{i=1}^{s} p_{i}^{∗}K_{Y_{i}} = ∑_{i=1}^{s} r_{i}p_{i}^{∗}H_{i},

which proves the assertion. (1.2.0) Obviously s ≤ N with equality if and only if n_{i} = 1 for all i, which means that X = (P^{1})^{N} by (1.1.(1)). Note that in this case r = 2, i.e., N = c + 1, so that s = c + 1. Now set

σ = \frac{N − ∑_{i=1}^{s} c_{i}}{N − c} ,

where c_{i} = n_{i} + 1 − r_{i} denotes the coindex of Y_{i}. We can prove the following

(1.2) Lemma. Let X be as in (1.0).

(a) If r = 1 then s ≤ c − 1, with equality if and only if X = (P^{1})^{N−2} × S, where S is either P^{2} or any Del Pezzo surface of index 1;

(b) if r ≥ 2, then s ≤ σ, with equality if and only if r = r_{1} = … = r_{s}.

Proof. (a) If r = 1 it cannot be s = N by what we said before; hence s ≤ N − 1 = c − 1 and of course s = c − 1 if and only if (s − 1) factors are P^{1} and the remaining one has dimension 2 and index ≠ 2. So it can only be either P^{2} or a Del Pezzo surface of index 1.

(b) Let r ≥ 2; by (1.1.(3)) we have r_{i} = rh_{i} for some positive integer h_{i}, i = 1, …, s. So

(1.2.1)  n_{i} = r_{i} + c_{i} − 1 = rh_{i} + c_{i} − 1.

By summing up and comparing with (1.0.1) we thus get

(1.2.2)  s = r ∑_{i=1}^{s} h_{i} + ∑_{i=1}^{s} c_{i} − N.

Since ∑_{i=1}^{s} h_{i} ≥ s, (1.2.2) gives

(1.2.3)  s(r − 1) ≤ N − ∑_{i=1}^{s} c_{i},

where the equality holds if and only if h_{i} = 1 for all i, i.e., if and only if r = r_{i} for all i; recalling (1.0.1) this proves the assertion.

As a first consequence of (1.2) we get

(1.3) Corollary. Let X be as in (1.0). Then c ≥ ∑_{i=1}^{s} c_{i}, equality holding if and only if r = r_{1} = … = r_{s} = 1.

Proof. If r > 1 we have that σ ≥ s > 1 by (1.2). Recalling the definition of σ this gives c > ∑_{i=1}^{s} c_{i}. So if

(1.3.1)  c = ∑_{i=1}^{s} c_{i},

then r = 1. But this is equivalent to c = N, so (1.3.1) gives

N = ∑_{i=1}^{s}(n_{i} + 1 − r_{i}) = ∑_{i=1}^{s} n_{i} + s − ∑_{i=1}^{s} r_{i},

which means that s = ∑_{i=1}^{s} r_{i} and so r_{i} = 1 for all i. To see the converse, let r = r_{1} = … = r_{s} = 1; then

c = N = ∑_{i=1}^{s} n_{i} = ∑_{i=1}^{s} c_{i}.

(1.4) Remark. Let r > 1. Then

σ ≤ 1 + \frac{c}{r − 1},

equality occurring if and only if Y_{i} = P^{n_{i}} for all i. In particular σ ≤ c + 1, equality implying r = 2 and Y_{i} = P^{n_{i}} for all i.

Proof. Actually by definition of σ,

σ ≤ \frac{N}{N − c} = 1 + \frac{c}{N − c} = 1 + \frac{c}{r − 1} ≤ 1 + c,

and the equality on the left holds if and only if c_{i} = 0 for all i, while the equality on the right occurs if and only if r = 2.

The material above gives the following

(1.5) Theorem. Let X be as in (1.0). Then s ≤ c + 1 with equality if and only if X = (P^{1})^{N}.

Proof. (1.2) combined with (1.4) gives s ≤ σ ≤ c + 1 and the equalities imply both r = r_{1} = … = r_{s} and Y_{i} = P^{n_{i}} with r = 2, hence r_{i} = 2, i.e., Y_{i} = P^{1} for all i.

The next step is to characterize those X as in (1.0) in case s = c.

(1.6) Theorem. Let X be as in (1.0). Then s = c if and only if X = P^{2} × P^{2}.

Proof. (1.2) combined with (1.4) gives

2 ≤ c = s ≤ σ ≤ 1 + \frac{c}{r − 1} ≤ c + 1;

then either s = σ = c or s = c and σ = s + 1 = c + 1. Let s = σ = c. By (1.2) we have r = r_{1} = … = r_{s}.

Claim. r = r_{1} = … = r_{s} > 2.

