SOME EXAMPLES OF 1-CONVEX NON-EMBEDDABLE THREEFOLDS

SOME EXAMPLES OF 1-CONVEX NON-EMBEDDABLE THREEFOLDS

GIOVANNI BASSANELLI and MARCO LEONI

We construct a family of 1-convex threefolds with exceptional curveC of type (0,−2), which are not embeddable in C^m×CPn. In order to show that they are not K¨ahler, we exhibit a real 3-dimensional chain A whose boundary is the complex curveC.

AMS 2000 Subject Classification: Primary 32F10.

Key words: embeddable variety, exceptional curve, threefold.

1. INTRODUCTION

In general, a 1-convex manifold with 1-dimensional exceptional set is embeddable, that is, it can be realized as an embedded subvariety ofC^m×CPn, for suitablem and n. The only possible exceptions are given by the following result.

Theorem 1.1 (see Theorem 3 in [C1]). Let X be a non-embeddable, 1- convex manifold whose exceptional setC has dimension 1. Then dim_CX= 3 andC has an irreducible component which is a rational curve of type(−1,−1), (0,−2) or (1,−3).

As regards the existence of the quoted exceptions, in [C1] there is an example of type (−1,−1). In [V1], p. 242 B there is an example of type (0,−2), but the argument is dubious (see [C1], Remark 4, but see also [V2]

and [C2]). For the case (1,−3) nothing is known.

Indeed, the first two cases are easier and we shall show

Theorem 1.2. For every integer k ≥ 1 there is a non-embeddable 1- convex threefoldX˜_k having as exceptional set a smooth rational curveCwhose sequence of normal bundles is

(0,−2), . . . ,(0,−2)

k−1

,(−1,−1).

In particular,N_C|X˜

1 =O(−1)⊕ O(−1) andN_C|X˜

k =O(0)⊕ O(−2) fork≥2.

REV. ROUMAINE MATH. PURES APPL.,52(2007),6, 611–617

For k = 1 we retrieve Colt¸oiu’s example. We point out that our con- struction is explicit and elementary. In order to see that ˜X_k is not K¨ahler, we shall exhibit a real 3-chainA whose boundary is the exceptional curveC.

2. THE PROOF OF THEOREM 1.2 Definition 2.1. Letk≥1 be an integer. The equations (2.1)





 w= ¹_z

y₁ =z²x₁+zx^k₂ y₂ =x₂

in the coordinates (z, x₁, x₂) and (w, y₁, y₂), define a fiber bundleWk onCP1, with fiberC².

The following results holds.

Proposition 2.2 (see [L], p. 263 and [P], p. 234). The null section of W_k is an exceptional rational curve C of type (−1,−1) for k= 1, or (0,−2) for k≥2.

There is a geometrical description of these threefolds.

Proposition2.3. Letk≥1be an integer. LetN_k−→^fk C⁴be the blow-up with center at the complex smooth surface

(2.2) S_k:={z∈C⁴;z₁−iz₂ =z₃−z₄^k= 0}. ThenW_k is the strict transform of the hypersurface (2.3) Yk:={z∈C⁴;z₁²+z²₂+z₃²−z₄^2k = 0},

which is singular only at the origin P_k = 0; the null section of W_k is C_k = f_k⁻¹(P_k).

Proof. It is enough to follow the outline of [P], Example 2.14.

Now, we shall investigate in more detail this geometric construction and build a suitable commutative diagram

(2.4)

N₀ −→^h1 N₁ −→ · · ·^{h2 h}−→^k−1 N_k−1 −→^hk N_k



^f0 ^f1 ^fk−1

 ^fk

M₀ −→^g1 M₁ −→ · · ·^{g2 g}−→^k−1 M_k−1 −→^gk M_k=C⁴.

Step 1. Applying the desingularization process to the hypersurface Y_k ⊂ C⁴ :=M_k we get the sequence

M₀−→^g1 M₁ −→ · · ·^{g2 g}−→^k−1 Mk−1 g_k

−→Mk=C⁴.

More precisely, this sequence is defined by induction: M_k−1−→^gk M_k=C⁴ is the blow-up atP_k:= 0. Then we define the chart (U_k−1;u₁, . . . , u₄) ofM_k−1 by requiring that in these coordinates the mapgk has the equations

(2.5)

zj =uju₄, forj = 1,2,3 z₄ =u₄.

LetY_k−1 ⊂M_k−1 be the strict transform ofY_k: we get that the only singular point ofY_k−1 isP_k−1:= 0∈U_k−1 and

(2.6) Y_k−1∩U_k−1 ={u²₁+u²₂+u²₃−u²⁽₄^k−1) = 0}.

Comparing (2.6) with (2.3), we notice that we can iterate the process: the map M_j−1 −→^gj M_j is the blow-up at P_j and Y_j−1 is the strict transform of Y_j. Finally, sinceY₀∩U₀={u²₁+u²₂+u²₃−1 = 0}, Y₀ is smooth, so that the process ends.

