Some Boundedness Results for

Fano-Like Moishezon Manifolds

Laurent Bonavero and Shigeharu Takayama

Received: February 2, 2000

Communicated by Thomas Peternell

Abstract. We prove finiteness of the number of smooth blow-downs on Fano manifolds and boundedness results for the geometry of non projective Fano-like manifolds. Our proofs use properness of Hilbert schemes and Mori theory.

2000 Mathematics Subject Classification: 14J45, 14E30, 14E05, 32J18. Keywords and Phrases: Fano manifolds, non projective manifold, smooth blow-down, Mori theory, Hilbert scheme.

Introduction

In this Note, we say that a compact complex manifold X is a Fano-like manifold if it becomes Fano after a finite sequence of blow-ups along smooth connected centers, i.e if there exist a Fano manifold X̃ and a finite sequence of blowups along smooth connected centers π : X̃ → X. We say that a Fano-like manifold X is simple if there exists a smooth submanifold Y of X (Y may not be connected) such that the blow-up of X along Y is Fano. If Z is a projective manifold, we call smooth blow-down of Z (with an s-dimensional center) a map π and a manifold Z′ such that π : Z → Z′ is the blow-up of Z′ along a smooth connected submanifold (of dimension s). We say that a smooth blow-down of Z is projective (resp. non projective) if Z′ is projective (resp. non projective).

It is well-known that any Moishezon manifold becomes projective after a finite sequence of blow-ups along smooth centers. Our aim is to bound the geometry of Moishezon manifolds becoming Fano after one blow-up along a smooth center, i.e the geometry of simple non projective Fano-like manifolds.

Our results in this direction are the following, the simple proof of Theorem 1 has been communicated to us by Daniel Huybrechts.

Theorem 1. Let Z be a Fano manifold of dimension n. Then, there is only a finite number of smooth blow-downs of Z.

Let us recall here that the assumption Z Fano is essential : there are projective smooth surfaces with infinitely many −1 rational curves, hence with infinitely many smooth blow-downs.

Since there is only a finite number of deformation types of Fano manifolds of dimension n (see [KMM92] and also [Deb97] for a recent survey on Fano manifolds) and since smooth blow-downs are stable under deformations [Kod63], we get the following corollary (see section 1 for a detailed proof) :

Corollary 1. There is only a finite number of deformation types of simple Fano-like manifolds of dimension n.

The next result is essentially due to Wiśniewski ([Wis91], prop. (3.4) and (3.5)). Before stating it, let us define

d_{n} = max{(−K_{Z})^{n} | Z is a Fano manifold of dimension n}

and

ρ_{n} = max{ρ(Z) := rk(Pic(Z)/ Pic^{0}(Z)) | Z is a Fano manifold of dimension n}.

The number ρ_{n} is well defined since there is only a finite number of deformation types of Fano manifolds of dimension n and we refer to [Deb97] for an explicit bound for d_{n}.

Theorem 2. Let X be an n-dimensional simple non projective Fano-like manifold, Y a smooth submanifold such that the blow-up π : X̃ → X of X along Y is Fano, and E the exceptional divisor of π. Then

(i) if each component of Y has Picard number equal to one, then each component of Y has ample conormal bundle in X and is Fano. Moreover

deg_{−KX̃} (E) := (−K_{X̃|E′})^{n−1} ≤ (ρ_{n} − 1)d_{n−1};

(ii) if Y is a curve, then (each component of) Y is a smooth rational curve with normal bundle O_{ℙ^{1}}(−1)^{⊕n−1}.

Finally, we prove here the following result :

Theorem 3. Let Z be a Fano manifold of dimension n and index r. Suppose there is a non projective smooth blow-down of Z with an s-dimensional center. Then

r ≤ (n − 1)/2 and s ≥ r.

Moreover, (i) if r > (n − 1)/3, then s = n − 1 − r ; (ii) if r < (n − 1)/2 and s = r, then Y ≃ ℙ^{r}.

Recall that the index of a Fano manifold Z is the largest integer m such that −K_{Z} = mL for L in the Picard group of Z.

Remarks.

