Rational curves on Calabi-Yau threefolds and a conjecture of Oguiso
aa r X i v : . [ m a t h . AG ] N ov RATIONAL CURVES ON CALABI-YAU THREEFOLDS AND ACONJECTURE OF OGUISO
SIMONE DIVERIO
Abstract.
This short note is an extended abstract of a talk given at theconference “Komplexe Analysis” at the Mathematisches ForschungsinstitutOberwolfach in September 2012. We explained some recent results about theexistence of rational curves on Calabi-Yau threefolds as well as a curvatureapproach to the non hyperbolicity of such manifolds.
Let X be a compact projective manifold over C , ω a K¨ahler metric on X , andconsider the following statements:(1) The holomorphic sectional curvature of ω is strictly negative.(2) The manifold X has non-degenerate negative k -jet curvature.(3) The manifold X is Kobayashi hyperbolic.(4) The manifold X is measure hyperbolic.(5) The manifold X is of general type.(6) The manifold X does not contain any rational curve.(7) The canonical bundle K X of X is nef.(8) The canonical bundle K X of X is ample.Perhaps only (2) needs some more explanations, which will be given subsequently.We have the following diagram of conjectural and actual implications:(7) (4) ? (5) t t (6) (8) t t x x (6) O O (3) O O o o ? ⑥⑥⑥⑥ > > ⑥⑥⑥⑥ ? (2) t t ? O O (1) o o ? O O As the diagram shows, the central conjecture here is the equivalence between (4)and (5) ( i.e. that measure hyperbolic implies general type), which is known to holdtrue whenever X is a projective surface. In spite of this, in the sequel we will mostlyconcentrate ourselves on the conjecture a latere , known as Kobayashi’s conjecture,which states that (3) should imply (5) (and hence (8)). We shall give some hints andrecent results both from a differential-geometric and algebro-geometric viewpoints;moreover, we shall fix our attention on threefolds, since it is the first unknown case.To begin with, observe that several powerful machineries from birational geom-etry —such as the characterization of uniruledness in terms of negativity of theKodaira dimension, the Iitaka fibration, the abundance conjecture (which is ac-tually a theorem in dimension three)— permit to reduce this conjecture to thefollowing statement: a projective threefold X of Kodaira dimension κ ( X ) = 0cannot be hyperbolic. By the Beauville-Bogomolov decomposition theorem and el-ementary properties of hyperbolic manifolds, in dimension three it suffices to showthat a Calabi-Yau threefold is not hyperbolic . Here, by a Calabi-Yau threefold we Mathematics Subject Classification.
Primary: 14J32; Secondary: 32Q45.The author is partially supported by the ANR project “POSITIVE”, ANR-2010-BLAN-0119-01. mean a simply connected compact projective threefold with trivial canonical class K X ≃ O X and h i ( X, O X ) = 0, i = 1 , Differential-geometric viewpoint.
A weaker form of (3) ⇒ (5) and (8), moredifferential-geometric in flavor, is to show that negative holomorphic sectional cur-vature implies ampleness of the canonical bundle. This is known up to dimensionthree, by the work of [4]. Again, the core here is to show that the negativity of theholomorphic sectional curvature of ( X, ω ) forces the real first Chern class c ( X ) R to be non-zero.A possible proof of this fact goes as follows. First, observe that an averagingargument shows that negative holomorphic sectional curvature implies negativescalar curvature. Now, suppose that c ( X ) R = 0. Then, there exists a smoothfunction f : X → R such that Ricci( ω ) = i∂ ¯ ∂f . Now, take the traces with respectto ω of both sides: modulo non-zero multiplicative constants, on the left we findthe scalar curvature and on the right the ω -Laplacian of f which must thereforebe always non-zero. Hence, f is constant and the scalar curvature must be zero,contradiction.Unfortunately, having negative holomorphic sectional curvature is much strongerthan being hyperbolic. In [1], it is conjectured that a weaker notion of negativecurvature, namely non-degenerate negative k -jet curvature, should be instead equiv-alent (and it is proved there that it actually implies hyperbolicity). Let us explainbriefly what this notion is, referring to [1] for more details.Let J k X → X be the holomorphic fiber bundle of k -jets of germs of holomor-phic curves γ : ( C , → X and J k X reg its subset of regular ones, i.e. such that γ ′ (0) = 0. There is a natural action of the group G k of k -jets of biholomorphismsof ( C ,
0) on J k X , and the quotient J k X reg / G k admits a nice geometric relativecompactification J k X reg / G k ֒ → X k . Here, X k is a tower of projective bundles over X . In particular, it is naturally endowed with a tautological line bundle O X k ( − V k ⊂ T X k of its tangent bundle. Definition.