Next, we need the following result, which is probably well known. Never- theless, for lack of reference, we shall give the proof.

Proposition 2.4. Let M be a complex manifold. Assume that S is a complex submanifold ofM andP is a complex submanifold ofS. The diagram

N −→^h N



^f ^f M −→^g M

is commutative. Here,g is the blow-up with center P, S the strict transform of S in M, f (resp. f) the blow-up with center S (resp. S), h the blow-up with centerf⁻¹(P).

Proof. The problem is local, therefore we may assume P = C^l× {0} ⊂ S=C^l+m× {0} ⊂M =C^l+m+n. Moreover, it is enough to consider the case l= 0, thus P is a point of S, which is ak-plane of M =Cⁿ. Now, it sufficies to prove the following

Claim. Proposition2.4holds ifP is a point of ak-planeS ofM =CPn. Proof of the claim. Assume 2 ≤ k ≤n−2, otherwise the statement is obvious. Fix a (n−k−1)-plane L of CPn such that L∩S = ∅. Thus, L parametrizes the family{R_x}x∈Lof all (k+ 1)-planes containing S.

LetR_x⊂N be the strict transform ofRx underg◦f. ThenN is the disjoint union∪x∈LR_x and an holomorphic projectionτ :N →Lis naturally defined.

LetE_g ⊂N be the strict transform of the exceptional set E_g CPn−1

of M →^g M =CPn. The mapτ induces a projection E_g −→^τ L whose fibers R_x ∩E_g are biholomorphic to CPk. Moreover, for each line r of R_x ∩E_g,

we get r._NE_g = −1. It follows that N can be contracted to a manifold Q which contains L and this contraction N −→^α Q is the blow-up of center L.

Moreover,α and τ agree on E_g.

LetE_f ⊂N be the exceptional set of N −→^f M. Thus τ⊗f :E_f → L×S is a biholomorphism (since each fiber ofEf τ

−→L intersects each fiber of E_f f

−→ S in only one point). But (τ ⊗f)(E_g ∩E_f) = L× E, where E CPk−1 is the exceptional set of S −→^g S. Therefore, the mapα acts on E_f as the projectionL×S→L×S. Thus,E :=α(E_f)L×S is a complex manifold.

As before, for every q ∈ S and every line s of L× {q} ⊂ E we have s._QE = −1. Therefore, Q can be contracted to a manifold T which contains S and this contraction is the blow-up Q→T. But now T and M =CPn are bimeromorphic and biholomorphic outside ofS, therefore they are biholomor- phic.

Now, recalling Proposition 2.3, we complete our construction.

Step 2. Define Sj−1 as the strict transform of Sj by means of the map M_j−1 −→^gj M_j, 1 ≤ j ≤ k. Let N_j −→^fj M_j be the blow-up of center S_j, 0≤j ≤k−1. Moreover, let Cj :=f_j⁻¹(Pj) and Nj−1 h_j

−→ Nj be the blow-up of center C_j, 1≤j≤k. Then diagram (2.4) is commutative.

Proof. By Proposition 2.4, it is enough to check thatP_j ∈S_j,j= 0, . . . , k.

But, as noted above,Pj = 0∈Uj and, using the chartUj, it is straighforward to check that

(2.7) S_j∩U_j ={u₁−iu₂=u₃−u^j₄}.

Corollary 2.5. Let X_j be the strict transform of Y_j under the map N_j −→^fj M_j, 0 ≤ j ≤ k. Considering restrictions of maps, from (2.4) we get the commutative diagram

(2.8)

X₀ −→^h1 X₁ −→^h2 . . . ^h−→^k−1 X_k−1 −→^hk X_k=W_k

↓^f0 ↓^f1 ↓^fk−1 ↓^fk

Y₀ −→^g1 Y₁ −→^g2 . . . ^g−→^k−1 Y_k−1 −→^gk Y_k. Moreover,

(i) X_j is smooth, the rational curve C_j (which is the center of h_j) is contained in Xj and there is a neighbourhood ofCj in Xj biholomorphic to a neighbourhood of the null section ofW_j, 1≤j≤k;

(ii)Cj+1=hj+1(Cj), j≥1;

(iii) X₀−→^f0 Y₀ is a biholomorphism;

(iv)the exceptional divisor E₀ of h₁ is biholomorphic toCP₁×CP₁ and the induced map E₀ CP₁ ×CP₁ −→^h1 C₁ CP₁ is one of the canonical projections.