Proof. As s = c we get r = r_{1} = … = r_{s} > 1 from (1.2). Suppose that r = 2; then

∑_{i=1}^{s} n_{i} − ∑_{i=1}^{s} c_{i} = ∑_{i=1}^{s} r_{i} − s = 2s − s = s = N − 1,

hence ∑_{i=1}^{s} c_{i} = 1. Up to reordering factors we can thus assume that c_{1} = … = c_{N−2} = 0 and c_{N−1} = 1; therefore n_{1} = … = n_{N−2} = 1 and n_{N−1} = 2, so X = (P^{1})^{N−2} × (P^{1} × P^{1}) = (P^{1})^{N}. But recalling (1.2.0) this leads to a contradiction, since s = N − 1. This proves the claim. From (1.4) we get

c = σ ≤ 1 + \frac{c}{r − 1},

hence

c ≤ (r − 1)/(r − 2).

By the claim this implies that either r = 3 with c = 1, 2 or r > 3 with c = 1. Recalling that s = c ≥ 2 we see that only case c = 2, r = 3 can occur, i.e., X is a Del Pezzo 4-fold. Then Fujita’s classification of Del Pezzo manifolds [2, Ch. I, §8] gives X = P^{2} × P^{2}.

Now let s = c and σ = s + 1 = c + 1. We prove that this case cannot occur. From (1.4) we get r = 2 and Y_{i} = P^{n_{i}} for all i, hence s = c = N − 1. It thus follows that s − 1 factors have dimension 1, hence they are P^{1}, and the last one has dimension 2, but since r = 2, it has to be P^{1} × P^{1}. Thus X = (P^{1})^{N}, which once again leads to a contradiction since s = N − 1.

Case s = c − 1 gives a far richer situation.

(1.7) Theorem. Let X be as in (1.0). Then s = c − 1 if and only if, up to reordering factors, X is one of the following products:

(1) (P^{1})^{N−2} × S, where S is either P^{2} or a Del Pezzo surface of index 1;

(2) (P^{1})^{N−3} × Y, where Y is either P^{3} or a Del Pezzo 3-fold of index 2;

(3) P^{2} × P^{2} × P^{2};

(4) P^{2} × Q^{3};

(5) P^{3} × P^{3}.

Proof. If r = 1 then X is as in (1), by (1.2.(a)). So let r ≥ 2. Keep the same notation as in the proof of (1.2) and recall that

(1.7.1)  ∑_{i=1}^{s} h_{i} ≥ s.

First assume that (1.7.1) is an equality. Then r = r_{1} = … = r_{s} and by (1.2.2) s = σ = c−1. Let r = 2; then N = c + 1, hence ∑_{i=1}^{s} c_{i} = 2. Therefore either c_{1} = … = c_{N−3} = 0, c_{N−2} = 2, or c_{1} = … = c_{N−4} = 0, c_{N−3} = c_{N−2} = 1. In the former case we have n_{1} = … = n_{N−3} = 1, n_{N−2} = 3, hence X is as in (2) with Y a Del Pezzo 3-fold of index 2. In the latter case we have n_{1} = … = n_{N−4} = 1, n_{N−3} = n_{N−2} = 2, hence X = (P^{1})^{N−4} × (P^{1} × P^{1}) × (P^{1} × P^{1}) = (P^{1})^{N}, but this is impossible, since s = N − 2. So this case does not occur. Now let r ≥ 3. Recalling (1.4) we get

c ≤ σ + 1 ≤ 2 + \frac{c}{r − 1}, hence

(1.7.2)  c ≤ \frac{2r − 2}{r − 2}.

For r = 3, (1.7.2) gives c ≤ 4 and σ = s ≤ 3, so that either c = 4, with σ = s = 3, or c = 3, with σ = s = 2. In the former case we get N = 6 and c_{1} + c_{2} + c_{3} = 0, hence c_{i} = 0 and n_{i} = 2 for i = 1, 2, 3. This exactly means that X is as in (3). In the latter case we get N = 5 and c_{1} + c_{2} = 1, hence we can assume c_{1} = 0, c_{2} = 1 and then n_{1} = 2, n_{2} = 3. So X is as in (4). For r = 4 (1.7.2) gives c ≤ 3; therefore c = 3 and σ = s = 2. Then N = 6 and c_{1} + c_{2} = 0; so c_{i} = 0, n_{i} = 3 for i = 1, 2, which leads to case (5). Note that it cannot be r ≥ 5; otherwise we would get c ≤ 2 from (1.7.2), which is impossible since s ≥ 2.

Now assume that (1.7.1) is a strict inequality. Then from (1.2.2) we get the inequality

s ≤ σ − \frac{r}{r − 1} < σ − 1.

