Hyperbolic algebraic varieties and holomorphic differential equations Jean-Pierre Demailly Institut Fourier, Universit´ e de Grenoble I, France & Acad´ emie des Sciences de Paris August 26, 2012 / VIASM Yearly Meeting, Hanoi Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations Entire curves 2/36 Definition. By an entire curve we mean a non constant holomorphic map f : C → X into a complex n-dimensional manifold. X is said to be (Brody) hyperbolic if ∃ such f : C → X . If X is a bounded open subset Ω ⊂ C n , then there are no entire curves f : C → Ω(Liouville’s theorem), ⇒ every bounded open set Ω ⊂ C n is hyperbolic X = C{0, 1, ∞} = C{0, 1} has no entire curves, so it is hyperbolic (Picard’s theorem) A complex torus X = C n /Λ (Λ lattice) has a lot of entire curves. As C simply connected, every f : C → X = C n /Λ lifts as ˜ f : C → C n , ˜ f (t )=( ˜ f 1 (t ),..., ˜ f n (t )), and ˜ f j : C → C can be arbitrary entire functions. Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
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Hyperbolic algebraic varieties and holomorphic differential equations
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Hyperbolic algebraic varieties and holomorphic
differential equations
Jean-Pierre Demailly
Institut Fourier, Universite de Grenoble I, France& Academie des Sciences de Paris
Definition. By an entire curve we mean a non constantholomorphic map f : C → X into a complex n-dimensionalmanifold.X is said to be (Brody) hyperbolic if 6 ∃ such f : C → X .
If X is a bounded open subset Ω ⊂ Cn, then there are noentire curves f : C → Ω (Liouville’s theorem),⇒ every bounded open set Ω ⊂ Cn is hyperbolic
X = Cr0, 1,∞ = Cr0, 1 has no entire curves,so it is hyperbolic (Picard’s theorem)
A complex torus X = Cn/Λ (Λ lattice) has a lot of entirecurves. As C simply connected, every f : C → X = Cn/Λ liftsas f : C → Cn, f (t) = (f1(t), . . . , fn(t)), and fj : C → C canbe arbitrary entire functions.
For a complex manifold, n = dimC X , one defines theKobayashi pseudo-metric : x ∈ X , ξ ∈ TX
κx(ξ) = infλ > 0 ; ∃f : D → X , f (0) = x , λf∗(0) = ξOn Cn, Pn or complex tori X = Cn/Λ, one has κX ≡ 0.
X is said to be hyperbolic in the sense of Kobayashi if theassociated integrated pseudo-distance is a distance(i.e. it separates points – i.e. has Hausdorff topology).
Examples. ∗ X = Ω/Γ, Ω bounded symmetric domain.∗ any product X = X1 × . . .× Xs where Xj hyperbolic.
Theorem (dimension n arbitrary) (Kobayashi, 1970)TX negatively curved (T ∗
X > 0, i.e. ample) ⇒ X hyperbolic.
Recall that a holomorphic vector bundle E is ample iff itssymmetric powers SmE have global sections which generate1-jets of (germs of) sections at any point x ∈ X .
The proof of the above Kobayashi result depends crucially on:
Ahlfors-Schwarz lemma. Let γ = i∑γjkdtj ∧ dtk be an almost
everywhere positive hermitian form on the ball B(0,R) ⊂ Cp, suchthat −Ricci(γ) := i ∂∂ log det γ ≥ Aγ in the sense of currents, forsome constant A > 0 (this means in particular thatdet γ = det(γjk) is such that log det γ is plurisubharmonic). Thenthe γ-volume form is controlled by the Poincare volume form :
Brody reparametrization Lemma. Assume that X is compact,
let ω be a hermitian metric on X and f : D → X a holomorphic
map. For every ε > 0, there exists a radius R ≥ (1− ε)‖f ′(0)‖ωand a homographic transformation ψ of the disk D(0,R) onto(1− ε)D such that ‖(f ψ)′(0)‖ω = 1 and
‖(f ψ)′(t)‖ω ≤ (1− |t|2/R2)−1 for every t ∈ D(0,R).⇒ if f ′ unbounded, ∃g = lim f ψν : C → X with ‖g ′‖ω ≤ 1.
Brody theorem (1978). If X is compact then X is Kobayashi
hyperbolic if and only if there are no entire holomorphic curves
f : C → X (Brody hyperbolicity).
Hyperbolic varieties are especially interesting for their expecteddiophantine properties :
Conjecture (S. Lang, 1986) An arithmetic projective variety X is
hyperbolic iff X (K) is finite for every number field K.
Using “jet technology” and deep results of McQuillan forcurve foliations on surfaces, D. – El Goul provedTheorem (solution of Kobayashi conjecture, 1998).A very generic surface X⊂P3 of degree ≥ 21 is hyperbolic.