Proof. Diagram (2.8) is well defined, because diagram (2.4) is commu- tative. Comparing equations (2.3) and (2.6), (2.2) and (2.7), we can ap- ply Proposition 2.3 for j = 1, . . . , k. Thus, we get that Wk = Xk, Wj = X_j ∩f_j⁻¹(U_j) and C_j is its null section, for 1 ≤ j ≤ k−1. By Proposition 2.2, N_C1|X1 = (−1,−1), while N_Cj|Xj = (0,−2) for j ≥ 2. Thus, the excep- tional divisorE_j−1 ofh_j is a rational ruled surface: E₀=F₀ (this proves (iv)) and Ej = F₂ for j ≥ 1. Now, the curve Cj is not a fiber of F₂, otherwise N_Cj|Xj = (0,−1), thus C_j is a section of E_j F₂. This shows (ii). Finally, sinceS₀ and Y₀ are smooth, X₀ −→^f0 Y₀ is a biholomorphism.

Remark 2.6. In the rational ruled surface F₂, there is only one curveC with negative self-intersection: C._F2C =−2. Since C_j is not a fiber ofE_j, it follows easily from the exact sequence

0→ O(Cj._EjCj)→N_Cj|Xj → O(Cj._XjEj)→0

thatC_j is the curve ofE_j with negative self-intersection. This means that the sequenceX₀ → · · · →X_k is the sequence of the blow-ups associated with the curveC_k.

Let us now state the following elementary result.

Lemma2.7. Let Q:={z∈CP₃;z₀²+z₁²+z₂²−z²₃ = 0}. Every line ofQ has a real point.

Proof. Let r ⊂ Q be a line and let P ∈ r. If P is not a real point, consider the linespassing throughP andP. Now,sis a real line ands∩RP₃ is external to the real sphereQ ∩RP₃, therefore there are exactly two planes passing throughs and tangent to Q, and the tangent points are real. One of these planes must be the planeα defined by the lines r and s(in fact, α∩ Q contains r and is thus a degenerate conic), therefore it is tangent to Q at a real pointQ, which must belong to r.

By the detailed description of the map X_k −→^fk Y_k given in (2.8), the following statement is a simple consequence.

Corollary 2.8. Let B := {z ∈ Y_k ∩R⁴;z₄ > 0} and A := f_k⁻¹(B).

ThenA is a real threefold with boundary ∂A=C_k.

Proof. LetD:= (g_k◦ · · · ◦g₁)⁻¹(B). Since diagram (2.8) is commutative and hj is surjective, for every j, A = hk◦ · · · ◦h₁(f₀⁻¹(D)). It follows from

(2.5) thatg_k⁻¹(B) ⊂ U_k−1, and iterating this argument we get that D ⊂U₀, more precisely,

D={x∈R⁴;x²₁+x²₂+x²₃−1 = 0, x₄>0}.

Therefore, the boundary∂D={x∈R⁴;x²₁+x²₂+x²₃−1 =x₄ = 0}is contained in{z ∈C⁴;z²₁+z²₂ +z²₃ −1 = z₄ = 0} =E₀∩U₀. By Corollary 2.5(iv), the fibers of the mapE₀ f₀⁻¹(E₀) −→^h1 C₁ are one of the two familyies of lines on the quadricE₀. By Lemma 2.7, each of these lines intersect∂D, therefore h₁(f₀⁻¹(∂D)) =C₁. Thus, by Corollary 2.5(ii), we have ∂A=C_k.

In order to obtain our example ˜X_k, we must perturb Y_k outside the origin.

Lemma 2.9. For every fixed integer k ≥1 there exist an integer N > k and0< ε≤1such that the origin is the only singular point of the hypersurface (2.9) Y˜_k:={z²₁+z₂²+z²₃−z₄^2k+ε(z₁^2N +z₂^2N +z₃^2N +z²₄^N) = 0}.

The equationsw_j :=z_j(1+εz_j^2N)^1/2, 1≤j≤3 andw₄ =z₄(1−εz²₄^N)^1/2k define a biholomorphic map V −→^Φ V˜ between two neighbourhoods of the origin of C⁴. Thus, we can define ˜X_k gluing f_k⁻¹(V)∩X_k and ˜Y_k\ {0} by means of Φ. The maps ˜X_j−1 →X˜_j and ˜Y_j−1 →Y˜_j are defined as above since nothing is changed nearPj and Cj while the maps ˜Xj →Y˜j are defined by a gluing process (these maps are not blow-ups).

Proposition 2.10. X˜_k is not embeddable.

Proof. Let ˜B := R⁴ ∩Y˜_k ∩ {x₄ > 0}. It follows from (2.9) that ˜B is relatively compact and from the definition of Φ that ˜B∩V˜ = Φ⁻¹(B ∩V).

Thus, the preimage ˜A of ˜B is again a 3-chain with boundary C_k. Hence the exceptional curveC_k=∂A˜ is a boundary in ˜X_k, which is not K¨ahler.

Acknowledgement.Research partially supported by the GNSAGA of the INdAM.

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Received 17 September 2006 Dipartimento di Matematica Universit`a degli Studi di Parma

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