Some Boundedness Results … 143

a) For a Fano manifold X of dimension n and index r with second Betti number greater than or equal to 2, it is known that 2r ≤ n + 2 [Wi91], with equality if and only if X ≃ ℙ^{r−1} × ℙ^{r−1}. b) Fano manifolds of even dimension (resp. odd dimension n) and middle

index (resp. index (n + 1)/2) with b_{2} ≥ 2 have been intensively studied, see for example [Wis93]. Our Theorem 3 shows that there are no non projective smooth blow-down of such a Fano manifold, without using any explicit classification.

c) The assumption that there is a non projective smooth blow-down of Z is essential in Theorem 3 (i) : the Fano manifold obtained by blowing-up ℙ^{2r−1} along a ℙ^{r−2} has index r.

1. Proof of Theorem 1 and Corollary 1. An example.

1.1. Proof of Theorem 1. Thanks to D. Huybrechts for the following proof. Let Z be a Fano manifold and π : Z → Z′ a smooth blow-down of Z with an sdimensional connected center. Let f be a line contained in a non trivial fiber of π. Then, the Hilbert polynomial P_{−K_{Z}} (m) := χ(f, m(−K_{Z})_{|f}) is determined by s and n since −K_{Z}·f = n−s−1 and f is a smooth rational curve. Since −K_{Z} is ample, the Hilbert scheme Hilb_{−K_{Z}} of curves in Z having P_{−K_{Z}} as Hilbert polynomial is a projective scheme, hence has a finite number of irreducible components. Since each curve being in the component H of Hilb_{−K_{Z}} containing f is contracted by π, there is only a finite number of smooth blow-downs of Z with an s-dimensional center.

1.2. Proof of Corollary 1. Let us first recall ([Deb97] section 5.2) that there exists an integer δ(n) such that every Fano n-fold can be realized as a smooth submanifold of ℙ^{2n+1} of degree at most δ(n). Let us denote by T a closed irreducible subvariety of the disjoint union of Chow varieties of ndimensional subvarieties of ℙ^{2n+1} of degree at most δ(n), and by π : X_{T} → T the universal family. Step 1 : Stability of smooth blow-downs. Fix t_{0} in the smooth locus T_{smooth} of T and suppose that X_{t_{0}} := π^{−1}(t_{0}) is a Fano n-fold and there exists a smooth blow-down of X_{t_{0}} (denote by E_{t_{0}} the exceptional divisor, P its Hilbert polynomial with respect to O_{ℙ^{2n+1}}(1)). Let S be the component of the Hilbert scheme of (n − 1)-dimensional subschemes of ℙ^{2n+1} with Hilbert polynomial P and u : E_{S} → S the universal family. Finally, let I be the following subscheme of T × S :

I = {(t, s) | u^{−1}(s) ⊂ X_{t}}

and p : I → T the proper algebraic map induced by the first projection. Thanks to the analytic stability of smooth blow-downs due to Kodaira (see [Kod63], Theorem 5), the image p(I) contains an analytic open neighbourhood of t_{0} hence it also contains a Zariski neighbourhood of t_{0}. Moreover, since exceptional divisors are rigid, the fiber p^{−1}(t) is a single point for t in a Zariski neighbourhood of t_{0}. Finally, we get algebraic stability of smooth blow-downs (the ℙ^{r}-fibered structure of exceptional divisor is also analytically stable - [Kod63], Theorem 4 - hence algebraically stable by the same kind of argument).

Step 2 : Stratification of T by the number of smooth blow-downs. For any integer k ≥ 0, let us define

U_{k}(T) = {t ∈ T_{smooth} | X_{t} is a Fano manifold and there exists at least

k smooth blow-downs of X_{t}};

and U_{−1}(T) = T_{smooth}. Thanks to Step 1, U_{k}(T) is Zariski open in T, and thanks to Theorem 1,

⋂_{k≥−1} U_{k}(T) = ∅.

Since {U_{k}(T)}_{k≥−1} is a decreasing sequence of Zariski open sets, by noetherian induction, we get that there exists an integer k such that U_{k}(T) = ∅ and we can thus define

k(T) := max{k ≥ −1 | U_{k}(T) ≠ ∅}, U(T) := U_{k(T)}(T).