Combining this with (1.4) and the assumption we obtain

c − 1 = s ≤ σ − 2 ≤ c + 1 − 2 = c − 1,

which gives σ = c + 1. By (1.4) this is equivalent to r = 2 and Y_{i} = P^{n_{i}} with n_{i} odd, for all i. Furthermore from (1.0.1) we get n_{1} + … + n_{s} + 1 − 2 = c = s + 1, hence

(1.7.3)  n_{1} + … + n_{s} = s + 2.

By what we said there exists an index i such that n_{i} ≥ 3. So, up to reordering the factors of X, we can assume n_{i} ≥ 3 for i = s − t, …, s − 1, s. Then by (1.7.3)

s − t − 1 ≤ n_{1} + … n_{s−(t+1)} ≤ s + 2 − 3(t + 1) = s − 3t − 1,

which shows that t = 0, n_{1} = … = n_{s−1} = 1, n_{s} = 3. So X is as in case (2) with Y = P^{3}. This concludes the proof.

Along the same line we can classify Fano products with s = c − 2. The proof is long but similar to the one of (1.7), and we omit it. The result is the following

(1.8) Theorem. Let X be as in (1.0). Then s = c − 2 if and only if, up to reordering factors, X is one of the following products:

(1) (P^{1})^{N−3} × Y, with Y either a quadric or a Fano 3-fold of index 1;

(2) (P^{1})^{N−4} × Y × Y′, where Y is either P^{2} or a Del Pezzo surface of index 1 and Y′ is a Del Pezzo surface of index 1; (3) (P^{1})^{N−4} × Y, where Y is either a quadric or a Fano 4-fold of index 2;

(4) P^{2} × Y, with Y a Del Pezzo 4-fold;

(5) Q^{3} × Q^{3};

(6) P^{2} × P^{2} × Q^{3};

(7) (P^{2})^{4};

(8) P^{3} × Q^{4};

(9) P^{4} × P^{4}.

For 4-folds occurring in case (3), which fit into the class of Mukai manifolds, see [4].

2. Special Fano P-bundles

Let P = P_{B}(E) be the projective bundle associated to a vector bundle E of rank r on a projective manifold B. Let π : P → B be the bundle projection and let ξ denote the tautological bundle O_{P(E)}(1). By the canonical bundle formula we have

(2.0.1)  K_{P} = −rξ + π^{∗}(K_{B} + det E).

If P is Fano we call it a Fano P-bundle. It follows from (2.0.1) that a Fano P-bundle has index ≤ r. In particular if

(2.0.2)  E is ample and K_{B} + det E = O_{B},

we immediately get from (2.0.1) that K_{P} = −rξ, where ξ is an ample line bundle with no proper submultiples in Pic(P). So P is a Fano manifold of index r.

We will call P a special Fano P-bundle if it corresponds to a vector bundle E satisfying (2.0.2).

Of course not every Fano P-bundle is special. E.g., for any integer t ≥ 1 let B = P^{3t−1} and E = O(1)^{2t}. Then for P = P_{B}(E) we have K_{P} = −t(2ξ + π^{∗}O(1)). As 2ξ + π^{∗}O(1) is ample with no proper submultiples in Pic(P) we thus see that P is a Fano P-bundle of index t < rk E, hence not special. For other examples in low dimensions, see [11], [12]. In fact we have

(2.1) Proposition. Let P = P_{B}(E) be a Fano P-bundle; P is special if and only if it has maximal index r.

Proof. We prove the “only if” part, the converse having been already shown. Let −K_{P} = rh, with h ∈ Pic(P) ample. By (2.0.1) we thus see that K_{B} + c_{1}(E) ∈ r Pic(B). Writing K_{B} + c_{1}(E) = rL for a suitable line bundle L on B, (2.0.1) gives −K_{P} = r(ξ − π^{∗}L): so h = ξ − π^{∗}L is the tautological bundle of E′ := E ⊗ L^{−1}, which is ample, since h is. Therefore E′ satisfies (2.0.2). On the other hand P = P_{B}(E) = P_{B}(E′).

(2.2) Recall that for an ample vector bundle E of rank r on a projective n-fold B the adjoint line bundle K_{B} + det E is always ample if r ≥ n + 1, unless (B, E) = (P^{n}, O(1)^{(n+1)}) [3]. This suggest to define the corank of P as

(2.2.1)  cork P = dim B + 1 − rk E

and the above result can be restated by saying that cork P ≥ 0, with equality if and only if

(2.2.2)  (B, E) = (P^{n}, O(1)^{(n+1)}).

Rephrasing known results of several authors [3], [8] and [10] in terms of corank we get the following lists of special Fano P-bundles with corank 1 and 2.

(2.3) Theorem. Let P = P_{B}(E) be a special Fano P-bundle. Then cork P = 1 if and only if (B, E) is one of the following pairs:

(1) (P^{n}, O(2) ⊕ O(1)^{(n−1)}),

(2) (P^{n}, T_{P^{n}}), where T_{P^{n}} is the tangent bundle,

(3) (Q^{n}, O(1)^{n}).

