On the cohomology of pseudoeffective line bundles

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1 On the cohomology of pseudoeffective line bundles Jean-Pierre D ly 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 1/21 [1:1]

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

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) Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21 [2:3]

4 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) Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21 [3:4]

5 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) Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21 [4:5]

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 ( ), analytic version of Shokurov s non vanishing theorem, finiteness of the canonical ring (2007), study of the abundance conjecture (2010)... Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21 [5:6]

7 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 ( ), 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21 [6:7]

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

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). Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21 [2:9]

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, Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21 [3:10]

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). Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21 [4:11]

12 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21 [5:12]

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, Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 4/21 [1:13]

14 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 ) Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 4/21 [2:14]

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 ( ). Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 4/21 [3:15]

16 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 4/21 [4:16]

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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 5/21 [1:17]

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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 5/21 [2:18]

19 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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 5/21 [3:19]

20 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 5/21 [4:20]

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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 6/21 [1:21]

22 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). Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 6/21 [2:22]

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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 6/21 [3:23]

24 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 6/21 [4:24]

25 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 7/21 [1:25]

26 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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 8/21 [1:26]

27 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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 8/21 [2:27]

28 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 8/21 [2:28]

29 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 9/21 [1:29]

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

31 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 < + }. Ω Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 10/21 [2:31]

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. Ω Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 10/21 [3:32]

33 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 10/21 [4:33] Ω

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 }. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 11/21 [1:34]

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 Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 11/21 [2:35]

36 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 11/21 [3:36]

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

38 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) Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 12/21 [2:38]

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, Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 12/21 [3:39]

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)}. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 12/21 [4:40]

41 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 12/21 [4:41]

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 }. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 13/21 [1:42]

43 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). Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 13/21 [2:43]

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!! Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 13/21 [3:44]

45 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 13/21 [3:45]

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 < + }. V Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 14/21 [1:46]

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. V Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 14/21 [2:47]

48 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 14/21 [2:48]

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). Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 15/21 [1:49]

50 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 15/21 [2:50]

51 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). Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 16/21 [1:51]

52 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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 16/21 [2:52]

53 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 16/21 [3:53]

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. X Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 17/21 [1:54]

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. X Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 17/21 [2:55]

56 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 17/21 [3:56] X

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). Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 18/21 [1:57]

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, Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 18/21 [2:58]

59 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 18/21 [3:59]

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

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 ( q ) u 2 ε + u 2 ε (λ j ε) u 2 e ϕε dv ω. X j=1 Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 19/21 [2:61]

62 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 19/21 [3:62]

63 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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 20/21 [1:63]

64 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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 20/21 [2:64]

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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 20/21 [3:65]

66 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 20/21 [4:66]

67 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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21 [1:67]

68 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) Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21 [2:68]

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 Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21 [3:69]

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 Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21 [4:70]

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. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21 [5:71]

72 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 D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21 [6:72]

73 References [BCHM10] Birkar, C., Cascini, P., Hacon, C., McKernan, J.: Existence of minimal models for varieties of log general type, arxiv: [math.ag], J. Amer. Math. Soc. 23 (2010) [Bou02] Boucksom, S.: Cônes positifs des variétés complexes compactes, Thesis, Grenoble, [BDPP13] Boucksom, S., D ly, J.-P., Paun M., Peternell, Th.: The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, arxiv: [math.ag]; J. Algebraic Geom. 22 (2013) [CDV13] Campana, F., D ly, J.-P., Verbitsky, M.: Compact Kähler 3-manifolds without non-trivial subvarieties, hal , arxiv: [math.cv]. Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles Ref. 1 [73]

74 [Cao12] Cao, J.: Numerical dimension and a Kawamata-Viehweg-Nadel type vanishing theorem on compact Kähler manifolds, arxiv: [math.ag] [Dem91] D ly, J.-P. : Transcendental proof of a generalized Kawamata-Viehweg vanishing theorem, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989) ; Proceedings of the Conference Geometrical and algebraical aspects in several complex variables held at Cetraro (Italy), June 1989, edited by C.A. Berenstein and D.C. Struppa, EditEl, Rende (1991). [Dem92] D ly, J.-P. : Regularization of closed positive currents and Intersection Theory, J. Alg. Geom. 1 (1992) [DHP10] D ly, J.-P., Hacon, C., Păun, M.: Extension theorems, Non-vanishing and the existence of good minimal models, arxiv: [math.ag], to appear in Acta Math Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles Ref. 2 [74]

75 [DP04] D ly, J.-P., Paun, M., Numerical characterization of the Kähler cone of a compact Kähler manifold, arxiv: [math.ag], Annals of Mathematics, 159 (2004), [DPS01] D ly, J.-P., Peternell, Th., Schneider M.: Pseudo-effective line bundles on compact Kähler manifolds, International Journal of Math. 6 (2001), [Mou95] Mourougane, Ch.: Versions kählériennes du théorème d annulation de Bogomolov-Sommese, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) [Mou97] Mourougane, Ch.: Notions de positivité et d amplitude des fibrés vectoriels, Théorèmes d annulation sur les variétés kähleriennes, Thèse Université de Grenoble I, [Mou98] Mourougane, Ch.: Versions kählériennes du théorème d annulation de Bogomolov, Dedicated to the memory of Fernando Serrano, Collect. Math. 49 (1998) Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles Ref. 3 [75]

76 [Nad90] Nadel, A.: Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Ann. of Math. 132 (1990) [OT87] Ohsawa, T., Takegoshi, K.: On the extension of L 2 holomorphic functions, Math. Zeitschrift, 195 (1987) [Pau07] Păun, M.: Siu s invariance of plurigenera: a one-tower proof, J. Differential Geom. 76 (2007), [Pau12] Păun, M.: Relative critical exponents, non-vanishing and metrics with minimal singularities, arxiv: [math.cv], Inventiones Math. 187 (2012) [Sho85] Shokurov, V.: The non-vanishing theorem, Izv. Akad. Nauk SSSR 49 (1985) & Math. USSR Izvestiya 26 (1986) [Siu98] Siu, Y.-T.: Invariance of plurigenera, Invent. Math. 134 (1998) Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles Ref. 4 [76]

77 [Siu00] Siu, Y.-T.: Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type, Complex Geometry (Göttingen, 2000), Springer, Berlin, 2002, [Siu08] Siu, Y.-T.: Finite Generation of Canonical Ring by Analytic Method, arxiv: [math.ag]. [Siu09] Siu, Y.-T.: Abundance conjecture, arxiv: [math.ag]. [Sko72] Skoda, H.: Application des techniques L 2 à la théorie des idéaux d une algèbre de fonctions holomorphes avec poids, Ann. Scient. Éc. Norm. Sup. 5 (1972) [Sko78] Skoda, H.: Morphismes surjectifs de fibrés vectoriels semi-positifs, Ann. Sci. Ecole Norm. Sup. 11 (1978) Jean-Pierre D ly Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles Ref. 5 [77]

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