R´ echauffement climatique et ´ energie nucl´ eaire du futur, aspects soci´ etaux, physiques et math´ ematique Jean-Pierre Demailly Institut Fourier, Universit´ e de Grenoble I & Acad´ emie des Sciences de Paris 29 septembre 2015 Conf´ erence grand public, IUT de Nancy J.-P. Demailly (Grenoble), IUT de Nancy R´ echauffement climatique et ´ energie nucl´ eaire du futur Kobayashi hyperbolicity and entire curves Definition A complex space X is said to be Kobayashi hyperbolic if the Kobayashi pseudodistance d Kob : X × X → R + is a distance (i.e. everywhere non degenerate). By an entire curve we mean a non constant holomorphic map f : C → X into a complex n -dimensional manifold. Theorem (Brody, 1978) For a compact complex manifold X , dim C X = n , TFAE: (i) X is Kobayashi hyperbolic (ii) X is Brody hyperbolic, i.e. �∃ entire curves f : C → X (iii) The Kobayashi infinitesimal pseudometric is everywhere non degenerate Our interest is the study of hyperbolicity for projective varieties. In dim 1, X is hyperbolic iff genus g ≥ 2. J.-P. Demailly (Grenoble), IUT de Nancy R´ echauffement climatique et ´ energie nucl´ eaire du futur
Main conjectures Conjecture of General Type (CGT) • A compact complex variety X is volume hyperbolic iff X is of general type, i.e. K X is big. • In particular, this is so if X is Kobayashi (or Brody) hyperbolic; one expects K X to be ample in that case. Green-Griffiths-Lang Conjecture (GGL) Let X be a projective variety/ C of general type. Then ∃ Y � X algebraic such that all entire curves f : C → X satisfy f ( C ) ⊂ Y . Consequence of CGT + GGL A compact complex manifold X should be Kobayashi hyperbolic iff it is projective and every subvariety Y of X is of general type. Arithmetic counterpart (Lang 1987): If X is projective and defined over a number field, the smallest locus Y = GGL ( X ) in GGL’s conjecture is also the smallest Y such that X ( K ) � Y is finite ∀ K . J.-P. Demailly (Grenoble), IUT de Nancy R´ echauffement climatique et ´ energie nucl´ eaire du futur Results on the Kobayashi conjecture Kobayashi conjecture (1970) • Let X ⊂ P n +1 be a (very) generic hypersurface of degree d ≥ d n large enough. Then X is Kobayashi hyperbolic. • By a result of M. Zaidenberg, the optimal bound must satisfy d n ≥ 2 n + 1, and one expects d n = 2 n + 1. Using “jet technology” and deep results of McQuillan for curve foliations on surfaces, the following has been proved: Theorem (D., El Goul, 1998) A very generic surface X ⊂ P 3 of degree d ≥ 21 is hyperbolic. Independently McQuillan got d ≥ 35. This was more recently improved to d ≥ 18 (P˘ aun, 2008). In 2012, Yum-Tong Siu announced a proof of the case of arbitrary dimension n , with a very large d n (and a rather involved proof). J.-P. Demailly (Grenoble), IUT de Nancy R´ echauffement climatique et ´ energie nucl´ eaire du futur
Results on the generic Green-Griffiths conjecture By a combination of an algebraic existence theorem for jet differentials and of Siu’s technique of “slanted vector fields” (itself derived from ideas of H. Clemens, L. Ein and C. Voisin), the following was proved: Theorem (S. Diverio, J. Merker, E. Rousseau, 2009) A generic hypersurface X ⊂ P n +1 of degree d ≥ d n := 2 n 5 satisfies the GGL conjecture. The bound was improved by (D-, 2012) to � � n � � n 4 = O (exp( n 1+ ε )) , d n = n log( n log(24 n )) ∀ ε > 0. 3 Theorem (S. Diverio, S. Trapani, 2009) Additionally, a generic hypersurface X ⊂ P 4 of degree d ≥ 593 is hyperbolic. J.-P. Demailly (Grenoble), IUT de Nancy R´ echauffement climatique et ´ energie nucl´ eaire du futur Category of directed manifolds Goal. More generally, we are interested in curves f : C → X such that f ′ ( C ) ⊂ V where V is a subbundle of T X (or singular linear subspace, i.e. a closed irreducible analytic subspace such that ∀ x ∈ X , V x := V ∩ T X , x linear) . Definition. Category of directed manifolds : – Objects : pairs ( X , V ), X manifold/ C and V ⊂ T X – Arrows ψ : ( X , V ) → ( Y , W ) holomorphic s.t. ψ ∗ V ⊂ W – “Absolute case” ( X , T X ), i.e. V = T X – “Relative case” ( X , T X / S ) where X → S – “Integrable case” when [ V , V ] ⊂ V (foliations) Fonctor “1-jet” : ( X , V ) �→ ( ˜ X , ˜ V ) where : ˜ X = P ( V ) = bundle of projective spaces of lines in V π : ˜ X = P ( V ) → X , ( x , [ v ]) �→ x , v ∈ V x � � ˜ V ( x , [ v ]) = ξ ∈ T ˜ X , ( x , [ v ]) ; π ∗ ξ ∈ C v ⊂ T X , x J.-P. Demailly (Grenoble), IUT de Nancy R´ echauffement climatique et ´ energie nucl´ eaire du futur
