On the cohomology of pseudoeffective line bundles

On the cohomology of pseudoeffective line bundles Jean-Pierre Demailly Institut Fourier, Université de Grenoble I, France & Académie 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 Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 1/21 [1:1]

Goals Study sections and cohomology of holomorphic line bundles L X on compact Kähler manifolds, without assuming any strict positivity of the curvature Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21 [1:2]

Goals Study sections and cohomology of holomorphic line bundles L X on compact Kähler manifolds, without assuming any strict positivity of the curvature Generalize the Nadel vanishing theorem (and therefore Kawamata-Viehweg) Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21 [2:3]

Goals Study sections and cohomology of holomorphic line bundles L X on compact Kähler 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)... Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21 [5:6]

Goals Study sections and cohomology of holomorphic line bundles L X on compact Kähler 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-Păun (2010) Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21 [6:7]

Basic concepts (1) Let X = compact Kähler manifold, L X holomorphic line bundle, h a hermitian metric on L. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21 [1:8]

Basic concepts (1) Let X = compact Kähler 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). Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21 [2:9]

Basic concepts (1) Let X = compact Kähler 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 2π ϕ = ddc logh, Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21 [3:10]

Basic concepts (1) Let X = compact Kähler 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 2π ϕ = ddc logh, c 1 (L) = { Θ L,h } H 2 (X,Z). Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21 [4:11]

Basic concepts (1) Let X = compact Kähler 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 2π ϕ = ddc logh, c 1 (L) = { Θ L,h } H 2 (X,Z). 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). Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21 [5:12]

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 h = ξ 2 ( σj (x) 2) 1/m, Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 4/21 [1:13]

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 h = ξ 2 ( σj (x) 2) 1/m, its weight is the plurisubharmonic (psh) function ϕ(x) = 1 m log ( σj (x) 2 ) and the curvature is Θ L,h = 1 m ddc logϕ 0 in the sense of currents, with logarithmic poles along the base locus B = σ 1 j (0) = ϕ 1 ( ). Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 4/21 [3:15]

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 h = ξ 2 ( σj (x) 2) 1/m, its weight is the plurisubharmonic (psh) function ϕ(x) = 1 m log ( σj (x) 2 ) and the curvature is Θ L,h = 1 m ddc logϕ 0 in the sense of currents, with logarithmic poles along the base locus B = σ 1 j (0) = ϕ 1 ( ). One has (Θ L,h ) X B = 1 m Φ ml ω FS where Φ ml : X B P(V m ) P Nm. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 4/21 [4:16]

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 logh 0 on X, in the sense of currents. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 5/21 [1:17]

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 logh 0 on X, in the sense of currents. L is semipositive if h = e ϕ smooth such that Θ L,h = dd c logh 0 on X. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 5/21 [2:18]

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 logh 0 on X, in the sense of currents. L is semipositive if h = e ϕ smooth such that Θ L,h = dd c logh 0 on X. L is positive if h = e ϕ smooth such that Θ L,h = dd c logh > 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. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 5/21 [4:20]

Positive cones Definitions Let X be a compact Kähler manifold. The Kähler cone is the (open) set K H 1,1 (X,R) of cohomology classes {ω} of positive Kähler forms. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 6/21 [1:21]

Positive cones Definitions Let X be a compact Kähler manifold. The Kähler cone is the (open) set K H 1,1 (X,R) of cohomology classes {ω} of positive Kähler forms. The pseudoeffective cone is the set E H 1,1 (X,R) of cohomology classes {T} of closed positive (1, 1) currents. This is a closed convex cone. (by weak compactness of bounded sets of currents). K is the cone of nef classes. One has K E. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 6/21 [3:23]

