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 2/74 Definition. By an entire curve we mean a non constant holomorphic map f : C → X into a complex n -dimensional manifold. Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Entire curves 3/74 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 . Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Entire curves 4/74 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 Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Entire curves 5/74 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) Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Entire curves 6/74 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
Projective algebraic varieties 7/74 Consider now the complex projective n -space P n = P n C = ( C n +1 � { 0 } ) / C ∗ , [ z ] = [ z 0 : z 1 : . . . : z n ] . Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Projective algebraic varieties 8/74 Consider now the complex projective n -space P n = P n C = ( C n +1 � { 0 } ) / C ∗ , [ z ] = [ z 0 : z 1 : . . . : z n ] . An entire curve f : C → P n is given by a map t �− → [ f 0 ( t ) : f 1 ( t ) : . . . : f n ( t )] where f j : C → C are holomorphic functions without common zeroes (so there are a lot of them). Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Projective algebraic varieties 9/74 Consider now the complex projective n -space P n = P n C = ( C n +1 � { 0 } ) / C ∗ , [ z ] = [ z 0 : z 1 : . . . : z n ] . An entire curve f : C → P n is given by a map t �− → [ f 0 ( t ) : f 1 ( t ) : . . . : f n ( t )] where f j : C → C are holomorphic functions without common zeroes (so there are a lot of them). More generally, look at a (complex) projective manifold, i.e. X n ⊂ P N , X = { [ z ] ; P 1 ( z ) = ... = P k ( z ) = 0 } where P j ( z ) = P j ( z 0 , z 1 , . . . , z N ) are homogeneous polynomials (of some degree d j ), such that X is non singular. Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Complex curves (genus 0 and 1) 10/74 Canonical bundle K X = Λ n T ∗ X (here K X = T ∗ X ) Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Complex curves (genus 0 and 1) 11/74 Canonical bundle K X = Λ n T ∗ X (here K X = T ∗ X ) g = 0 : X = P 1 courbure T X > 0 not hyperbolic Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Complex curves (genus 0 and 1) 12/74 Canonical bundle K X = Λ n T ∗ X (here K X = T ∗ X ) g = 0 : X = P 1 courbure T X > 0 not hyperbolic g = 1 : X = C / ( Z + Z τ ) courbure T X = 0 not hyperbolic Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Complex curves (genus g ≥ 2) 13/74 deg K X = 2 g − 2 If g ≥ 2 , X ≃ D / Γ ( T X < 0) ⇒ X is hyperbolic. Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Complex curves (genus g ≥ 2) 14/74 deg K X = 2 g − 2 If g ≥ 2 , X ≃ D / Γ ( T X < 0) ⇒ X is hyperbolic. In fact every curve f : C → X ≃ D / Γ lifts to � f : C → D , and so must be constant by Liouville. Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Kobayashi metric / hyperbolic manifolds 15/74 For a complex manifold, n = dim C X , one defines the Kobayashi pseudo-metric : x ∈ X , ξ ∈ T X κ x ( ξ ) = inf { λ > 0 ; ∃ f : D → X , f (0) = x , λ f ∗ (0) = ξ } On C n , P n or complex tori X = C n / Λ, one has κ X ≡ 0 . Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Kobayashi metric / hyperbolic manifolds 16/74 For a complex manifold, n = dim C X , one defines the Kobayashi pseudo-metric : x ∈ X , ξ ∈ T X κ x ( ξ ) = inf { λ > 0 ; ∃ f : D → X , f (0) = x , λ f ∗ (0) = ξ } On C n , P n or complex tori X = C n / Λ, one has κ X ≡ 0 . X is said to be hyperbolic in the sense of Kobayashi if the associated integrated pseudo-distance is a distance (i.e. it separates points – i.e. has Hausdorff topology). Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Kobayashi metric / hyperbolic manifolds 17/74 For a complex manifold, n = dim C X , one defines the Kobayashi pseudo-metric : x ∈ X , ξ ∈ T X κ x ( ξ ) = inf { λ > 0 ; ∃ f : D → X , f (0) = x , λ f ∗ (0) = ξ } On C n , P n or complex tori X = C n / Λ, one has κ X ≡ 0 . X is said to be hyperbolic in the sense of Kobayashi if the associated 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 = X 1 × . . . × X s where X j hyperbolic. Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Kobayashi metric / hyperbolic manifolds 18/74 For a complex manifold, n = dim C X , one defines the Kobayashi pseudo-metric : x ∈ X , ξ ∈ T X κ x ( ξ ) = inf { λ > 0 ; ∃ f : D → X , f (0) = x , λ f ∗ (0) = ξ } On C n , P n or complex tori X = C n / Λ, one has κ X ≡ 0 . X is said to be hyperbolic in the sense of Kobayashi if the associated 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 = X 1 × . . . × X s where X j hyperbolic. Theorem (dimension n arbitrary) (Kobayashi, 1970) T X negatively curved ( T ∗ X > 0 , i.e. ample ) ⇒ X hyperbolic. Recall that a holomorphic vector bundle E is ample iff its symmetric powers S m E have global sections which generate 1-jets of (germs of) sections at any point x ∈ X . Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
Ahlfors-Schwarz lemma 19/74 The proof of the above Kobayashi result depends crucially on: Ahlfors-Schwarz lemma. Let γ = i � γ jk dt j ∧ dt k be an almost everywhere positive hermitian form on the ball B (0 , R ) ⊂ C p , such that − Ricci ( γ ) := i ∂∂ log det γ ≥ A γ in the sense of currents, for some constant A > 0 (this means in particular that det γ = det( γ jk ) is such that log det γ is plurisubharmonic). Then the γ -volume form is controlled by the Poincar´ e volume form : � p + 1 � p 1 det( γ ) ≤ (1 − | t | 2 / R 2 ) p +1 . AR 2 Jean-Pierre Demailly (Grenoble), VIASM, Hanoi, 26/08/2012 Hyperbolic algebraic varieties & holomorphic differential equations
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