Semple jet bundles For every entire curve f : ( C , T C ) → ( X , V ) tangent to V f [1] ( t ) := ( f ( t ) , [ f ′ ( t )]) ∈ P ( V f ( t ) ) ⊂ ˜ X f [1] : ( C , T C ) → ( ˜ X , ˜ V ) (projectivized 1st-jet) Definition. Semple jet bundles : – ( X k , V k ) = k -th iteration of fonctor ( X , V ) �→ ( ˜ X , ˜ V ) – f [ k ] : ( C , T C ) → ( X k , V k ) is the projectivized k -jet of f . Basic exact sequences π ⋆ X / X → ˜ ⇒ rk ˜ 0 → T ˜ → O ˜ X ( − 1) → 0 V = r = rk V V X → π ⋆ V ⊗ O ˜ 0 → O ˜ X (1) → T ˜ X / X → 0 (Euler) ( π k ) ⋆ 0 → T X k / X k − 1 → V k → O X k ( − 1) → 0 ⇒ rk V k = r 0 → O X k → π ⋆ k V k − 1 ⊗ O X k (1) → T X k / X k − 1 → 0 (Euler) J.-P. Demailly (Grenoble), IUT de Nancy R´ echauffement climatique et ´ energie nucl´ eaire du futur Direct image formula For n = dim X and r = rk V , one gets a tower of P r − 1 -bundles π k π 1 π k , 0 : X k → X k − 1 → · · · → X 1 → X 0 = X with dim X k = n + k ( r − 1), rk V k = r , and tautological line bundles O X k (1) on X k = P ( V k − 1 ). Theorem X k is a smooth compactification of X GG , reg / G k = J GG , reg / G k , where k k G k is the group of k -jets of germs of biholomorphisms of ( C , 0), acting on the right by reparametrization: ( f , ϕ ) �→ f ◦ ϕ , and J reg is the space k of k -jets of regular curves. Direct image formula ( π k , 0 ) ∗ O X k ( m ) = E k , m V ∗ = invariant algebraic differential operators f �→ P ( f [ k ] ) acting on germs of curves f : ( C , T C ) → ( X , V ). J.-P. Demailly (Grenoble), IUT de Nancy R´ echauffement climatique et ´ energie nucl´ eaire du futur
Definition of algebraic differential operators Let ( C , T C ) → ( X , V ) , t �→ f ( t ) = ( f 1 ( t ) , . . . , f n ( t )) be a curve written in some local holomorphic coordinates ( z 1 , . . . , z n ) on X . It has a local Taylor expansion ξ s = 1 f ( t ) = x + t ξ 1 + . . . + t k ξ k + O ( t k +1 ) , s ! ∇ s f (0) for some connection ∇ on V . k , m V ∗ of polynomials of One considers the Green-Griffiths bundle E GG weighted degree m written locally in coordinate charts as � a α 1 α 2 ...α k ( x ) ξ α 1 1 . . . ξ α k P ( x ; ξ 1 , . . . , ξ k ) = k , ξ s ∈ V , also viewed as algebraic differential operators P ( f ′ , f ′′ , . . . , f ( k ) ) P ( f [ k ] ) = � a α 1 α 2 ...α k ( f ( t )) f ′ ( t ) α 1 f ′′ ( t ) α 2 . . . f ( k ) ( t ) α k . = J.-P. Demailly (Grenoble), IUT de Nancy R´ echauffement climatique et ´ energie nucl´ eaire du futur Definition of algebraic differential operators [cont.] Here t �→ z = f ( t ) is a curve, f [ k ] = ( f ′ , f ′′ , . . . , f ( k ) ) its k -jet, and a α 1 α 2 ...α k ( z ) are supposed to holomorphic functions on X . The G k -action : ( f , ϕ ) �→ f ◦ ϕ , yields in particular, ϕ λ ( t ) = λ t ⇒ ( f ◦ ϕ λ ) ( k ) ( t ) = λ k f ( k ) ( λ t ), whence a C ∗ -action λ · ( ξ 1 , ξ 1 , . . . , ξ k ) = ( λξ 1 , λ 2 ξ 2 , . . . , λ k ξ k ) . E GG k , m is precisely the set of polynomials of weighted degree m , corresponding to coefficients a α 1 ...α k with m = | α 1 | + 2 | α 2 | + . . . + k | α k | . E k , m V ∗ ⊂ E GG k , m V ∗ is the bundle of G k -”invariant” operators, i.e. such that P (( f ◦ ϕ ) [ k ] ) = ϕ ′ m P ( f [ k ] ) ◦ ϕ, ∀ ϕ ∈ G k . J.-P. Demailly (Grenoble), IUT de Nancy R´ echauffement climatique et ´ energie nucl´ eaire du futur
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