Positive cones Definitions Let X be a compact Kähler manifold. The Kähler cone is the (open) set K H 1,1 (X,R) of cohomology classes {ω} of positive Kähler forms. The pseudoeffective cone is the set E H 1,1 (X,R) of cohomology classes {T} of closed positive (1, 1) currents. This is a closed convex cone. (by weak compactness of bounded sets of currents). K is the cone of nef classes. One has K E. It may happen that K E: if X is the surface obtained by blowing-up P 2 in one point, then the exceptional divisor E P 1 has a cohomology class {α} such that E α = E2 = 1, hence {α} / K, although {α} = {[E]} E. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 6/21 [4:24]

Ample / nef / effective / big divisors Positive cones can be visualized as follows : H 1,1 (X,R) (containing Neron-Severi space NS R (X)) K Kähler cone K NS ample divisors: K NS nef divisors: K NS E psef cone NS R (X) E NS big divisors: E NS effective & psef: E NS K NS = K NS R (X) E NS = E NS R (X) where NS R (X) = (H 1,1 (X,R) H 2 (X,Z)) Z R Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 7/21 [1:25]

Approximation of currents, Zariski decomposition Definition On X compact Kähler, a Kähler current T is a closed positive (1,1)-current T such that T δω for some smooth hermitian metric ω and a constant δ 1. Easy observation α E (interior of E) α = {T}, T = a Kähler current. We say that E is the cone of big (1,1)-classes. Theorem on approximate Zariski decomposition (D, 92) Any Kähler current can be written T = limt m where T m {T} has analytic singularities & logarithmic poles, i.e. modification µ m : X m X such that µ m T m = [E m ]+β m where E m is an effective Q-divisor on X m with coefficients in 1 m Z and β m is a Kähler form on X m. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 8/21 [2:28]

Schematic picture of Zariski decomposition K( X m ) K(X) ω µ m T m T β m E( X m ) T m E(X) E m T δω 0 NS R ( X m ) H 1,1 ( X m,r) µ m : X m X (blow-up) NS R (X) H 1,1 (X,R) Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 9/21 [1:29]

Idea of proof of analytic Zariski decomposition Write locally T = i ϕ for some strictly plurisubharmonic psh potential ϕ on X. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 10/21 [1:30]

Idea of proof of analytic Zariski decomposition Write locally T = i ϕ for some strictly plurisubharmonic psh potential ϕ on X. Approximate T (again locally) as T m = i ϕ m, ϕ m (z) = 1 2m log g l,m (z) 2 l where (g l,m ) is a Hilbert basis of the space H(Ω,mϕ) = { f O(Ω); f 2 e 2mϕ dv < + }. The Ohsawa-Takegoshi L 2 extension theorem (extending from a single isolated point) implies that there are enough such holomorphic functions, and thus ϕ m ϕ C/m. Ω Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 10/21 [3:32]

Idea of proof of analytic Zariski decomposition Write locally T = i ϕ for some strictly plurisubharmonic psh potential ϕ on X. Approximate T (again locally) as T m = i ϕ m, ϕ m (z) = 1 2m log g l,m (z) 2 l where (g l,m ) is a Hilbert basis of the space H(Ω,mϕ) = { f O(Ω); f 2 e 2mϕ dv < + }. The Ohsawa-Takegoshi L 2 extension theorem (extending from a single isolated point) implies that there are enough such holomorphic functions, and thus ϕ m ϕ C/m. Further, ϕ = lim m + ϕ m by the mean value inequality. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 10/21 [4:33] Ω

Movable intersection of currents Let P(X) = closed positive (1,1)-currents on X H k,k 0 (X) = { {T} H k,k (X,R); T closed 0 }. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 11/21 [1:34]

Movable intersection of currents Let P(X) = closed positive (1,1)-currents on X H k,k 0 (X) = { {T} H k,k (X,R); T closed 0 }. Theorem (Boucksom PhD 2002, Junyan Cao PhD 2012) k = 1,2,...,n, canonical movable intersection product P P H k,k 0 (X), (T 1,...,T k ) T 1 T 2 T k Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 11/21 [2:35]

