On the cohomology of pseudoeffective line bundles Jean-Pierre Demailly Institut Fourier, Universit´ e de Grenoble I, France & Acad´ emie des Sciences de Paris in honor of Professor Yum-Tong Siu on the occasion of his 70th birthday Abel Symposium, NTNU Trondheim, July 2–5, 2013 1/21 [1:1] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Goals Study sections and cohomology of holomorphic line bundles L → X on compact K¨ ahler manifolds, without assuming any strict positivity of the curvature 2/21 [1:2] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Goals Study sections and cohomology of holomorphic line bundles L → X on compact K¨ ahler manifolds, without assuming any strict positivity of the curvature Generalize the Nadel vanishing theorem (and therefore Kawamata-Viehweg) 2/21 [2:3] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Goals Study sections and cohomology of holomorphic line bundles L → X on compact K¨ ahler manifolds, without assuming any strict positivity of the curvature Generalize the Nadel vanishing theorem (and therefore Kawamata-Viehweg) Several known results already in this direction: – Skoda division theorem (1972) 2/21 [3:4] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Goals Study sections and cohomology of holomorphic line bundles L → X on compact K¨ ahler manifolds, without assuming any strict positivity of the curvature Generalize the Nadel vanishing theorem (and therefore Kawamata-Viehweg) Several known results already in this direction: – Skoda division theorem (1972) – Ohsawa-Takegoshi L 2 extension theorem (1987) 2/21 [4:5] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Goals Study sections and cohomology of holomorphic line bundles L → X on compact K¨ ahler manifolds, without assuming any strict positivity of the curvature Generalize the Nadel vanishing theorem (and therefore Kawamata-Viehweg) Several known results already in this direction: – Skoda division theorem (1972) – Ohsawa-Takegoshi L 2 extension theorem (1987) – more recent work of Yum-Tong Siu: invariance of plurigenera (1998 → 2000), analytic version of Shokurov’s non vanishing theorem, finiteness of the canonical ring (2007), study of the abundance conjecture (2010) ... 2/21 [5:6] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Goals Study sections and cohomology of holomorphic line bundles L → X on compact K¨ ahler manifolds, without assuming any strict positivity of the curvature Generalize the Nadel vanishing theorem (and therefore Kawamata-Viehweg) Several known results already in this direction: – Skoda division theorem (1972) – Ohsawa-Takegoshi L 2 extension theorem (1987) – more recent work of Yum-Tong Siu: invariance of plurigenera (1998 → 2000), analytic version of Shokurov’s non vanishing theorem, finiteness of the canonical ring (2007), study of the abundance conjecture (2010) ... – solution of MMP (BCHM 2006), D-Hacon-P˘ aun (2010) 2/21 [6:7] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (1) Let X = compact K¨ ahler manifold, L → X holomorphic line bundle, h a hermitian metric on L . 3/21 [1:8] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (1) Let X = compact K¨ ahler manifold, L → X holomorphic line bundle, h a hermitian metric on L . Locally L | U ≃ U × C and for ξ ∈ L x ≃ C , � ξ � 2 h = | ξ | 2 e − ϕ ( x ) . 3/21 [2:9] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (1) Let X = compact K¨ ahler manifold, L → X holomorphic line bundle, h a hermitian metric on L . Locally L | U ≃ U × C and for ξ ∈ L x ≃ C , � ξ � 2 h = | ξ | 2 e − ϕ ( x ) . Writing h = e − ϕ locally, one defines the curvature form of L to be the real (1 , 1)-form Θ L , h = i 2 π∂∂ϕ = − dd c log h , 3/21 [3:10] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (1) Let X = compact K¨ ahler manifold, L → X holomorphic line bundle, h a hermitian metric on L . Locally L | U ≃ U × C and for ξ ∈ L x ≃ C , � ξ � 2 h = | ξ | 2 e − ϕ ( x ) . Writing h = e − ϕ locally, one defines the