Finally, we have proved that U(T) is a non empty Zariski open set of T_{smooth} such that for every t ∈ U(T), Z_{t} is a Fano n-fold with exactly k(T) smooth blow-downs (k(T) = −1 means that for every t ∈ T_{smooth}, X_{t} is not a Fano manifold). Now let T_{0} = T, and T_{1} be any closed irreducible component of T_{0} \ U(T_{0}). We get U(T_{1}) as before and denote by T_{2} any closed irreducible component of T_{1} \ U(T_{1}), and so-on. Again by noetherian induction, this process terminates after finitely many steps and we get a finite stratification of T such that each strata corresponds to an algebraic family of Fano n-folds with the same number of smooth blow-downs.

Step 3 : Conclusion. Since there is only a finite number of irreducible components in the Chow variety of Fano n-folds, each being finitely stratified by Step 2, we get a finite number of deformation types of simple Fano-like n-folds.

As it has been noticed by Kodaira, it is essential to consider only smooth blowdowns. A −2 rational smooth curve on a surface is, in general, not stable under deformations of the surface.

1.3. An example. Before going further, let us recall the following well known example. Let Z be the projective 3-fold obtained by blowing-up ℙ^{3} along a smooth curve of type (3, 3) contained in a smooth quadric Q of ℙ^{3}. Let π denotes the blow-up Z → ℙ^{3}. Then Z is a Fano manifold of index one and there are at least three smooth blow-downs of Z : π, which is projective, and two non projective smooth blow-downs consisting in contracting the strict transform Q′ of the quadric Q along one of its two rulings (the normal bundle of Q′ in Z is O(−1, −1)).

Lemma 1. There are exactly three smooth blow-downs of Z.

Proof : the Mori cone NE(Z) is a 2-dimensional closed cone, one of its two extremal rays being generated by the class of a line f_{π} contained in a non trivial fiber of π, the other one, denoted by [R], by the class of one of the two rulings of Q′ (the two rulings are numerically equivalent, the corresponding extremal contraction consists in contracting Q′ to a singular point in a projective variety, hence is not a smooth blow-down). If E is the exceptional divisor of π, we have

E · [f_{π}] = −1, E · [R] = 3, Q′ · [f_{π}] = 1, Q′ · [R] = −1.

Now suppose there exists a smooth blow-down τ of Z with a 1-dimensional center, which is not one of the three previously described. Let L be a line contained in a non trivial fiber of τ, then since −K_{Z} · [L] = 1, we have [L] = a[f_{π}] + b[R] for some strictly positive numbers such that a + b = 1. Since we have moreover

Q′ · [L] = a − b = 2a − 1 ∈ ℤ and E · [L] = 3b − a = 3 − 4a,

we get a = b = 1/2. Therefore Q′ · [L] = 0 hence L is disjoint from Q′ (it can not be contained in Q′ since Q′_{|Q′} = O(−1, −1)). It implies that there are two smooth blow-downs of Z with disjoint exceptional divisors, which is impossible since ρ(Z) = 2. Finally, if there is a smooth blow-down τ : Z → Z′ of Z with a 0-dimensional center, then Z′ is projective and τ is a Mori extremal contraction, which is again impossible since we already met the two Mori extremal contractions on Z.

2. Non projective smooth blow-downs on a center with Picard number 1. Proof of Theorem 2.

The proof of Theorem 2 we will give is close to Wiśniewski’s one but we give two intermediate results of independant interest.

2.1. On the normal bundle of the center. Let us recall that a smooth submanifold A in a complex manifold W is contractible to a point (i.e. there exists a complex space W′ and a map μ : W → W′ which is an isomorphism outside A and such that μ(A) is a point) if and only if N_{A/W}^{∗} is ample (Grauert’s criterion [Gra62]). The following proposition was proved by Campana [Cam89] in the case where Y is a curve and dim(X) = 3.

Proposition 1. Let X be a non projective manifold, Y a smooth submanifold of X such that the blow-up π : X̃ → X of X along Y is projective. Then, for each connected component Y′ of Y with ρ(Y′) = 1, the conormal bundle N_{Y′/X}^{∗} is ample.

Before the proof, let us remark that Y is projective since the exceptional divisor of π is.

