Connecting Certain Rigid Birational Non-homeomorphic Calabi–yau Threefolds via Hilbert Scheme

We shall give an explicit pair of birational projective Calabi–Yau threefolds which are rigid, non-homeomorphic, but are connected by projective flat deformation over some connected base scheme. 0. Introduction A Calabi–Yau manifold is a compact Kähler simply-connected manifold with a nowhere vanishing global n-form but no global i-from with 0 < i < n = dimX . By Kodaira’s criterion, it is projective if dimension n ≥ 3. As well known, Calabi–Yau manifolds, hyperkähler manifolds and complex tori form the building blocks of compact Kähler manifolds with vanishing first Chern class ([Be1], [GHJ]). A famous theorem of Huybrechts states that two bimeromorphic hyperkähler manifolds are equivalent under smooth deformation ([Hu1], [Hu2]). In particular, they are homeomorphic to each other, having the same Betti numbers and Hodge numbers. Clearly, the same holds true for complex tori. Other famous theorem, called Kontsevich’s theorem, says that two birational Calabi–Yau manifolds have the same Betti numbers and Hodge numbers ([Ba], [DL], [It], [Ya], [Wa]). However, there are rigid birational non-isomorphic Calabi–Yau manifolds (cf. Theorem 0.1). Obviously they are not equivalent under any smooth deformation. The aim of this note is to remark that there nevertheless exist birational Calabi– Yau threefolds which are rigid, non-homeomorphic, but are connected by (necessarily non-smooth) projective flat deformation: Theorem 0.1. There are Calabi–Yau threefolds X and Y such that: (1) X and Y are birational and rigid, (2) X and Y are not homeomorphic but, (3) X and Y are connected by projective flat deformation over some connected scheme. This work is motivated by the famous fantasy of Miles Reid [Re] – especially the question what this fantasy would be like for rigid Calabi–Yau threefolds – and by the first named author’s recent result on the equivalence of certain Calabi–Yau threefolds with Picard number one, of different topological type, under projective flat deformation [Le]. In the proof of our main theorem (Theorem 0.1), the following deep theorem of Hartshorne [Ha] (see also [PS]) plays an important role: 2000 Mathematics Subject Classification. 14J32, 14D06. The second named author was supported by JSPS. 1 2 N.-H. LEE AND K. OGUISO Theorem 0.2 (R. Hartshorne). A Hilbert scheme Hilb P (x) PN of P with fixed Hilbert polynomial P (x) is connected. So, if two varieties belong to the same Hilbert scheme Hilb P (x) PN , then they appear as fibers of the universal family u : U −→ Hilb P (x) PN , in which Hilb P (x) PN is connected. In this way, they are connected by projective flat deformation. Let Z be a Calabi–Yau threefold and let H be an ample divisor on Z. Then, by the Kodaira vanishing theorem and the Riemann-Roch formula, we have dimH(OZ(nH)) = χ(OZ(nH)) = H 6 n + H · c2(Z) 12 n . Here c2(Z) = c2(TZ) is the second Chern class of Z. It is also known that 10H is always very ample on Z ([OP]). Therefore, as a special case of Theorem 0.2, one obtains the following: Theorem 0.3. Two Calabi–Yau threefolds have the same Hilbert polynomial, belong to the same Hilbert scheme of some projective space, and accordingly connected by projective flat deformation, if and only if they have ample divisors that have the same values of H and H · c2. In general, two Calabi–Yau threefolds are unlikely to be connected by projective flat deformation, especially if they are of different topological type. Let X and Y be a complete intersection of two cubics in P and a quintic hypersurface in P respectively. Then