Clemens ’ Conjecture : Part I

This is the first of a series of two papers: Clemens’ conjecture: part I, Clemens’ conjecture: part II. In these two papers we solve the Clemens’ conjecture: there are finitely many smooth rational curves of each degree in a generic quintic threefold. In this paper, we deal with a family of smooth Calabi-Yau threefolds fǫ for a small complex number ǫ. If fǫ contains a smooth rational curve cǫ, then its normal bundle is Ncǫ (fǫ) = Ocǫ (k)⊕ Ocǫ (−2− k) for k ≥ −1. In a lot of examples k = −1, then c is rigid. But in many other examples, k ≥ 0. Then the question is: can cǫ move in fǫ if k ≥ 0 ? or can cǫ be deformed in fǫ in this case ? In this paper we give an geometric obstruction, deviated quasi-regular deformations Bb of cǫ, to a deformation of the rational curve cǫ in a Calabi-Yau threefold fǫ. Overview of two papers: part I, part II 1 History of the Clemens’ conjecture Let f be a smooth Calabi-Yau threefold and c be a smooth rational curve in f . Then the first Chern number c1(Nc(f)) of its normal bundle is −2. Thus the rank 2 normal bundle Nc(f) = Oc(k)⊕Oc(−2− k) with k ≥ −1. A wishful scenario is that the splitting of this bundle is symmetric, i.e. k = −1. In this case, c can’t be deformed in f , i.e there is no one dimensional family cf (small complex numbers s) of smooth rational curves in f with c 0 f = c. But this is only a wish. There are many examples in which c can be deformed. Research partially supported by NSF grant DMS-0070409 Typeset by AMS-TEX 1 2 BIN WANG OCT, 2005 Twenty years ago, Herb Clemens and Sheldon Katz([C1], [K]) proved in a general hypersurface f of degree 5 in CP , which is a popular type of Calabi-Yau threefolds, a smooth rational curves c exists in any degree. At the meantime, Clemens conjectured c can’t be deformed in f(Later in international congress of mathematics in 1986, Clemens added more statements in the conjecture[C2]). This seemingly accessible conjecture has been outstanding ever since. Immediately after Clemens made the conjecture, Katz considered it in a stronger form: the incidence scheme Id = {c ⊂ f} between the set of all quintic threefolds f and the set of smooth rational curves c of degree d in CP 4 is irreducible and the normal bundle Nc(f) = Oc(−1)⊕Oc(−1). He proved then the conjecture in this stronger form is true for d 6 7, furthermore if Id is irreducible then the conjecture is true. Ten years later, S.Kleiman and P.Nijsse improved Katz’s result to establish the irreducibility of Id in degree 8, 9. Jonhsen and Kleiman later in [JK1], [JK2] proved the conjecture for 10 6 d 6 24 assuming a likely condition. In this series of papers, “Clemens’ conjecture, part I and part II”, we would like to prove a positive dimensional family of rational curves of any degree in a generic quintic threefold does not exist. We’ll refer these two papers by “part I” and “part II”. The following is our main result Theorem 1.1. For each d > 0, there is no one parameter family cf ( for a small complex number s) of smooth rational curves of degree d in a generic quintic three fold f . It is the same to say that any irreducible component of the incidence scheme Id = {c ⊂ f} which dominates the space CP 125 of quintic threefolds, must have the same dimension 125 as the space of quintic threefolds. 2 A general description of the proof The idea of the proof comes from intersection theory. It can be naturally divided into two steps (part I and part II). One deals with all calculations in a general setting, the other is all about a concrete construction of a family of quasi-regular deformations Bb in a universal quintic threefold. CLEMENS’ CONJECTURE: PART I 3 2.1 Step one (part I). Let’s start with a family of smooth Calabi-Yau threefolds fǫ. Let ∆ be an open set of C that contains 0. Let π: X π −−−−→ ∆ be a smooth morphism such that for each ǫ ∈ T , π(ǫ), denoted by fǫ, is a smooth Calabi-Yau threefold, i.e. c1(T (fǫ) = 0. Assume there is a surface C ⊂ X such that the restriction map C π −−−−→ ∆ is also smooth and for each ǫ, π(ǫ), denoted by cǫ, is a smooth rational curve. Furthermore we assume the normal bundle of cǫ in fǫ has the following splitting (2.2) Ncǫ(fǫ) = Ocǫ(k)⊕Ocǫ(−2− k), where k ≥ 0. Hence NC(X)|cǫ is also equal to (2.3) Ocǫ(k)⊕Ocǫ(−2− k). Suppose the deformation cǫ of the rational curve cǫ exists in each quintic threefold fǫ (where c 0 ǫ = cǫ). We can construct a correspondence R using this family c s ǫ as follows: Choose a small neighborhood ∆ of 0 in C, such that the parameters s, ǫ lie in ∆. Let R ⊂ ∆×∆×X R = {(s, ǫ, x) : x ∈ csǫ}. So R is the universal curve of the family cǫ . Let B be a quasi-projective variety. Let Bb ⊂ X, b ∈ B be the restriction of a family of surfaces B ′ b in X , to a tubular neighborhood of c0 in X such that R (Bb) is a curve around the origin 0, that contains 0. Let A be the specific curve (2.4) s = ǫ in ∆ such that A meets R(Bb) at one point 0 and Bb meets R∗(A) only at the entire rational curve c0, where the power r depends on certain order of Bb. In this work, r is always 3. Let (2.5) δ1(b) = #0(A ·R (Bb)), δ2(b) = #c0(Bb ·R∗(A)), where #O denotes the intersection number supported on the set O. We let I(b) = δ2(b)− δ1(b). 4 BIN WANG OCT, 2005 By the incidence paring formula in intersection theory, we have