Proof of Proposition 1 : (following Campana) we can suppose that Y is connected. Let E be the exceptional divisor of π and f a line contained in a non trivial fiber of π. Since E · f = −1, there is an extremal ray R of the Mori cone \overline{NE}(X̃) such that E · R < 0. Since E · R < 0, R defines an extremal ray of the Mori cone \overline{NE}(E) which we still denote by R (even if \overline{NE}(E) is not a subcone of \overline{NE}(X̃) in general !). Since ρ(Y) = 1, we have ρ(E) = 2, hence \overline{NE}(E) is a 2-dimensional closed cone, one of its two extremal rays being generated by f. Then :

- either R is not generated by f and E_{|E} is strictly negative on \overline{NE}(E)\\{0\}. In that case, −E_{|E} = O_{E}(1) is ample by Kleiman’s criterion, which means that N_{Y/X}^{∗} is ample.

- or, R is generated by f. In that case, the Mori contraction φ_{R} : X̃ → Z

factorizes through π :

\xymatrix{X̃ \ar[d]_{π} \ar[r]^{φ_{R}} & Z \\ X \ar[ur]_{ψ}}

where ψ : X → Z is an isomorphism outside Y. Since the variety Z is projective and X is not, ψ is not an isomorphism and since ρ(Y) = 1, Y is contracted to a point by ψ, hence N_{Y/X}^{∗} is ample by Grauert’s criterion.

Let us prove the following consequence of Proposition 1:

Proposition 2. Let X be a non projective manifold, Y a smooth submanifold of X such that the blow-up π : X̃ → X of X along Y is projective with −KX̃ numerically effective (nef). Then, each connected component Y′ of Y with ρ(Y′) = 1 is a Fano manifold.

Proof : we can suppose that Y is connected. Let E be the exceptional divisor of π. Since −E_{|E} is ample by Proposition 1, the adjunction formula −K_{E} = −K_{X̃|E} − E_{|E} shows that −K_{E} is ample, hence E is Fano. By a result of Szurek and Wiśniewski [SzW90], Y is itself Fano.

2.2. Proof of Theorem 2. For the first assertion, we only have to prove that

deg_{−KX̃} (E) ≤ (ρ_{n} − 1)d_{n−1}.

Let Y′ be a connected component of Y and E′ = π^{−1}(Y′). Then, since −E_{|E′} is ample :

deg_{−KX̃} (E′) = (−K_{X̃|E′})^{n−1} = (−K_{E′} + E_{|E′})^{n−1} ≤ (−K_{E′})^{n−1} ≤ d_{n−1}.

Now, if m is the number of connected components of Y, then

ρ_{n} ≥ ρ(X̃) = m + ρ(X) ≥ m + 1.

Putting all together, we get

deg_{−KX̃} (E) ≤ (ρ_{n} − 1)d_{n−1},

which ends the proof of the first point. We refer to [Wis91] prop. (3.5) for the second point.

3. On the dimension of the center of non projective smooth blow-downs. Proof of Theorem 3.

Theorem 3 is a by-product of the more precise following statement and of Proposition 3 below :

Theorem 4. Let Z be a Fano manifold of dimension n and index r, π : Z → Z′ be a non projective smooth blow-down of Z, Y ⊂ Z′ the center of π. Let f be a line contained in a non trivial fiber of π, then

(i) if f generates an extremal ray of NE(Z), then dim(Y) ≥ (n − 1)/2. (ii) if f does not generate an extremal ray of NE(Z), then dim(Y) ≥ r.

Moreover, if dim(Y) = r, then Y is isomorphic to ℙ^{r}. In both cases (i) and (ii), Y contains a rational curve.

The proof relies on Wiśniewski’s inequality (see [Wis91] and [AnW95]), which we recall now for the reader’s convenience : let φ : X → Y be a Fano-Mori contraction (i.e −K_{X} is φ-ample) on a projective manifold X, Exc(φ) its exceptional locus and

l(φ) := min{−K_{X} · C; C rational curve contained in Exc(φ)}

its length, then for every non trivial fiber F :

dim Exc(φ) + dim(F) ≥ dim(X) − 1 + l(φ).