Definition of algebraic differential operators 13/36
The main idea in order to attack GGL is to use differentialequations. Let
C → X , t 7→ f (t) = (f1(t), . . . , fn(t))
be a curve written in some local holomorphic coordinates(z1, . . . , zn) on X .Consider algebraic differential operators which can be writtenlocally in multi-index notation
P(f[k]) = P(f ′, f ′′, . . . , f (k))
=∑
aα1α2...αk(f (t)) f ′(t)α1 f ′′(t)α2 . . . f (k)(t)αk
where aα1α2...αk(z) are holomorphic coefficients on X and
t 7→ z = f (t) is a curve, f[k] = (f ′, f ′′, . . . , f (k)) its k-jet. ObviousC∗-action :
λ · f (t) = f (λt), (λ · f )(k)(t) = λk f (k)(λt)
⇒ weighted degree m = |α1|+ 2|α2|+ . . .+ k |αk |.
Vanishing theorem for differential operators 14/36
Definition. EGGk,m is the sheaf (bundle) of algebraic differential
operators of order k and weighted degree m.
Fundamental vanishing theorem[Green-Griffiths 1979], [Demailly 1995], [Siu-Yeung 1996]Let P ∈ H0(X ,EGG
k,m ⊗O(−A)) be a global algebraic
differential operator whose coefficients vanish on some ample
divisor A. Then ∀f : C → X, P(f[k]) ≡ 0.
Proof. One can assume that A is very ample and intersectsf (C). Also assume f ′ bounded (this is not so restrictive byBrody !). Then all f (k) are bounded by Cauchy inequality.Hence
C ∋ t 7→ P(f ′, f ′′, . . . , f (k))(t)
is a bounded holomorphic function on C which vanishes atsome point. Apply Liouville’s theorem !
Generalized GGL conjecture. If (X ,V ) is directed manifold
of general type, i.e. detV ∗ big, then ∃Y ( X such that
∀f : (C,TC) → (X ,V ) non const., f (C) ⊂ Y .
Remark. Elementary by Ahlfors-Schwarz if r = rkV = 1.t 7→ log ‖f ′(t)‖V ,h is strictly subharmonic if r = 1 and(V ∗, h∗) has > 0 curvature in the sense of currents.
Strategy. Try some sort of induction on r = rkV .First try to get differential equations f[k](C) ⊂ Z ( Xk .Take minimal such k . If k = 0, we are done! Otherwise k ≥ 1and πk,k−1(Z ) = Xk−1, thus V
′ = Vk ∩ TZ hasrank < rkVk = r and should have again detV ′∗ big (unlesssome unprobable geometry situation occurs ?).
Needed induction step. If (X ,V ) has detV ∗ big and
Z ⊂ Xk irreducible with πk,k−1(Z ) = Xk−1, then (Z ,V ′),V ′ = Vk ∩ TZ has OZℓ
Probabilistic interpretation of the curvature 26/36
In such polar coordinates, one gets the formula
ΘLk ,hk = ωFS,p,k(ξ) +i
2π
∑
1≤s≤k
1
sxs
∑
i ,j ,α,β
cijαβ(z) usαusβ dzi ∧ dz j
where ωFS,k(ξ) is positive definite in ξ. The other terms are aweighted average of the values of the curvature tensor ΘV ,h onvectors us in the unit sphere bundle SV ⊂ V . The weightedprojective space can be viewed as a circle quotient of thepseudosphere
∑|ξs |
2p/s = 1, so we can take here xs ≥ 0,∑xs = 1. This is essentially a sum of the form
∑ 1sγ(us) where
us are random points of the sphere, and so as k → +∞ this canbe estimated by a “Monte-Carlo” integral
It follows that the leading term in the estimate only involves thetrace of ΘV ∗,h∗ , i.e. the curvature of (detV ∗, det h∗), which can betaken to be > 0 if detV ∗ is big.
Corollary (D-, 2010) Let (X ,V ) be a directed manifold, F → X aQ-line bundle, (V , h) and (F , hF ) hermitian. Define
Lk = OXGG
k(1)⊗ π∗kO
( 1
kr
(1 +
1
2+ . . .+
1
k
)F),
η = ΘdetV ∗,det h∗ +ΘF ,hF .
Then for all q ≥ 0 and all m ≫ k ≫ 1 such that m is sufficientlydivisible, we have
Using the above cohomological estimate, we obtain:
Theorem (D-, 2010) Let (X ,V ) be of general type, i.e.KV = (detV )∗ is a big line bundle. Then there exists k ≥ 1 andan algebraic hypersurface Z ( Xk such that every entire curvef : (C,TC) 7→ (X ,V ) satisfies f[k](C) ⊂ Z (in other words, fsatisfies an algebraic differential equation of order k).
Another important consequence is:
Theorem (D-, 2012) A generic hypersurface X ⊂ Pn+1 of degreed ≥ dn with
d2 = 286, d3 = 7316, dn =
⌊n4
3
(n log(n log(24n))
)n⌋
(for n ≥ 4) satisfies the Green-Griffiths conjecture.
The proof of the last result uses an important idea due toYum-Tong Siu, itself based on ideas of Claire Voisin and HerbClemens, and then refined by M. Paun [Pau08], E. Rousseau[Rou06b] and J. Merker [Mer09].The idea consists of studying vector fields on the relative jet spaceof the universal family of hypersurfaces of Pn+1.
Let X ⊂ Pn+1 × PNd be the universal hypersurface, i.e.
X = (z , a) ; a = (aα) s.t. Pa(z) =∑
aαzα = 0,
let Ω ⊂ PNd be the open subset of a’s for which Xa = Pa(z) = 0is smooth, and let
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