(2.4) Theorem. Let P = P_{B}(E) be a special Fano P-bundle. Then cork P = 2 if and only if (B, E) is one of the following pairs:

(1) (P^{3}, N(2)), where N is the null correlation bundle,

(2) (P^{2} × P^{1}, p^{∗}T_{P^{2}} ⊗ O(0, 1)), where p stands for the first projection,

(3) (P^{2} × P^{1}, O(2, 1) ⊕ O(1, 1)),

(4) (Q^{3}, S(2)), where S is the spinor bundle,

(5) (Q^{4}, S(2) ⊕ O(1)), where S is one of the two spinor bundles,

(6) (Y, H^{(n−1)}), where Y is any Del Pezzo manifold and −K_{Y} = (n − 1)H,

(7) (P^{n}, O(2)^{2} ⊕ O(1)^{(n−3)}),

(8) (P^{n}, O(3) ⊕ O(1)^{(n−2)}),

(9) (Q^{n}, O(2) ⊕ O(1)^{(n−2)}), or, possibly, P has two P^{2}-bundle structures on Fano 4-folds with Picard number 1, index 1 and pseudoindex ≥ 3.

As in section 1, Del Pezzo manifolds are intended in the sense of Fujita [2]. Note that case (6) includes the 6 types of Del Pezzo manifolds with Pic(B) ≅ Z and the following 4 pairs

(P^{2} × P^{2}, O(1, 1)), (P(T_{P^{2}}), ξ), ((P^{1})^{3}, O(1, 1, 1)), (B_{p}(P^{3}), σ^{∗}O(2) − E),

where σ : B_{p}(P^{3}) → P^{3} is the blowing-up at the point p and E is the exceptional divisor [2, p. 72]. For the definitions and the main properties of the bundles N and S appearing in (2.4) we refer to [5, p. 76] and to [6].

In the range cork P ≥ 3 no classification result is known. Let us produce some examples.

(2.5) Examples. Obvious special Fano P-bundles with cork P = 3 are given by the following pairs:

(1) B = P^{n}, with E = O(4) ⊕ O(1)^{n−3}, O(3) ⊕ O(2) ⊕ O(1)^{n−4}, or O(2)^{3} ⊕ O(1)^{n−5};

(2) B = Q^{n}, with E = O(3) ⊕ O(1)^{n−3}, or O(2)^{2} ⊕ O(1)^{n−4}.

(3) A less obvious example is given by the pair (Q^{6}, S(2)), where S is one of the two spinor bundles on Q^{6}. Actually rk S = 4 and S(2) = S(1) ⊗ O(1) is ample, since S(1) is spanned, as a quotient of O^{8} [6, (2.8), (ii)]. Moreover c_{1}(S(2)) = c_{1}(S) + 8 = 6. Therefore E := S(2) satisfies (2.0.2).

In fact it is easy to show that if (Q^{n}, E) gives rise to a special Fano P-bundle, E being or having as a summand a twisted spinor bundle, then only cases (2.4.(4)), (2.4.(5)) or (2.5.(3)) can occur.

(4) Another example related to spinor bundles is the following. Let S^{∗} be the dual of the spinor bundle S on Q^{5} and consider the exact sequence [6, (3.1)]

0 → O → S^{∗} → F := S^{∗}/O → 0.

F has rank 3 and is spanned, as S^{∗} is; actually S^{∗} ≃ S(1), which is a quotient of O^{8} [6, (2.8), (i)]. Moreover c_{1}(F) = c_{1}(S^{∗}) = −c_{1}(S) = 2 [6, (2.9)]. Therefore E := F (1) satisfies (2.0.2) and the corresponding P has corank 3.

(2.6) Let B be any Fano manifold of index r and coindex c and let E = H^{r}, where H ∈ Pic(B) is the ample line bundle such that −K_{B} = rH. Then obviously P = P_{B}(E) is a special Fano P-bundle with cork P = c. In particular, taking as B any Mukai manifold [4], we get in this way a lot of special Fano P-bundles of corank 3.

To conclude this section let us produce a non-obvious example of special Fano Pbundle of corank 4.

(2.7) Example. Let F be as in (2.5.(4)). By [7, (2.2)] there is an exact sequence

0 → O → F^{∗}(1) → C(1) → 0

defining a rank-2 bundle C on Q^{5} called the Cayley bundle. From this sequence we get c_{1}(C(1)) = 1. Moreover C(2) is spanned [7, (3.7)]. It thus follows that the rank-2 bundle E := C(3) satisfies (2.0.2), hence the corresponding P is a special Fano P-bundle of corank 4.