Proof of Theorem 4. The method of proof is taken from Andreatta’s recent paper [And99] (see also [Bon96]). First case : suppose that a line f contained in a non trivial fiber of π generates an extremal ray R of NE(Z). Then the Mori contraction φ_{R} : Z → W factorizes through π :

\xymatrix{Z \ar[d]_{π} \ar[r]^{φ_{R}} & W \\ Z′ \ar[ur]_{ψ}}

where ψ is an isomorphism outside Y. In particular, the exceptional locus E of π is equal to the exceptional locus of the extremal contraction φ_{R}. Let us now denote by ψ_{Y} the restriction of ψ to Y, s = dim(Y), π_{E} and φ_{R,E} the restriction of π and φ_{R} to E. Since Z′ is not projective, ψ_{Y} is not a finite map. Since φ_{R} is birational, W is ℚ-Gorenstein, hence K_{W} is ℚ-Cartier and K_{Z′} = ψ^{∗}K_{W}. Therefore, K_{Z′} is ψ-trivial, hence K_{Y} + det N_{Y/Z′}^{∗} is ψ_{Y} -trivial. Moreover, O_{E}(1) = −E_{|E} is φ_{R,E}-ample by Kleiman’s criterion, hence N_{Y/Z′}^{∗} is ψ_{Y} -ample. Finally, ψ_{Y} is a Fano-Mori contraction, of length greater or equal to n − s = rk(N_{Y/Z′}^{∗}). Together with Wiśniewski’s inequality applied on Y, we get that for every non trivial fiber F of ψ_{Y}

2s ≥ dim(F) + dim Exc(ψ_{Y}) ≥ n − s + s − 1

hence 2s ≥ n − 1. Moreover, Exc(ψ_{Y}) is covered by rational curves, hence Y contains a rational curve.

Second case : suppose that a line f contained in a non trivial fiber of π does not generate an extremal ray R of NE(Z). In that case, since E · f = −1, there is an extremal ray R of NE(Z) such that E · R < 0. In particular, the exceptional locus Exc(R) of the extremal contraction φ_{R} is contained in E, and since f is not on R, we get for any fiber F of φ_{R} :

dim(F) ≤ s = dim(Y).

By the adjunction formula, −K_{E} = −K_{Z|E} − E_{|E}, the length l_{E}(R) of R as an extremal ray of E satisfies

l_{E}(R) ≥ r + 1,

where r is the index of Z. Together with Wiśniewski’s inequality applied on E, we get :

r + 1 + (n − 1) − 1 ≤ s + dim(Exc(R)) ≤ s + n − 1.

Finally, we get r ≤ s, and since the fibers of φ_{R} are covered by rational curves, there is a rational curve in Y. Suppose now (up to the end) that r = s. Then E is the exceptional locus of the Mori extremal contraction φ_{R}. Moreover, K_{Z}+r(−E) is a good supporting divisor for φ_{R}, and since every non trivial fiber of φ_{R} has dimension r, φ_{R} is a smooth projective blow-down. In particular, the restriction of π to a non trivial fiber F ≃ ℙ^{r} induces a finite surjective map π : F ≃ ℙ^{r} → Y hence Y ≃ ℙ^{r} by a result of Lazarsfeld [Laz83]. This ends the proof of Theorem 4.

The proof of Theorem 4 does not use the hypothesis Z Fano in the first case. We therefore have the following :

Corollary 2. Let Z be a projective manifold of dimension n, π : Z → Z′ be a non projective smooth blow-down of Z, Y ⊂ Z′ the center of π. Let f be a line contained in a non trivial fiber of π and suppose f generates an extremal ray of \overline{NE}(Z). Then dim(Y) ≥ (n − 1)/2. Moreover, if dim(Y) = (n − 1)/2, then Y is contractible on a point.

We finish this section by the following easy proposition, which combined with Theorem 4 implies Theorem 3 of the Introduction :

Proposition 3. Let Z be a Fano manifold of dimension n and index r, π : Z → Z′ be a smooth blow-down of Z, Y ⊂ Z′ the center of π. Then n − 1 − dim(Y) is a multiple of r.

Proof. Write

−K_{Z} = rL and − K_{Z} = −π^{∗}K_{Z′} − (n − 1 − dim(Y))E

where E is the exceptional divisor of π. Let f be a line contained in a fiber of π. Then rL · f = n − 1 − dim(Y), which ends the proof.