curvature form of L to be the real (1 , 1)-form Θ L , h = i 2 π∂∂ϕ = − dd c log h , � � ∈ H 2 ( X , Z ) . c 1 ( L ) = Θ L , h 3/21 [4:11] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (1) Let X = compact K¨ ahler manifold, L → X holomorphic line bundle, h a hermitian metric on L . Locally L | U ≃ U × C and for ξ ∈ L x ≃ C , � ξ � 2 h = | ξ | 2 e − ϕ ( x ) . Writing h = e − ϕ locally, one defines the curvature form of L to be the real (1 , 1)-form Θ L , h = i 2 π∂∂ϕ = − dd c log h , � � ∈ H 2 ( X , Z ) . c 1 ( L ) = Θ L , h Any subspace V m ⊂ H 0 ( X , L ⊗ m ) define a meromorphic map − → Φ mL : X � Z m P ( V m ) (hyperplanes of V m ) � � x �− → H x = σ ∈ V m ; σ ( x ) = 0 where Z m = base locus B ( mL ) = � σ − 1 (0). 3/21 [5:12] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (2) Given sections σ 1 , . . . , σ n ∈ H 0 ( X , L ⊗ m ), one gets a singular hermitian metric on L defined by | ξ | 2 | ξ | 2 h = � � | σ j ( x ) | 2 � 1 / m , 4/21 [1:13] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (2) Given sections σ 1 , . . . , σ n ∈ H 0 ( X , L ⊗ m ), one gets a singular hermitian metric on L defined by | ξ | 2 | ξ | 2 h = � � | σ j ( x ) | 2 � 1 / m , its weight is the plurisubharmonic (psh) function � � | σ j ( x ) | 2 � ϕ ( x ) = 1 m log 4/21 [2:14] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (2) Given sections σ 1 , . . . , σ n ∈ H 0 ( X , L ⊗ m ), one gets a singular hermitian metric on L defined by | ξ | 2 | ξ | 2 h = � � | σ j ( x ) | 2 � 1 / m , its weight is the plurisubharmonic (psh) function � � | σ j ( x ) | 2 � ϕ ( x ) = 1 m log m dd c log ϕ ≥ 0 and the curvature is Θ L , h = 1 in the sense of currents, with logarithmic poles along the base locus � σ − 1 j (0) = ϕ − 1 ( −∞ ) . B = 4/21 [3:15] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (2) Given sections σ 1 , . . . , σ n ∈ H 0 ( X , L ⊗ m ), one gets a singular hermitian metric on L defined by | ξ | 2 | ξ | 2 h = � � | σ j ( x ) | 2 � 1 / m , its weight is the plurisubharmonic (psh) function � � | σ j ( x ) | 2 � ϕ ( x ) = 1 m log m dd c log ϕ ≥ 0 and the curvature is Θ L , h = 1 in the sense of currents, with logarithmic poles along the base locus � σ − 1 j (0) = ϕ − 1 ( −∞ ) . B = One has (Θ L , h ) | X � B = 1 m Φ ∗ mL ω FS where Φ mL : X � B → P ( V m ) ≃ P N m . 4/21 [4:16] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (3) Definition L is pseudoeffective (psef) if ∃ h = e − ϕ , ϕ ∈ L 1 loc , (possibly singular) such that Θ L , h = − dd c log h ≥ 0 on X , in the sense of currents. 5/21 [1:17] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (3) Definition L is pseudoeffective (psef) if ∃ h = e − ϕ , ϕ ∈ L 1 loc , (possibly singular) such that Θ L , h = − dd c log h ≥ 0 on X , in the sense of currents. L is semipositive if ∃ h = e − ϕ smooth such that Θ L , h = − dd c log h ≥ 0 on X . 5/21 [2:18] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (3) Definition L is pseudoeffective (psef) if ∃ h = e − ϕ , ϕ ∈ L 1 loc , (possibly singular) such that Θ L , h = − dd c log h ≥ 0 on X , in the sense of currents. L is semipositive if ∃ h = e − ϕ smooth such that Θ L , h = − dd c log h ≥ 0 on X . L is positive if ∃ h = e − ϕ smooth such that Θ L , h = − dd c log h > 0 on X . 5/21 [3:19] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
Basic concepts (3) Definition L is pseudoeffective (psef) if ∃ h = e − ϕ , ϕ ∈ L 1 loc , (possibly singular) such that Θ L , h = − dd c log h ≥ 0 on X , in the sense of currents. L is semipositive if ∃ h = e − ϕ smooth such that Θ L , h = − dd c log h ≥ 0 on X . L is positive if ∃ h = e − ϕ smooth such that Θ L , h = − dd c log h > 0 on X . The well-known Kodaira embedding theorem states that L is positive if and only if L is ample, namely: Z m = B ( mL ) = ∅ and Φ | mL | : X → P ( H 0 ( X , L ⊗ m )) is an embedding for m ≥ m 0 large enough. 5/21 [4:20] Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
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