3. Special Fano P-bundles of corank 3 on products

Let P = P_{B}(E) be a special Fano P-bundle over a product manifold B = B_{1} × B_{2}. Let r = rk E, b = dim B, b_{i} = dim B_{i} and let p_{i} be the projection of B onto the i-th factor.

We say that P is standard to mean that

(3.0.1)  E = (p_{1}^{∗}E_{1} ⊗ p_{2}^{∗}L_{2}) ⊕ (p_{2}^{∗}E_{2} ⊗ p_{1}^{∗}L_{1}),

where E_{i} and L_{i} are ample vector bundles of rank r_{i} and ample line bundles on B_{i} respectively, i = 1, 2, one of the two summands being possibly zero. The aim of this section is to determine all standard special Fano P-bundles satisfying the condition

(3.0.2)  cork P = 3.

In the simpler case cork P = 2 the same analysis we are going to do leads to the pairs (2.4.(2)), (2.4.(3)) and the two products fitting in case (2.4.(6)). Our result is the following

(3.1) Theorem. Up to exchanging the factors of B, the pairs (B, E) corresponding to the standard special Fano P-bundles satisfying (3.0.2) are the following:

(1) (P^{3} × P^{3}, O(1, 1)^{4}),

(2) (P^{1} × P^{3}, O(1, 2)^{2}),
(3) (P^{2} × Q^{3}, O(1, 1)^{3}),
(4) (P^{3} × P^{2}, O(2, 1) ⊕ O(1, 1)^{2}),
(5) (P^{3} × P^{2}, p_{1}^{∗}T_{P^{3}} ⊗ O(0, 1)),
(6) (P^{2} × P^{1} × P^{1}, O(2, 1, 1) ⊕ O(1, 1, 1)),
(7) (P^{2} × P^{1} × P^{1}, p_{1}^{∗}T_{P^{2}} ⊗ O(0, 1, 1)),
(8) (P^{3} × P^{1}, p_{1}^{∗}N(2) ⊗ O(0, 1)),
(9) (Q^{3} × P^{1}, p_{1}^{∗}S(2) ⊗ O(0, 1)),
(10) (Y × P^{1}, p_{1}^{∗}H^{2} ⊗ O(0, 1)), (Y, H) being any Del Pezzo threefold,
(11) (P^{3} × P^{1}, O(3, 1) ⊕ O(1, 1)),
(12) (Q^{3} × P^{1}, O(2, 1) ⊕ O(1, 1)),
(13) (P^{2} × P^{2}, O(1, 2) ⊕ O(2, 1)),
(14) (P^{2} × P^{2}, O(1, 1) ⊕ O(2, 2)).

Note that (10) includes just one case in which B is the product of three or more factors: it corresponds to the pair

(3.1.0)  ((P^{1})^{4}, O(1, 1, 1, 1)^{2}).

The proof of Theorem (3.1) takes the remainder of this section. First note that B is Fano, by (2.0.2), hence

(3.1.1)  Pic(B) = p_{1}^{∗} Pic(B_{1}) × p_{2}^{∗} Pic(B_{2}),

by (1.1.(2)). Moreover K_{B} = p_{1}^{∗}K_{B_{1}} + p_{2}^{∗}K_{B_{2}}.

(3.2) First assume that the second summand in (3.0.1) is zero and set L = L_{2}. Then, in view of (3.1.1), the fact that P is special gives

(3.2.1)  K_{B_{1}} + det E_{1} = O_{B_{1}},

i.e. P_{B_{1}}(E_{1}) itself is special, and

(3.2.2)  K_{B_{2}} + rL = O_{B_{2}}.

Condition (3.0.2) means that r = b − 2 = b_{1} + (b_{2} − 2). This combined with (3.2.1) and the results of section 2 shows that b_{2} ≤ 3 and

if b_{2} = 3, then r = b_{1} + 1 and (B_{1}, E_{1}) is as in (2.2.2),

if b_{2} = 2, then r = b_{1} and (B_{1}, E_{1}) is as in (2.3),

if b_{2} = 1, then r = b_{1} − 1 and (B_{1}, E_{1}) is as in (2.4).

Note however that not all pairs in (2.4) can occur; actually we see from (3.2.2) that r divides the index of B_{2}. Since r ≥ 2 we thus conclude that

if b_{2} = 1, then r = 2 and (B_{2}, L) = (P^{1}, O(1));

if b_{2} = 2, then r = 3 or 2 and (B_{2}, L) = (P^{2}, O(1)) or (P^{1} × P^{1}, O(1, 1)) accordingly, if b_{2} = 3, then either

B_{2} = P^{3} with L = O(1), O(2) and r = 4, 2 respectively, (B_{2}, L) = (Q^{3}, O(1)) with r = 3, or (B_{2}, L) is a Del Pezzo threefold.