Proof of Theorem 3. Let Z be a Fano manifold of dimension n and index r and suppose there is a non projective smooth blow-down of Z with an sdimensional center. By Proposition 3, there is a strictly positive integer k such that n − 1 − kr = s. By Theorem 4, either n − 1 − kr ≥ (n − 1)/2 or n − 1 − kr ≥ r. In both cases, it implies that r ≤ (n − 1)/2 and therefore s ≥ r. If r > (n − 1)/3, since n − 1 ≥ (k + 1)r > (k + 1)(n − 1)/3, we get k = 1 and s = n − 1 − r.

4. Rational curves on simple Moishezon manifolds.

The arguments of the previous section can be used to deal with the following well-known question : does every non projective Moishezon manifold contain a rational curve ? The answer is positive in dimension three (it is due to Peternell [Pet86], see also [CKM88] p. 49 for a proof using the completion of Mori’s program in dimension three).

Proposition 4. Let Z be a projective manifold, π : Z → Z′ be a non projective smooth blow-down of Z. Then Z′ contains a rational curve.

Proof. With the notations of the previous section, it is clear in the first case where a line f contained in a non trivial fiber of π generates an extremal ray R of \overline{NE}(Z) (in that case, the center of π contains a rational curve). In the second case, since f is not extremal and K_{Z} is not nef, there is a Mori contraction φ on Z such that any rational curve contained in a fiber of φ is mapped by π to a non constant rational curve in Z′.

References

[And99] M. Andreatta. Moishezon manifolds. Math. Zeitschrift, 230, 713-726 (1999).

[AnW95] M. Andreatta, J. Wiśniewski. A view on contractions of higherdimensional varieties. Algebraic geometry—Santa Cruz, 153–183, (1995) Proc. Sympos. Pure Math., 62, Part 1, Amer. Math. Soc., Providence, RI, (1997).

[Bon96] L. Bonavero. Sur des variétés de Moishezon dont le groupe de Picard est de rang un. Bull. Soc. Math. Fr. 124, No.3, 503-521 (1996).

[Cam89] F. Campana. Critère de projectivité de la transformation élémentaire d’une variété projective de dimension 3. C. R. Acad. Sci., Paris, Ser. I 309, No.14, 863-866 (1989).

[CKM88] H. Clemens, J. Kollár, S. Mori. Higher dimensional complex geometry. Société Mathématique de France, Astérisque 166 (1988).

[Deb97] O. Debarre. Variétés de Fano. Séminaire Bourbaki. Volume 1996/97. Exposés 820–834. Paris, Société Mathématique de France, Astérisque 245, 197-221, Exp. No.827 (1997).

[Gra62] H. Grauert. Ueber Modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331-368 (1962).

[Kod63] K. Kodaira. On stability of compact submanifolds of complex manifolds. Amer. J. Math., 85, 79-94 (1963).

[KMM92] J. Kollár, Y. Miyaoka, S. Mori. Rational connectedness and boundedness of Fano manifolds. J. Diff. Geom. 36, 765-769 (1992).

[Laz83] R. Lazarsfeld. Some applications of the theory of positive vector bundles. Complete intersections, Lect. 1st Sess. C.I.M.E., Acireale/Italy 1983, Lect. Notes Math. 1092, 29-61 (1984).

[Pet86] T. Peternell. Rational curves on a Moishezon 3-fold. Complex Analysis and Algebraic Geometry. Springer LN 1194, 133-144 (1986).

[SzW90] M. Szurek, J. Wiśniewski. Fano bundles over ℙ^{3} and Q^{3}. Pac. J. Math. 141, No.1, 197-208 (1990).

[Wis91] J. Wiśniewski. On contractions of extremal rays of Fano manifolds. J. Reine Angew. Math. 417, 141-157 (1991).

[Wi91] J. Wiśniewski. On Fano manifolds of large index. Manuscripta Math. 70, no. 2, 145–152 (1991).

[Wis93] J. Wiśniewski. A report on Fano manifolds of middle index and b_{2} ≥ 2. Mathematica Gottingensis, Schriftenreihe des Sonderforschungsbereichs Geometrie und Analysis. Heft 16 (1993).

Laurent Bonavero Institut Fourier, UMR 5582 Université de Grenoble 1 BP 74. 38402 Saint Martin d’Hères France bonavero@ujf-grenoble.fr Shigeharu Takayama Department of Mathematics Graduate School of Science Osaka University Toyonaka, Osaka, 560-0043 Japan taka@math.sci.osaka-u.ac.jp