Movable intersection of currents Let P(X) = closed positive (1,1)-currents on X H k,k 0 (X) = { {T} H k,k (X,R); T closed 0 }. Theorem (Boucksom PhD 2002, Junyan Cao PhD 2012) k = 1,2,...,n, canonical movable intersection product P P H k,k 0 (X), (T 1,...,T k ) T 1 T 2 T k Method. T j = lim ε 0 T j +εω, can assume T j Kähler. Approximate each T j by Kähler currents T j,m with logarithmic poles,take a simultaneous log-resolution µ m : X m X such that µ m T j = [E j,m ]+β j,m. and define T 1 T 2 T k = lim {(µ m ) (β 1,m β 2,m... β k,m )}. m + Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 11/21 [3:36]

Volume and numerical dimension of currents Remark. The limit exists a weak limit of currents thanks to uniform boundedness in mass. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 12/21 [1:37]

Volume and numerical dimension of currents Remark. The limit exists a weak limit of currents thanks to uniform boundedness in mass. Uniqueness comes from monotonicity (β j,m increases with m) Special case. The volume of a class α H 1,1 (X,R) is Vol(α) = sup T n if α E (big class), T α Vol(α) = 0 if α E, Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 12/21 [3:39]

Volume and numerical dimension of currents Remark. The limit exists a weak limit of currents thanks to uniform boundedness in mass. Uniqueness comes from monotonicity (β j,m increases with m) Special case. The volume of a class α H 1,1 (X,R) is Vol(α) = sup T n if α E (big class), T α Vol(α) = 0 if α E, Numerical dimension of a current nd(t) = max { p N; T p 0 in H p,p 0 (X)}. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 12/21 [4:40]

Volume and numerical dimension of currents Remark. The limit exists a weak limit of currents thanks to uniform boundedness in mass. Uniqueness comes from monotonicity (β j,m increases with m) Special case. The volume of a class α H 1,1 (X,R) is Vol(α) = sup T n if α E (big class), T α Vol(α) = 0 if α E, Numerical dimension of a current nd(t) = max { p N; T p 0 in H p,p 0 (X)}. Numerical dimension of a hermitian line bundle (L, h) nd(l,h) = nd(θ L,h ). Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 12/21 [4:41]

Generalized abundance conjecture Numerical dimension of a class α H 1,1 (X,R) If α is not pseudoeffective, set nd(α) =, otherwise nd(α) = max { p N; T ε {α+εω}, lim ε 0 T p ε ωn p C>0 }. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 13/21 [1:42]

Generalized abundance conjecture Numerical dimension of a class α H 1,1 (X,R) If α is not pseudoeffective, set nd(α) =, otherwise nd(α) = max { p N; T ε {α+εω}, lim ε 0 T p ε ωn p C>0 }. Numerical dimension of a pseudo-effective line bundle nd(l) = nd(c 1 (L)). L is said to be abundant if κ(l) = nd(l). Subtlety! Let E be the rank 2 v.b. = non trivial extension 0 O C E O C 0 on C = elliptic curve, let X = P(E) (ruled surface over C) and L = O P(E) (1). Then nd(l) = 1 but! positive current T = [σ(c)] c 1 (L) and nd(t) = 0!! Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 13/21 [3:44]

Generalized abundance conjecture Numerical dimension of a class α H 1,1 (X,R) If α is not pseudoeffective, set nd(α) =, otherwise nd(α) = max { p N; T ε {α+εω}, lim ε 0 T p ε ωn p C>0 }. Numerical dimension of a pseudo-effective line bundle nd(l) = nd(c 1 (L)). L is said to be abundant if κ(l) = nd(l). Subtlety! Let E be the rank 2 v.b. = non trivial extension 0 O C E O C 0 on C = elliptic curve, let X = P(E) (ruled surface over C) and L = O P(E) (1). Then nd(l) = 1 but! positive current T = [σ(c)] c 1 (L) and nd(t) = 0!! Generalized abundance conjecture For X compact Kähler, K X is abundant, i.e. κ(x) = nd(k X ). Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 13/21 [3:45]

