Algebro-differential embeddings of compact almost complex structures Jean-Pierre Demailly Institut Fourier, Universit´ e de Grenoble Alpes & Acad´ emie des Sciences de Paris Conference at ETH Z¨ urich “Analysis in the large, Calculus of Variations, Dynamics, Geometry, ...” in honour of Helmut Hofer, June 6–10, 2016 J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 1/16
Symplectic geometry and almost complex geometry An important part of Helmut Hofer’s work deals with symplectic geometry, and the geometry of pseudoholomorphic curves. J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 2/16
Symplectic geometry and almost complex geometry An important part of Helmut Hofer’s work deals with symplectic geometry, and the geometry of pseudoholomorphic curves. The subject was given a strong impetus by Mikhail Gromov in his famous Inventiones paper from 1985. J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 2/16
Symplectic geometry and almost complex geometry An important part of Helmut Hofer’s work deals with symplectic geometry, and the geometry of pseudoholomorphic curves. The subject was given a strong impetus by Mikhail Gromov in his famous Inventiones paper from 1985. Since then, many interconnections between symplectic geometry, topology and algebraic geometry have been developed, e.g. through the study of Gromov-Witten invariants. J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 2/16
Symplectic geometry and almost complex geometry An important part of Helmut Hofer’s work deals with symplectic geometry, and the geometry of pseudoholomorphic curves. The subject was given a strong impetus by Mikhail Gromov in his famous Inventiones paper from 1985. Since then, many interconnections between symplectic geometry, topology and algebraic geometry have been developed, e.g. through the study of Gromov-Witten invariants. Basic question Let ( M 2 n , ω ) be a compact symplectic manifold and J a compatible almost complex structure. Assume that � c 1 ( M , J ) ∧ ω n − 1 > 0 . M Is it true that there exists a differentiable family of mobile pseudoholomorphic curves ( f t ) t ∈ S : P 1 → M , i.e. generically injective and covering an open set in M ? J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 2/16
Existence of rational curves Related question Let ( X n , ω ) be a compact K¨ ahler manifold. Assume that that c 1 ( K X ) · ω n − 1 < 0 or more generally that K X is not pseudoeffective (this means that the class c 1 ( K X ) does not contain any closed (1 , 1)-current T ≥ 0. Can one conclude that X is covered by rational curves ? J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 3/16
Existence of rational curves Related question Let ( X n , ω ) be a compact K¨ ahler manifold. Assume that that c 1 ( K X ) · ω n − 1 < 0 or more generally that K X is not pseudoeffective (this means that the class c 1 ( K X ) does not contain any closed (1 , 1)-current T ≥ 0. Can one conclude that X is covered by rational curves ? This would be crucial for the theory of compact K¨ ahler manifolds. J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 3/16
Existence of rational curves Related question Let ( X n , ω ) be a compact K¨ ahler manifold. Assume that that c 1 ( K X ) · ω n − 1 < 0 or more generally that K X is not pseudoeffective (this means that the class c 1 ( K X ) does not contain any closed (1 , 1)-current T ≥ 0. Can one conclude that X is covered by rational curves ? This would be crucial for the theory of compact K¨ ahler manifolds. Theorem (BDPP = Boucksom - D - Peternell - Paun, 2002) The answer is positive when X is a complex projective manifold. J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 3/16
Existence of rational curves Related question Let ( X n , ω ) be a compact K¨ ahler manifold. Assume that that c 1 ( K X ) · ω n − 1 < 0 or more generally that K X is not pseudoeffective (this means that the class c 1 ( K X ) does not contain any closed (1 , 1)-current T ≥ 0. Can one conclude that X is covered by rational curves ? This would be crucial for the theory of compact K¨ ahler manifolds. Theorem (BDPP = Boucksom - D - Peternell - Paun, 2002) The answer is positive when X is a complex projective manifold. The proof uses intersection theory of currents and characteristic p techniques due to Mori. J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 3/16