By combining the possibilities for (B_{1}, E_{1}) with those for (B_{2}, L) and taking into account the admissible values for r, we thus get the pairs (1)–(12) in (3.1).

(3.3) Now assume that the summands in (3.0.1) are nontrivial. Then, recalling (3.1.1), the fact that P is special gives rise to the following two symmetric conditions:

(3.3.1)  K_{B_{i}} + det E_{i} + r_{j}L_{i} = O_{B_{i}}, (i, j) = (1, 2) or (2, 1).

In particular this means that K_{B_{i}} + det E_{i} is not nef. By [14, Theorem 1] this implies that r_{i} ≤ b_{i} + 1, i = 1, 2, which, combined with the condition r_{1} + r_{2} = b_{1} + b_{2} − 2 coming from (3.0.2), leads, up to exchanging the factors, to the following possibilities:

A) r_{1} = b_{1} and r_{2} = b_{2} − 2,

B) r_{i} = b_{i} − 1 for i = 1, 2.

Moreover, by [14, Theorem 2], we see that in case A)

(3.3.2)  (B_{1}, E_{1}) = (P^{b_{1}}, O(1)^{b_{1}}).

Let us continue the study of

Case A). From (3.3.1) with i = 1 and (3.3.2) it follows that r_{2} = 1 and L_{1} = O_{P}(1). Hence b_{2} = 3. Now (3.3.1) with i = 2 becomes

K_{B_{2}} + det E_{2} + b_{1}L_{2} = O_{B_{2}}.

This says that E_{2} ⊕L_{2}^{b_{1}} is an ample vector bundle of rank (1+b_{1}) on B_{2} with trivial adjoint bundle. So recalling what we said in section 2, we get 1 + b_{1} ≤ b_{2} + 1 = 4, i.e. b_{1} ≤ 3. Moreover recalling (2.2.2), (2.3), (2.4) and noting that E_{2} ⊕ L_{2}^{b_{1}} is split, we get:

(3.3.3) if b_{1} = 3 then B_{2} = P^{3}, E_{2} = L_{2} = O_{P^{3}}(1);

(3.3.4) if b_{1} = 2, then either (B_{2}, E_{2}, L_{2}) = (P^{3}, O(2), O(1)), or B_{2} = Q^{3} and E_{2} = L_{2} = O(1);

(3.3.5) if b_{1} = 1, then we have 5 possibilities: either B_{2} is a Del Pezzo threefold with −K_{B_{2}} = 2H and E_{2} = L_{2} = H, or B_{2} = P^{3} and E_{2} = L_{2} = O(2), or, up to exchanging E_{2} and L_{2}, the triple (B_{2}, E_{2}, L_{2}) is one of the following:

(P^{1} × P^{2}, O(1, 2), O(1, 1)), (P^{3}, O(3), O(1)), (Q^{3}, O(2), O(1)).

Taking into account these possibilities we thus get for (B, E) the pairs (1)–(4), (6), (10)–(12) in (3.1), all of which already found. Now come to

Case B). It follows from [14, Theorem 3] that both pairs (B_{i}, E_{i}) belong to the following list:

1) (P^{(n−1)}-bundle over a smooth curve, E_{i} restricting to any fibre as O(1)^{(n−1)}),

2) (P^{n}, O(1)^{(n−1)}),

3) (P^{n}, O(2) ⊕ O(1)^{(n−2)}),

4) (Q^{n}, O(1)^{(n−1)}).

Let us first discuss case 1).

(3.4) Lemma. Let cork P = 3 and assume that (B_{1}, E_{1}) is as in case 1). Then B_{1} = P^{1} × P^{1}, E_{1} = L_{1} = O(1, 1) and b_{2} = 2.

Proof. Set m = b_{1} − 1. Since B_{1} is Fano, its base curve has to be P^{1} and so B_{1} = P_{P^{1}}(F), where F is a rank-(m + 1) vector bundle on P^{1}, which we can suppose to be normalized in the form

F = O_{P^{1}} ⊕ O_{P^{1}}(a_{1}) ⊕ … ⊕ O_{P^{1}}(a_{m}), with 0 ≤ a_{1} ≤ … ≤ a_{m}.

Let δ = deg F = a_{1} + … + a_{m}; thus δ ≥ 0. As

(3.4.1)  −K_{B_{1}} = (m + 1)ξ + (2 − δ)F,

where ξ and F stand for the tautological bundle of F and a fibre respectively, the ampleness condition for −K_{B_{1}} gives 2 − δ > 0 [1, p. 74, Lemma 3.2.4]. From 1) we have that (det E_{1})_{F} = O_{P^{m}}(m); so since ξ and F generate Pic(B_{1}) we have

(3.4.2)  det E_{1} = mξ + nF,

with n > 0 since det E_{1} is ample. Similarly for the ample line bundle L_{1} we have

(3.4.3)  L_{1} = αξ + βF,

with α > 0 and β > 0. Now recall that cork P = 3. Then putting together (3.4.1), (3.4.2), (3.4.3), condition (3.3.1) with i = 1 gives

−(m + 1)ξ + (δ − 2)F + mξ + nF + r_{2}(αξ + βF) = 0.