Hard Lefschetz theorem with pseudoeffective coefficients Let (L,h) be a pseudo-effective line bundle on a compact Kähler manifold (X,ω) of dimension n, and for h = e ϕ, let I(h) = I(ϕ) be the multiplier ideal sheaf: I(ϕ) x := { f O X,x ; V x, f 2 e ϕ dv ω < + }. V Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 14/21 [1:46]

Hard Lefschetz theorem with pseudoeffective coefficients Let (L,h) be a pseudo-effective line bundle on a compact Kähler manifold (X,ω) of dimension n, and for h = e ϕ, let I(h) = I(ϕ) be the multiplier ideal sheaf: I(ϕ) x := { f O X,x ; V x, f 2 e ϕ dv ω < + }. The Nadel vanishing theorem claims that Θ L,h εω = H q (X,K X L I(h) = 0 for q 1. V Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 14/21 [2:47]

Hard Lefschetz theorem with pseudoeffective coefficients Let (L,h) be a pseudo-effective line bundle on a compact Kähler manifold (X,ω) of dimension n, and for h = e ϕ, let I(h) = I(ϕ) be the multiplier ideal sheaf: I(ϕ) x := { f O X,x ; V x, f 2 e ϕ dv ω < + }. The Nadel vanishing theorem claims that Θ L,h εω = H q (X,K X L I(h) = 0 for q 1. Hard Lefschetz theorem (D-Peternell-Schneider 2001) Assume merely Θ L,h 0. Then, the Lefschetz map : u ω q u induces a surjective morphism: Φ q ω,h : H0 (X,Ω n q X V L I(h)) H q (X,Ω n X L I(h)). Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 14/21 [2:48]

Idea of proof of Hard Lefschetz theorem Main tool. Equisingular approximation theorem : ϕ = lim ϕ ν h = limh ν with: ϕ ν C (X Z ν ), where Z ν is an increasing sequence of analytic sets, I(h ν ) = I(h), ν, Θ L,hν ε ν ω. (Again, the proof uses in several ways the Ohsawa-Takegoshi theorem). Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 15/21 [1:49]

Idea of proof of Hard Lefschetz theorem Main tool. Equisingular approximation theorem : ϕ = lim ϕ ν h = limh ν with: ϕ ν C (X Z ν ), where Z ν is an increasing sequence of analytic sets, I(h ν ) = I(h), ν, Θ L,hν ε ν ω. (Again, the proof uses in several ways the Ohsawa-Takegoshi theorem). Then, use the fact that X Z ν is Kähler complete, so one can apply (non compact) harmonic form theory on X Z ν, and pass to the limit to get rid of the errors ε ν. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 15/21 [2:50]

Generalized Nadel vanishing theorem Theorem (Junyan Cao, PhD 2012) Let X be compact Kähler, and let (L,h) be pseudoeffective on X. Then where H q (X,K X L I + (h)) = 0 for q n nd(l,h)+1, I + (h) = lim ε 0 I(h 1+ε ) = lim ε 0 I((1+ε)ϕ) is the upper semicontinuous regularization of I(h). Remark 1. Conjecturally I + (h) = I(h). This might follow from recent work by Bo Berndtsson on the openness conjecture. Remark 2. In the projective case, one can use a hyperplane section argument, provided one first shows that nd(l, h) coincides with H. Tsuji s algebraic definition (dimy = p) : nd(l,h) = max { p N; Y p X, h 0 (Y,(L m I(h m )) V ) cm p}. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 16/21 [3:53]

Proof of generalized Nadel vanishing(projective case) Hyperplane section argument (projective case). Take A = very ample divisor, ω = Θ A,hA > 0, and Y = A 1... A n p, A j A. Then Θ p L,h Y = Θ p L,h Y = Θ p L,h ωn p > 0. X From this one concludes that (Θ L,h ) Y is big. X Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 17/21 [1:54]