we always have 9k = (kHX) 3 6= (lHY ) 3 = 5l for any positive integers k, l, where HX and HY are the ample generators of the Picard groups of X and Y respectively. So X and Y can not be connected by any projective flat deformation. Our Calabi–Yau threefolds in Theorem 0.1 are the famous rigid Calabi–Yau threefold Xφ constructed by Beauville [Be2] and its birational modification XT studied by the second named author [Og] (See also Section 2). The structure of this note is as follows: We discuss some toy case of elliptic curves in Section 1. This explains some idea behind our consideration. In Section 2, we recall Beauville’s rigid Calabi–Yau threefold Xφ and its birational modificationXT . Sections 3 and 4 are devoted to the proof of Theorem 0.1. Acknowledgement. We would like to express our thanks to Professors J.M. Hwang, J.H. Keum, B. Kim for valuable discussions. 1. Toy example: connecting two elliptic curves in two ways Let Cλ (λ 6= 0, 1) be the elliptic curve defined by the Weierstrass equation y = x(x − 1)(x− λ) . Obviously, any two elliptic curves Cλ1 and Cλ2 are connected by the following projective smooth family: ψ : X = {([x0 : x1 : x2], λ) ∈ P 2 × B ∣∣ x1x2 − x0(x0 − x2)(x0 − λx2) = 0} −→ B . Here and hereafter, we put B = P \ {0, 1,∞}. CONNECTING RIGID CALABI–YAU THREEFOLDS 3 Yet, we can connect Cλ1 and Cλ2 by another way. Let D be a hyperelliptic curve with a hyperelliptic involution ι and let Ξ be the set of the branch points of ι in D/〈ι〉 ≃ P. We consider the natural morphisms, φ1 : ̃ Cλ1 ×D / 〈(−1, ι)〉 −→ D/〈ι〉 ≃ P and φ2 : ̃ Cλ2 ×D / 〈(−1, ι)〉 −→ D/〈ι〉 ≃ P. Here ̃ ’s are the minimal resolutions. We regard φ1 and φ2 as projective flat deformations. Then, for q ∈ Ξ, the scheme-theoretic fiber φ−1 1 (q) = 2l + l0 + l1 + l∞ + lλ1 consists of 5 P ’s, intersecting like: • 0 l0 • 1 l1 • ∞ l∞ • λ1 lλ1 2l and φ−1 1 (p) ≃ Cλ1 for p / ∈ Ξ. Similarly the scheme-theoretic fiber φ −1 2 (q) for q ∈ Ξ is like: • 0 l0 • 1 l1 • ∞ l∞ • λ2 lλ2 2l and φ−1 2 (p) ≃ Cλ2 for p / ∈ Ξ. The singular schemes φ −1 1 (q) and φ −1 2 (q) can be put into a projective flat family, in which the fibers are of the form: • 0 • 1 • ∞ • λ For example, the natural projection ψ : Y −→ B, where Y = {([x0 : x1], [y0 : y1], λ) ∈ P 1 × P × B ∣∣ x0y0y1(y0 − y1)(y0 − λ1y1) = 0} is such a family. In this way, Cλ1 and Cλ2 are connected by a chain of three projective flat deformations. In the second method, smooth fibers in families are only Cλ1 and Cλ2 and they are connected through very singular spaces. So, the method suggests some possibilities to connect two rigid manifolds of different topological structure. This is the idea behind our construction. 2. Beauville’s rigid Calabi–Yau threefold and its modification We briefly recall the two rigid Calabi–Yau threefolds X and Y that appear in Theorem 0.1. Let ζ = e √ −1/3. By Eζ , we denote the elliptic curve whose period is ζ and by E ζ /〈ζ〉 the quotient n-fold of the product manifold E n ζ by the scalar multiplication by ζ. Let Q0 = 0, Q1 = (1− ζ)/3 and Q2 = −(1− ζ)/3 in Eζ . These are exactly the fixed points of the scalar multiplication by ζ on Eζ . For ik = 0, 1, 2, let Qi1i2···in = (Qi1 , Qi2 , · · · , Qin) ∈ E n ζ and let Qi1i2···in be its image in E n ζ /〈ζ〉. Then X = E 3 ζ/〈ζ〉 has singularities of type 1 3 (1, 1, 1) at Qijk’s and the blow-up π : Xφ −→ X at these 27 singular points gives a Calabi–Yau threefold Xφ. This is the famous rigid Calabi–Yau threefold found 4 N.-H. LEE AND K. OGUISO by Beauville [Be2]. We denote by Eijk the exceptional divisor lying over Qijk . The surfaces Eijk is isomorphic to P . • Eij0 Eζ • Eij1 Xφ • Eij2 lij