The manifold X is said to have non-degenerate negative k -jet curvature if there exists a singular hermitian metric on O X k ( −
1) whose Chern curvaturecurrent is negative definite along V k and whose degeneration set is contained in X k \ ( J k X reg / G k ).Observe that if X has negative holomorphic sectional curvature, then it natu-rally has a non-degenerate negative 1-jet curvature. The following question seemstherefore particularly appropriate. Question.
Is it true that if X has non-degenerate negative k -jet curvature then c ( X ) R = 0 ? Algebro-geometric viewpoint.
Algebraic geometers expect more than the nonhyperbolicity of Calabi-Yau’s: a folklore conjecture states that every Calabi-Yaumanifold should contain a rational curve. For threefolds, let us cite a couple ofresults in this direction: • The article [3] is the culmination of a series of papers by Wilson in which hestudies in a systematic way the geometry of Calabi-Yau threefolds; amongmany other things, it is shown there that if the Picard number ρ ( X ) > X . • Following somehow the same circle of ideas, it was proven in [6] (see also[5]) that a Calabi-Yau threefold X has a rational curve provided there existson X a non-zero effective non-ample line bundle on X .By the Cone Theorem, if there exists on a Calabi-Yau manifold X a non-zeroeffective non-nef line bundle, then there exists on X a rational curve (generating an ATIONAL CURVES ON CALABI-YAU THREEFOLDS 3 extremal ray). Therefore, we can always suppose that such an effective line bundle isnef. Remark, on the other hand, that in Peternell’s result, the effectivity hypothesisis crucial (regarding it in a more modern way) in order to make the machinery ofthe logMMP work. In this spirit, Oguiso asked in [5] the following question: isit true that if a Calabi-Yau threefold X possesses a non-zero nef non-ample linebundle, then there exists a rational curve on X ? Here is a positive answer, under a mild condition on the Picard number of X . Theorem (Diverio, Ferretti [2]) . Let X be a Calabi-Yau threefold and L → X a non-zero nef non-ample line bundle. Then, X has a rational curve provided ρ ( X ) > . In order to give a rough idea of the techniques involved in this kind of business,let us state (a special case of) the Kawamata-Morrison Cone conjecture and explainhow it would almost imply the Kobayashi conjecture.
Conjecture (Kawamata-Morrison) . Let X be a Calabi-Yau manifold. Then, theaction of Aut( X ) on the nef-effective cone of X has a rational polyhedral funda-mental domain. Proposition.
Suppose that the Kawamata-Morrison conjecture holds. Then, theKobayashi conjecture is true in dimension three, except possibly if there exists aCalabi-Yau threefold of Picard number one which is hyperbolic.Proof.
We shall suppose that the Kawamata-Morrison conjecture holds true andthat there exists a hyperbolic Calabi-Yau threefold X with ρ ( X ) ≥ X is supposed to be hyperbolic, it does not contain any rational curve andAut( X ) is finite. The Kawamata-Morrison conjecture implies therefore that the nefcone of X is rational polyhedral.Now, since it is rational polyhedral, rational points are dense on each face of thenef boundary. Moreover, at most one of these faces (which are at least in number of ρ ( X ) ≥
2) can be contained in the hyperplane given by ( c ( X ) · D ) = 0: in fact thisis a “true” hyperplane since if c ( X ) = c ( X ) = 0, then X would be a finite ´etalequotient of a complex torus, so that X would not be hyperbolic (nor a Calabi-Yaumanifold in our strict sense).Therefore, there exists on X a (in fact plenty of) nef Q -divisor D such that c ( X ) · D >
0. Computing its Euler characteristic and using Kawamata–Viehwegvanishing, D can be shown to be effective. But then, since there exists on X a non-zero effective non-ample divisor, there exists a rational curve on X , contradictingits hyperbolicity. (cid:3) References [1] J. P. Demailly;
Algebraic criteria for Kobayashi hyperbolic projective varieties and jet dif-ferentials , Algebraic geometry—Santa Cruz 1995, 285–360, Proc. Sympos. Pure Math., 62,Part 2, Amer. Math. Soc., Providence, RI, 1997.[2] S. Diverio, A. Ferretti;
On a conjecture of Oguiso about rational curves on Calabi-Yau three-folds , to appear on Comment. Math. Helv.[3] D. R. Heath-Brown, P. M. H. Wilson;
Calabi-Yau threefolds with ρ >
13, Math. Ann. (1992), no. 1, 49–57.[4] G. Heier, S. Y. Lu, B. Wong;
On the canonical line bundle and negative holomorphic sectionalcurvature , Math. Res. Lett. (2010), no. 6, 1101–1110.[5] K. Oguiso; On algebraic fiber space structures on a Calabi-Yau 3-fold , Internat. J. Math. (1993), no. 3, 439–465.[6] T. Peternell; Calabi-Yau manifolds and a conjecture of Kobayashi , Math. Z. (1991), no.2, 305–318.
SIMONE DIVERIO
Simone Diverio, CNRS et Institut de Math´ematiques de Jussieu – Paris Rive Gauche.
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