This shows that α = r_{2} = 1 and δ −2+n+β = 0. Therefore δ = 0, n = β = 1. This proves that B_{1} = P^{1} × P^{m}, L_{1} = O(1, 1) and b_{2} = r_{1} + r_{2} + 2 − b_{1} = m + 1 + 2 − (m + 1) = 2. Now note that E_{1} ⊗ ξ^{−1} restricts trivially to every fibre F, by 1). Hence E_{1} ⊗ ξ^{−1} = ρ^{∗}G, where ρ : B_{1} → P^{1} is the projection and G is a rank-m vector bundle on P^{1}. Since

G = O_{P^{1}}(d_{1}) ⊕ … ⊕ O_{P^{1}}(d_{m}),

we thus get

(3.4.4)  E_{1} = ρ^{∗}G ⊗ ξ = O(d_{1}, 1) ⊕ … ⊕ O(d_{m}, 1)

and the ampleness of E_{1} implies that each summand in (3.4.4) is ample. So d_{i} ≥ 1 for all i. On the other hand from the expression of det E_{1} we immediately see that deg G = 1. Hence from

m ≤ d_{1} + … + d_{m} = deg G = 1

we conclude that m = 1. Thus E_{1} = det E_{1} = ξ + F = O(1, 1). This concludes the proof.

In other words (3.4) says that case 1) fits into case 4) for n = 2. Now come back to (3.3.1) and note that E_{i} ⊕ L_{i}^{r_{j}} is an ample vector bundle of rank r = r_{1} + r_{2} on B_{i} with trivial adjoint bundle. Therefore, by section 2 we get b_{1} + b_{2} − 2 = r ≤ b_{i} + 1, which gives b_{i} ≤ 3 for i = 1, 2. On the other hand from r_{i} ≥ 1 we also get b_{i} ≥ 2, so that b_{i} = 2 or 3 for i = 1, 2. Looking again at (3.3.1) and taking into account (3.4) we can now further refine the list of possibilities for (B_{i}, E_{i}), L_{i} and b_{j} = r_{j} + 1; they are the following:

(1) (P^{1} × P^{1}, O(1, 1)), L_{i} = O(1, 1), b_{j} = 2,

(2) (P^{2}, O(1)), L_{i} = O(1), b_{j} = 3,

(3) (P^{2}, O(1)), L_{i} = O(2), b_{j} = 2,

(4) (P^{3}, O(1)^{2}), L_{i} = O(1), b_{j} = 3,

(5) (P^{3}, O(1)^{2}), L_{i} = O(2), b_{j} = 2,

(6) (P^{2}, O(2)), L_{i} = O(1), b_{j} = 2,

(7) (P^{3}, O(2) ⊕ O(1)), L_{i} = O(1), b_{j} = 2,

(8) (Q^{3}, O(1)^{2}), L_{i} = O(1), b_{j} = 2.

Let (h,k) denote the pair (B, E) obtained by combining a factor of type (h) with a factor of type (k) in the above list. Then it is immediate to check that up to exchanging h and k, only the following cases can occur: (1,1), (1,3), (1,6), (2,5), (2,7), (2,8), (3,3), (3,6), (4,4), (6,6). Case (1,1) corresponds to the pair (3.1.0), while the remaining ones give rise to the pairs (6), (6), (4), (4), (3), (13), (14), (1), (13) in (3.1), two of which not already found.

4. On the index of the base of a special Fano P-bundle

In this section we assume that (4.0) P = P(E) is a special Fano P-bundle over a projective manifold B and r = rk E.

Then by (2.0.2) B is Fano. This is more generally true if P is simply any Fano Pbundle over B [11, (1.6)]. Let i be the index of B; then −K_{B} = iH, with H ∈ Pic(B) ample. A rational curve l ⊂ B is said to be a line of (B, H) if lH = 1. The following fact is well known.

(4.1) Proposition. Let P be as in (4.0). If (B, H) contains a line, then

(4.1.1)  i ≥ r.

Proof. Let l be a line of (B, H). Since l is a rational curve we have c_{1}(E_{l}) ≥ r; hence from (2.0.2) we get

r ≤ c_{1}(E_{l}) = −K_{B}l = iHl = i.

If B has a large index then certainly (B, H) contains lines. In fact

(4.2) Proposition. Let B be a Fano manifold of dimension b and index i > (b + 1)/2. Then (B, H) contains a line.