Proof of generalized Nadel vanishing(projective case) Hyperplane section argument (projective case). Take A = very ample divisor, ω = Θ A,hA > 0, and Y = A 1... A n p, A j A. Then Θ p L,h Y = Θ p L,h Y = Θ p L,h ωn p > 0. X From this one concludes that (Θ L,h ) Y is big. Lemma (J. Cao) When (L,h) is big, i.e. Θ n L,h > 0, there exists a metric h such that I( h) = I + (h) with Θ L, h εω [Riemann-Roch]. Then Nadel H q (X,K X L I + (h)) = 0 for q 1. X Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 17/21 [2:55]

Proof of generalized Nadel vanishing(projective case) Hyperplane section argument (projective case). Take A = very ample divisor, ω = Θ A,hA > 0, and Y = A 1... A n p, A j A. Then Θ p L,h Y = Θ p L,h Y = Θ p L,h ωn p > 0. X From this one concludes that (Θ L,h ) Y is big. Lemma (J. Cao) When (L,h) is big, i.e. Θ n L,h > 0, there exists a metric h such that I( h) = I + (h) with Θ L, h εω [Riemann-Roch]. Then Nadel H q (X,K X L I + (h)) = 0 for q 1. Conclude by induction on dim X and the exact cohomology sequence for the restriction to a hyperplane section. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 17/21 [3:56] X

Proof of generalized Nadel vanishing (Kähler case) Kähler case. Assume c 1 (L) nef for simplicity. Then c 1 (L)+εω Kähler. By Yau s theorem, solve Monge-Ampère equation: h ε on L, (Θ L,hε +εω) n = C ε ω n. Here C ε ( ) n p Θ p L,h (εω)n p Cε n p, p = nd(l,h). Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 18/21 [1:57]

Proof of generalized Nadel vanishing (Kähler case) Kähler case. Assume c 1 (L) nef for simplicity. Then c 1 (L)+εω Kähler. By Yau s theorem, solve Monge-Ampère equation: h ε on L, (Θ L,hε +εω) n = C ε ω n. Here C ε ( ) n p Θ p L,h (εω)n p Cε n p, p = nd(l,h). Ch. Mourougane argument (PhD 1996). Let λ 1... λ n be the eigenvalues of Θ L,h +εω w.r.to ω. Then and X λ q+1...λ n ω n = λ 1...λ n = C ε Const ε n p X Θ n q L,h ωq Const, q 1, Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 18/21 [2:58]

Proof of generalized Nadel vanishing (Kähler case) Kähler case. Assume c 1 (L) nef for simplicity. Then c 1 (L)+εω Kähler. By Yau s theorem, solve Monge-Ampère equation: h ε on L, (Θ L,hε +εω) n = C ε ω n. Here C ε ( ) n p Θ p L,h (εω)n p Cε n p, p = nd(l,h). Ch. Mourougane argument (PhD 1996). Let λ 1... λ n be the eigenvalues of Θ L,h +εω w.r.to ω. Then and X λ q+1...λ n ω n = λ 1...λ n = C ε Const ε n p X Θ n q L,h ωq Const, q 1, so λ q+1...λ n C on a large open set U X and λ q q λ 1...λ q cε n p λ q cε (n p)/q on U, q j=1 (λ j ε) λ q qε cε (n p)/q qε > 0 for q > n p. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 18/21 [3:59]

Final step: use Bochner-Kodaira formula λ j = eigenvalues of (Θ L,hε +εω) (eigenvalues of Θ L,hε ) = λ j ε. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 19/21 [1:60]

Final step: use Bochner-Kodaira formula λ j = eigenvalues of (Θ L,hε +εω) (eigenvalues of Θ L,hε ) = λ j ε. Bochner-Kodaira formula yields ( q ) u 2 ε + u 2 ε (λ j ε) u 2 e ϕε dv ω. X j=1 Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 19/21 [2:61]