Existence of rational curves Related question Let ( X n , ω ) be a compact K¨ ahler manifold. Assume that that c 1 ( K X ) · ω n − 1 < 0 or more generally that K X is not pseudoeffective (this means that the class c 1 ( K X ) does not contain any closed (1 , 1)-current T ≥ 0. Can one conclude that X is covered by rational curves ? This would be crucial for the theory of compact K¨ ahler manifolds. Theorem (BDPP = Boucksom - D - Peternell - Paun, 2002) The answer is positive when X is a complex projective manifold. The proof uses intersection theory of currents and characteristic p techniques due to Mori. It would be nice to have a “symplectic proof”, especially in the K¨ ahler case. J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 3/16
A question raised by Fedor Bogomolov Rough question Can one produce an arbitrary compact complex manifold X / an arbitrary compact K¨ ahler manifold X by means of a “purely algebraic construction” ? J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 4/16
A question raised by Fedor Bogomolov Rough question Can one produce an arbitrary compact complex manifold X / an arbitrary compact K¨ ahler manifold X by means of a “purely algebraic construction” ? Let Z be a projective algebraic manifold, dim C Z = N , equipped with a subbundle (or rather subsheaf) D ⊂ O Z ( T Z ). J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 4/16
A question raised by Fedor Bogomolov Rough question Can one produce an arbitrary compact complex manifold X / an arbitrary compact K¨ ahler manifold X by means of a “purely algebraic construction” ? Let Z be a projective algebraic manifold, dim C Z = N , equipped with a subbundle (or rather subsheaf) D ⊂ O Z ( T Z ). Assume that X 2 n is a compact C ∞ real even dimensional manifold that is embedded in Z , as follows: J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 4/16
A question raised by Fedor Bogomolov Rough question Can one produce an arbitrary compact complex manifold X / an arbitrary compact K¨ ahler manifold X by means of a “purely algebraic construction” ? Let Z be a projective algebraic manifold, dim C Z = N , equipped with a subbundle (or rather subsheaf) D ⊂ O Z ( T Z ). Assume that X 2 n is a compact C ∞ real even dimensional manifold that is embedded in Z , as follows: (i) f : X ֒ → Z is a smooth (say C ∞ ) embedding (ii) ∀ x ∈ X , f ∗ T X , x ⊕ D f ( x ) = T Z , f ( x ) . (iii) f ( X ) ∩ D sing = ∅ . We say that X ֒ → ( Z , D ) is a transverse embedding. J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 4/16
A conjecture of Bogomolov f ∗ T X , x = T M , f ( x ) ≃ T Z , f ( x ) / D f ( x ) Observation 1 If D ⊂ T Z is an algebraic foliation, i.e. [ D , D ] ⊂ D , then the almost complex structure J f on X induced by ( Z , D ) is integrable. J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 5/16
A conjecture of Bogomolov f ∗ T X , x = T M , f ( x ) ≃ T Z , f ( x ) / D f ( x ) Observation 1 If D ⊂ T Z is an algebraic foliation, i.e. [ D , D ] ⊂ D , then the almost complex structure J f on X induced by ( Z , D ) is integrable. Proof: J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 5/16
A conjecture of Bogomolov (2) Observation 2 If D ⊂ T Z is an algebraic foliation and f t : X ֒ → ( Z , D ) is an isotopy of transverse embeddings, t ∈ [0 , 1], then all complex structures ( X , J f t ) are biholomorphic. J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 6/16
A conjecture of Bogomolov (2) Observation 2 If D ⊂ T Z is an algebraic foliation and f t : X ֒ → ( Z , D ) is an isotopy of transverse embeddings, t ∈ [0 , 1], then all complex structures ( X , J f t ) are biholomorphic. Proof: J.-P. Demailly (Grenoble), Helmut Hofer’s 60th birthday, Z¨ urich Embeddings of compact almost complex structures 6/16
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