Proof. As B is Fano, by Mori’s cone theorem, it contains a rational extremal curve, say C. Since −K_{B}C ≤ b+1 we get b+1 ≥ −K_{B}C = iHC, i.e., HC ≤ (b+1)/i. So if i > (b+1)/2 we conclude that HC = 1.

In fact the argument shows that all extremal rational curves of B are lines of (B, H). By combining (4.1) with (4.2) we get the following

(4.3) Corollary. Let P be as in (4.0). If i > (b + 1)/2, then i ≥ r.

Special Fano P-bundles with i < r are known, e.g., see (2.4.(2)), (2.4.(3)). However it is conjectured ([10, p. 152], see also [9, (2.12), (3)]) that

(4.3.1)  if Pic(B) ≅ Z, then i ≥ r.

Corollary (4.3) provides a support for this conjecture since B having large index implies that Pic(B) ≅ Z, apart from very special cases [13]. What can be said about the equality i = r? Of course for any Fano manifold B of index i the vector bundle E = H^{i} gives a special Fano P-bundle satisfying i = r and the known classification results outlined in section 2 show that this obvious way is the only one to produce special Fano P-bundles with i = r when cork P ≤ 2. However this is no longer true if cork P ≥ 3, as the following example suggested by (3.1.(11)) shows.

(4.4) Example. Let B = P^{2n−1} × P^{1}, n ≥ 3 odd. Then i = 2 with H = O(n, 1). Let E = O(p, 1) ⊕ O(q, 1), with p, q positive, p ≠ q, p + q = 2n. Then E satisfies (2.0.2), but E ≠ H^{2}. This also shows the irrelevance of the existence of lines on (B, H) for the obvious way being the only one to produce special Fano P-bundles with i = r. Similarly we have

(4.5) Example. Let B = P^{5} × P^{3}; then i = 2 with H = O(3, 2). Thus E = O(4, 3) ⊕ O(2, 1) satisfies (2.0.2) but E ≠ H^{2}. Note that the pair (B, H) does not contain lines. So this shows in particular that the converse to (4.1) does not hold. Note that B has small index in all our examples. We would like to conclude this paper by appending the following question to conjecture (4.3.1).

(4.6) Question. Let things be as above and assume that i = r. Is it true that E = H^{r}

(1) if Pic(B) ≅ Z ?

(2) (weaker form) if i > (b + 1)/2 ?

References

[1] Beltrametti, M. C.; Sommese, A. J.: The Adjunction Theory of Complex Projective Varieties. Expositions in Mathematics, vol. 16, De Gruyter, Berlin 1995.

[2] Fujita, T.: Classification Theories of Polarized Varieties. London Math. Soc. Lect. Note Ser. 155, Cambridge Univ. Press, Cambridge 1990.

[3] Fujita, T.: On adjoint bundles of ample vector bundles. Complex Algebraic Varieties, Proc. Bayreuth 1990, Lect. Notes in Math., vol. 1507, Springer-Verlag Berlin-Heidelberg-New York 1992, 105–112.

[4] Mukai, S.: Birational classification of Fano 3-folds and Fano manifolds of coindex 3. Proc. Natl. Acad. Sci. U.S.A. 86 (1989), 3000–3002.

[5] Okonek, Ch.; Schneider, M.; Spindler, H.: Vector Bundles on Complex Projective Spaces. Progress in Math., Birkhäuser Boston-Basel-Stuttgart 1980.

[6] Ottaviani, G.: Spinor bundles on quadrics. Trans. Amer. Math. Soc. 307 (1988), 301–316.

[7] Ottaviani, G.: On Cayley bundles on the five dimensional quadric. Boll. Un. Mat. Ital., (7) 4-A (1990), 87–100.

[8] Peternell, Th.: Ample vector bundles on Fano manifolds. Intern. J. Math. 2 (1990), 311–322.

[9] Peternell, Th.: Manifolds of semipositive curvature. Notes of CIME Session “Transcendental Methods in Algebraic Geometry”, Cetraro 1994.

[10] Peternell, Th.; Szurek, M.; Wisniewski, J. A.: Fano manifolds and vector bundles. Math. Ann. 294 (1992), 151–165.

[11] Szurek, M.; Wisniewski, J. A.: Fano bundles of rank 2 on surfaces. Compositio Math. 76 (1990), 295–305.

[12] Szurek, M.; Wisniewski, J. A.: Fano bundles over P^{3} and Q^{3}. Pac. J. Math. 141 (1990), 197–208.

[13] Wisniewski, J. A.: On a conjecture of Mukai. Manuscripta Math. 68 (1990), 135–141.

[14] Ye, Y.; Zhang, Q.: On ample vector bundles whose adjunction bundles are not numerically effective. Duke Math. J. 60 (1990), 671–687.

Received October 18, 1994