Final step: use Bochner-Kodaira formula λ j = eigenvalues of (Θ L,hε +εω) (eigenvalues of Θ L,hε ) = λ j ε. Bochner-Kodaira formula yields u 2 ε + u 2 ε X ( q ) (λ j ε) u 2 e ϕε dv ω. j=1 Then one has to show that one can take the limit by assuming integrability with e (1+δ)ϕ, thus introducing I + (h). Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 19/21 [3:62]

Application to Kähler geometry Definition (Campana) A compact Kähler manifold is said to be simple if there are no positive dimensional analytic sets A x X through a very generic point x X. Well-known fact A complex torus X = C n /Λ defined by a sufficiently generic lattice Λ C n is simple, and in fact has no positive dimensional analytic subset A X at all. In fact [A] would define a non zero (p,p)-cohomology class with integral periods, and there are no such classes in general. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 20/21 [3:65]

Application to Kähler geometry Definition (Campana) A compact Kähler manifold is said to be simple if there are no positive dimensional analytic sets A x X through a very generic point x X. Well-known fact A complex torus X = C n /Λ defined by a sufficiently generic lattice Λ C n is simple, and in fact has no positive dimensional analytic subset A X at all. In fact [A] would define a non zero (p,p)-cohomology class with integral periods, and there are no such classes in general. It is expected that simple compact Kähler manifolds are either generic complex tori, generic hyperkähler manifolds and their finite quotients, up to modification. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 20/21 [4:66]

On simple Kähler 3-folds Theorem (Campana - D - Verbitsky, 2013) Let X be a compact Kähler 3-fold without any positive dimensional analytic subset A X. Then X is a complex 3-dimensional torus. Sketch of proof Every pseudoeffective class is nef, i.e. K = E (D, 90) K X is pseudoeffective: otherwise X would be covered by rational curves (Brunella 2008), hence in fact nef. All multiplier ideal sheaves I(h) are trivial Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21 [3:69]

On simple Kähler 3-folds Theorem (Campana - D - Verbitsky, 2013) Let X be a compact Kähler 3-fold without any positive dimensional analytic subset A X. Then X is a complex 3-dimensional torus. Sketch of proof Every pseudoeffective class is nef, i.e. K = E (D, 90) K X is pseudoeffective: otherwise X would be covered by rational curves (Brunella 2008), hence in fact nef. All multiplier ideal sheaves I(h) are trivial H 0 (X,Ω n q X K m 1 X ) H q (X,K m X ) is surjective Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21 [4:70]

On simple Kähler 3-folds Theorem (Campana - D - Verbitsky, 2013) Let X be a compact Kähler 3-fold without any positive dimensional analytic subset A X. Then X is a complex 3-dimensional torus. Sketch of proof Every pseudoeffective class is nef, i.e. K = E (D, 90) K X is pseudoeffective: otherwise X would be covered by rational curves (Brunella 2008), hence in fact nef. All multiplier ideal sheaves I(h) are trivial H 0 (X,Ω n q X K m 1 X ) H q (X,K m X ) is surjective Hilbert polynomial P(m) = χ(x,k m X ) is bounded, hence χ(x,o X ) = 0. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21 [5:71]

On simple Kähler 3-folds Theorem (Campana - D - Verbitsky, 2013) Let X be a compact Kähler 3-fold without any positive dimensional analytic subset A X. Then X is a complex 3-dimensional torus. Sketch of proof Every pseudoeffective class is nef, i.e. K = E (D, 90) K X is pseudoeffective: otherwise X would be covered by rational curves (Brunella 2008), hence in fact nef. All multiplier ideal sheaves I(h) are trivial H 0 (X,Ω n q X K m 1 X ) H q (X,K m X ) is surjective Hilbert polynomial P(m) = χ(x,k m X ) is bounded, hence χ(x,o X ) = 0. Albanese map α : X Alb(X) is a biholomorphism. Jean-Pierre Demailly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21 [6:72]

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