Clemens ’ S Conjecture : Part I

This is the first of a series of two papers in which we solve the Clemens conjecture: there are only 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, so c is rigid. But in many other examples, k ≥ 0. Then the question is: can cǫ move in fǫ if k ≥ 0, that is, can cǫ deform in fǫ in this case ? In this paper we give a 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) for k ≥ −1. A wishful scenario is that the splitting of this bundle is symmetric, i.e. k = −1. In this case, c can’t deform in f , i.e there is no one dimensional family csf (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 deform. 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, that smooth rational curves c exist in any degree. At the meantime, Clemens conjectured c can’t be deformed in f .(Later in the international congress Research partially supported by NSF grant DMS-0070409 Typeset by AMS-TEX 1 2 BIN WANG OCT, 2005 of mathematicians 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} ⊂ CP 125 ×M in the Cartitian product of the set CP 125 of all quintic threefolds f and the Hilbert scheme M of smooth rational curves c of degree d in CP , is irreducible and the normal bundle is Nc(f) = Oc(−1)⊕Oc(−1). He proved that the conjecture in this stronger form is true for d 6 7, furthermore if Id is irreducible, then the Clemens 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 prove that a positive dimensional family of rational curves of any degree in a generic quintic threefold does not exist. In these two papers, the names “part I” and “part II” refer to part I and part II of these two papers. The following is our main result Theorem 1.1. For each d > 0, there is no one-parameter family csf ( for a small complex number s) of smooth rational curves of degree d in a generic quintic threefold 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 as the space of quintic threefolds. 2 A general description of the proof The idea of the proof comes from intersection theory. Specifically, we study the incidence relation of rational curves and surfaces. The method stems from our interest in the intersection theory of a less dimension (or incidence relation on algebraic cycles), a lot of which is till a novelty. It can be briefly described as follows. Let X be a smooth projective variety. Let Bb, b ∈ B be a family of cycles on X , and Aa, a ∈ ∆ be another family of cycles on X , where B,∆ are quasi-projective or projective. Let dim(Aa) + dim(Bb)− dim(X) = N . CLEMENS’S CONJECTURE: PART I 3 In the classical intersection theory, we require N ≥ 0 in order to have the intersection Ab ∩Bb. Our proposal of the “intersection theory of a less dimension” concentrates on the rest of missing cases where: N ≦ −1 (the name “less dimension” comes from the condition N < 0). An extensive research plan on this was stated in our proposal of NSF grant DMS-0070409. Roughly speaking, we define incidence cycles R(Bb) = {a ∈ ∆ : Aa meets Bb} ⊂ ∆. Our philosophy in the proposal is that the properties of family of incidence cycles R(Bb) should reveal deeper information of the underlying families of cycles Aa and Bb. Here we should not discuss this general case in a detail. However our proof of Clemens conjecture is exactly developed from this philosophy applied to the case: