On the cohomology of pseudoeffective line bundles Jean-Pierre - - PowerPoint PPT Presentation
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
Jean-Pierre Demailly
Institut Fourier, Universit´ e de Grenoble I, France & Acad´ emie des Sciences de Paris
in honor of Professor Yum-Tong Siu
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]
Study sections and cohomology of holomorphic line bundles L → X on compact K¨ ahler 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]
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)
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21[2:3]
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)
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21[3:4]
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 L2 extension theorem (1987)
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21[4:5]
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 L2 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]
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 L2 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)
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 2/21[6:7]
Let X = compact K¨ ahler 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]
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 ∈ Lx ≃ C, ξ2
h = |ξ|2e−ϕ(x).
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21[2:9]
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 ∈ Lx ≃ C, ξ2
h = |ξ|2e−ϕ(x).
Writing h = e−ϕ locally, one defines the curvature form of L to be the real (1, 1)-form ΘL,h = i 2π∂∂ϕ = −dd c log h,
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21[3:10]
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 ∈ Lx ≃ C, ξ2
h = |ξ|2e−ϕ(x).
Writing h = e−ϕ locally, one defines the curvature form of L to be the real (1, 1)-form ΘL,h = i 2π∂∂ϕ = −dd c log h, c1(L) =
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21[4:11]
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 ∈ Lx ≃ C, ξ2
h = |ξ|2e−ϕ(x).
Writing h = e−ϕ locally, one defines the curvature form of L to be the real (1, 1)-form ΘL,h = i 2π∂∂ϕ = −dd c log h, c1(L) =
Any subspace Vm ⊂ H0(X, L⊗m) define a meromorphic map ΦmL : X Zm − → P(Vm) (hyperplanes of Vm) x − → Hx =
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 3/21[5:12]
Given sections σ1, . . . , σn ∈ H0(X, L⊗m), one gets a singular hermitian metric on L defined by |ξ|2
h =
|ξ|2 |σj(x)|21/m ,
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 4/21[1:13]
Given sections σ1, . . . , σn ∈ H0(X, L⊗m), one gets a singular hermitian metric on L defined by |ξ|2
h =
|ξ|2 |σj(x)|21/m , its weight is the plurisubharmonic (psh) function ϕ(x) = 1 m log |σj(x)|2
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 4/21[2:14]
Given sections σ1, . . . , σn ∈ H0(X, L⊗m), one gets a singular hermitian metric on L defined by |ξ|2
h =
|ξ|2 |σj(x)|21/m , its weight is the plurisubharmonic (psh) function ϕ(x) = 1 m log |σj(x)|2 and the curvature is ΘL,h = 1
mdd c log ϕ ≥ 0
in the sense of currents, with logarithmic poles along the base locus B =
j (0) = ϕ−1(−∞).
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 4/21[3:15]
Given sections σ1, . . . , σn ∈ H0(X, L⊗m), one gets a singular hermitian metric on L defined by |ξ|2
h =
|ξ|2 |σj(x)|21/m , its weight is the plurisubharmonic (psh) function ϕ(x) = 1 m log |σj(x)|2 and the curvature is ΘL,h = 1
mdd c log ϕ ≥ 0
in the sense of currents, with logarithmic poles along the base locus B =
j (0) = ϕ−1(−∞).
One has (ΘL,h)|X B = 1 mΦ∗
mLωFS where ΦmL : X B → P(Vm) ≃ PNm.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 4/21[4:16]
Definition L is pseudoeffective (psef) if ∃h = e−ϕ, ϕ ∈ L1
loc,
(possibly singular) such that ΘL,h = −dd c log h ≥ 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]
Definition L is pseudoeffective (psef) if ∃h = e−ϕ, ϕ ∈ L1
loc,
(possibly singular) such that ΘL,h = −dd c log h ≥ 0 on X, in the sense of currents. L is semipositive if ∃h = e−ϕ smooth such that ΘL,h = −dd c log h ≥ 0 on X.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 5/21[2:18]
Definition L is pseudoeffective (psef) if ∃h = e−ϕ, ϕ ∈ L1
loc,
(possibly singular) such that ΘL,h = −dd c log h ≥ 0 on X, in the sense of currents. L is semipositive if ∃h = e−ϕ smooth such that ΘL,h = −dd c log h ≥ 0 on X. L is positive if ∃h = e−ϕ smooth such that ΘL,h = −dd c log h > 0 on X.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 5/21[3:19]
Definition L is pseudoeffective (psef) if ∃h = e−ϕ, ϕ ∈ L1
loc,
(possibly singular) such that ΘL,h = −dd c log h ≥ 0 on X, in the sense of currents. L is semipositive if ∃h = e−ϕ smooth such that ΘL,h = −dd c log h ≥ 0 on X. L is positive if ∃h = e−ϕ smooth such that ΘL,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: Zm = B(mL) = ∅ and Φ|mL| : X → P(H0(X, L⊗m)) is an embedding for m ≥ m0 large enough.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 5/21[4:20]
Definitions Let X be a compact K¨ ahler manifold. The K¨ ahler cone is the (open) set K ⊂ H1,1(X, R) of cohomology classes {ω} of positive K¨ ahler forms.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 6/21[1:21]
Definitions Let X be a compact K¨ ahler manifold. The K¨ ahler cone is the (open) set K ⊂ H1,1(X, R) of cohomology classes {ω} of positive K¨ ahler forms. The pseudoeffective cone is the set E ⊂ H1,1(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 Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 6/21[2:22]
Definitions Let X be a compact K¨ ahler manifold. The K¨ ahler cone is the (open) set K ⊂ H1,1(X, R) of cohomology classes {ω} of positive K¨ ahler forms. The pseudoeffective cone is the set E ⊂ H1,1(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]
Definitions Let X be a compact K¨ ahler manifold. The K¨ ahler cone is the (open) set K ⊂ H1,1(X, R) of cohomology classes {ω} of positive K¨ ahler forms. The pseudoeffective cone is the set E ⊂ H1,1(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 P2 in one point, then the exceptional divisor E ≃ P1 has a cohomology class {α} such that
{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]
Positive cones can be visualized as follows : H1,1(X, R) (containing Neron-Severi space NSR(X)) K KNS E ENS K¨ ahler cone psef cone ample divisors: KNS nef divisors: KNS big divisors: E◦
NS
effective & psef: ENS KNS = K ∩ NSR(X) ENS = E ∩ NSR(X) where NSR(X) = (H1,1(X, R) ∩ H2(X, Z)) ⊗Z R NSR(X)
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 7/21[1:25]
Definition On X compact K¨ ahler, a K¨ ahler 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 Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 8/21[1:26]
Definition On X compact K¨ ahler, a K¨ ahler 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 K¨ ahler current. We say that E◦ is the cone of big (1, 1)-classes.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 8/21[2:27]
Definition On X compact K¨ ahler, a K¨ ahler 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 K¨ ahler current. We say that E◦ is the cone of big (1, 1)-classes. Theorem on approximate Zariski decomposition (D, ’92) Any K¨ ahler current can be written T = lim Tm where Tm ∈ {T} has analytic singularities & logarithmic poles, i.e. ∃ modification µm : Xm → X such that µ⋆
mTm = [Em] + βm
where Em is an effective Q-divisor on Xm with coefficients in
1 mZ and βm is a K¨
ahler form on Xm.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 8/21[2:28]
NSR( Xm) ⊂ H1,1( Xm, R) NSR(X) ⊂ H1,1(X, R) K( Xm) K(X) ω Em βm µ∗
mTm
E(X) E( Xm) T−δω≥0 Tm T µm : Xm − → X (blow-up)
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 9/21[1:29]
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]
T = i∂∂ϕ for some strictly plurisubharmonic psh potential ϕ on X.
Tm = i∂∂ϕm, ϕm(z) = 1 2m log
|gℓ,m(z)|2 where (gℓ,m) is a Hilbert basis of the space H(Ω, mϕ) =
|f |2e−2mϕdV < +∞
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 10/21[2:31]
T = i∂∂ϕ for some strictly plurisubharmonic psh potential ϕ on X.
Tm = i∂∂ϕm, ϕm(z) = 1 2m log
|gℓ,m(z)|2 where (gℓ,m) is a Hilbert basis of the space H(Ω, mϕ) =
|f |2e−2mϕdV < +∞
from a single isolated point) implies that there are enough such holomorphic functions, and thus ϕm ≥ ϕ − C/m.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 10/21[3:32]
T = i∂∂ϕ for some strictly plurisubharmonic psh potential ϕ on X.
Tm = i∂∂ϕm, ϕm(z) = 1 2m log
|gℓ,m(z)|2 where (gℓ,m) is a Hilbert basis of the space H(Ω, mϕ) =
|f |2e−2mϕdV < +∞
from a single isolated point) implies that there are enough such holomorphic functions, and thus ϕm ≥ ϕ − C/m.
lim
m→+∞ ϕm by the mean value inequality.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 10/21[4:33]
Let P(X) = closed positive (1, 1)-currents on X Hk,k
≥0 (X) =
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 11/21[1:34]
Let P(X) = closed positive (1, 1)-currents on X Hk,k
≥0 (X) =
Theorem (Boucksom PhD 2002, Junyan Cao PhD 2012) ∀k = 1, 2, . . . , n, ∃ canonical “movable intersection product” P × · · · × P → Hk,k
≥0 (X),
(T1, . . . , Tk) → T1 · T2 · · · Tk
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 11/21[2:35]
Let P(X) = closed positive (1, 1)-currents on X Hk,k
≥0 (X) =
Theorem (Boucksom PhD 2002, Junyan Cao PhD 2012) ∀k = 1, 2, . . . , n, ∃ canonical “movable intersection product” P × · · · × P → Hk,k
≥0 (X),
(T1, . . . , Tk) → T1 · T2 · · · Tk
ahler. Approximate each Tj by K¨ ahler currents Tj,m with logarithmic poles,take a simultaneous log-resolution µm : Xm → X such that µ⋆
mTj = [Ej,m] + βj,m.
and define T1 · T2 · · ·Tk = lim ↑
m→+∞{(µm)⋆(β1,m ∧ β2,m ∧ . . . ∧ βk,m)}.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 11/21[3:36]
uniform boundedness in mass.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 12/21[1:37]
uniform boundedness in mass. Uniqueness comes from monotonicity (βj,m “increases” with m)
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 12/21[2:38]
uniform boundedness in mass. Uniqueness comes from monotonicity (βj,m “increases” with m) Special case. The volume of a class α ∈ H1,1(X, R) is Vol(α) = sup
T∈α
T n if α ∈ E◦ (big class), 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]
uniform boundedness in mass. Uniqueness comes from monotonicity (βj,m “increases” with m) Special case. The volume of a class α ∈ H1,1(X, R) is Vol(α) = sup
T∈α
T n if α ∈ E◦ (big class), Vol(α) = 0 if α ∈ E◦, Numerical dimension of a current nd(T) = max
in Hp,p
≥0 (X)
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 12/21[4:40]
uniform boundedness in mass. Uniqueness comes from monotonicity (βj,m “increases” with m) Special case. The volume of a class α ∈ H1,1(X, R) is Vol(α) = sup
T∈α
T n if α ∈ E◦ (big class), Vol(α) = 0 if α ∈ E◦, Numerical dimension of a current nd(T) = max
in Hp,p
≥0 (X)
Numerical dimension of a hermitian line bundle (L, h) nd(L, h) = nd(ΘL,h).
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 12/21[4:41]
Numerical dimension of a class α ∈ H1,1(X, R) If α is not pseudoeffective, set nd(α) = −∞, otherwise nd(α) = max
ε→0T 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]
Numerical dimension of a class α ∈ H1,1(X, R) If α is not pseudoeffective, set nd(α) = −∞, otherwise nd(α) = max
ε→0T p ε ∧ ωn−p ≥ C>0
Numerical dimension of a pseudo-effective line bundle nd(L) = nd(c1(L)). L is said to be abundant if κ(L) = nd(L).
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 13/21[2:43]
Numerical dimension of a class α ∈ H1,1(X, R) If α is not pseudoeffective, set nd(α) = −∞, otherwise nd(α) = max
ε→0T p ε ∧ ωn−p ≥ C>0
Numerical dimension of a pseudo-effective line bundle nd(L) = nd(c1(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 → OC → E → OC → 0 on C = elliptic curve, let X = P(E) (ruled surface over C) and L = OP(E)(1). Then nd(L) = 1 but ∃ ! positive current T = [σ(C)] ∈ c1(L) and nd(T) = 0 !!
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 13/21[3:44]
Numerical dimension of a class α ∈ H1,1(X, R) If α is not pseudoeffective, set nd(α) = −∞, otherwise nd(α) = max
ε→0T p ε ∧ ωn−p ≥ C>0
Numerical dimension of a pseudo-effective line bundle nd(L) = nd(c1(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 → OC → E → OC → 0 on C = elliptic curve, let X = P(E) (ruled surface over C) and L = OP(E)(1). Then nd(L) = 1 but ∃ ! positive current T = [σ(C)] ∈ c1(L) and nd(T) = 0 !! Generalized abundance conjecture For X compact K¨ ahler, KX is abundant, i.e. κ(X) = nd(KX).
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 13/21[3:45]
Let (L, h) be a pseudo-effective line bundle on a compact K¨ ahler 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 |2e−ϕdVω < +∞
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 14/21[1:46]
Let (L, h) be a pseudo-effective line bundle on a compact K¨ ahler 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 |2e−ϕdVω < +∞
The Nadel vanishing theorem claims that ΘL,h ≥ εω = ⇒ Hq(X, KX ⊗ L ⊗ I(h) = 0 for q ≥ 1.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 14/21[2:47]
Let (L, h) be a pseudo-effective line bundle on a compact K¨ ahler 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 |2e−ϕdVω < +∞
The Nadel vanishing theorem claims that ΘL,h ≥ εω = ⇒ Hq(X, KX ⊗ L ⊗ I(h) = 0 for q ≥ 1. Hard Lefschetz theorem (D-Peternell-Schneider 2001) Assume merely ΘL,h ≥ 0. Then, the Lefschetz map : u → ωq ∧ u induces a surjective morphism : Φq
ω,h : H0(X, Ωn−q X
⊗ L ⊗ I(h)) − → Hq(X, Ωn
X ⊗ L ⊗ I(h)).
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 14/21[2:48]
Main tool. “Equisingular approximation theorem”: ϕ = lim ↓ ϕν ⇒ h = lim hν with: ϕν ∈ C ∞(X Zν), where Zν is an increasing sequence of analytic sets, I(hν) = I(h), ∀ν, ΘL,hν ≥ −ενω. (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]
Main tool. “Equisingular approximation theorem”: ϕ = lim ↓ ϕν ⇒ h = lim hν with: ϕν ∈ C ∞(X Zν), where Zν is an increasing sequence of analytic sets, I(hν) = I(h), ∀ν, ΘL,hν ≥ −ενω. (Again, the proof uses in several ways the Ohsawa-Takegoshi theorem). Then, use the fact that X Zν is K¨ ahler complete, so one can apply (non compact) harmonic form theory on X Zν, and pass to the limit to get rid of the errors εν.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 15/21[2:50]
Theorem (Junyan Cao, PhD 2012) Let X be compact K¨ ahler, and let (L, h) be pseudoeffective
Hq(X, KX ⊗ L ⊗ I+(h)) = 0 for q ≥ n − nd(L, h) + 1, where I+(h) = limε→0 I(h1+ε) = limε→0 I((1 + ε)ϕ) is the “upper semicontinuous regularization” of I(h).
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 16/21[1:51]
Theorem (Junyan Cao, PhD 2012) Let X be compact K¨ ahler, and let (L, h) be pseudoeffective
Hq(X, KX ⊗ L ⊗ I+(h)) = 0 for q ≥ n − nd(L, h) + 1, where I+(h) = limε→0 I(h1+ε) = limε→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 Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 16/21[2:52]
Theorem (Junyan Cao, PhD 2012) Let X be compact K¨ ahler, and let (L, h) be pseudoeffective
Hq(X, KX ⊗ L ⊗ I+(h)) = 0 for q ≥ n − nd(L, h) + 1, where I+(h) = limε→0 I(h1+ε) = limε→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 (dim Y = p) : nd(L, h) = max
.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 16/21[3:53]
Hyperplane section argument (projective case). Take A = very ample divisor, ω = ΘA,hA > 0, and Y = A1 ∩ . . . ∩ An−p, Aj ∈ |A|. Then Θp
L,h · Y =
Θp
L,h · Y =
Θp
L,h ∧ ωn−p > 0.
From this one concludes that (ΘL,h)|Y is big.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 17/21[1:54]
Hyperplane section argument (projective case). Take A = very ample divisor, ω = ΘA,hA > 0, and Y = A1 ∩ . . . ∩ An−p, Aj ∈ |A|. Then Θp
L,h · Y =
Θp
L,h · Y =
Θp
L,h ∧ ωn−p > 0.
From this one concludes that (ΘL,h)|Y is big. Lemma (J. Cao) When (L, h) is big, i.e. Θn
L,h > 0, there exists a metric
h such that I( h) = I+(h) with ΘL,
h ≥ εω
[Riemann-Roch]. Then Nadel ⇒ Hq(X, KX ⊗ L ⊗ I+(h)) = 0 for q ≥ 1.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 17/21[2:55]
Hyperplane section argument (projective case). Take A = very ample divisor, ω = ΘA,hA > 0, and Y = A1 ∩ . . . ∩ An−p, Aj ∈ |A|. Then Θp
L,h · Y =
Θp
L,h · Y =
Θp
L,h ∧ ωn−p > 0.
From this one concludes that (ΘL,h)|Y is big. Lemma (J. Cao) When (L, h) is big, i.e. Θn
L,h > 0, there exists a metric
h such that I( h) = I+(h) with ΘL,
h ≥ εω
[Riemann-Roch]. Then Nadel ⇒ Hq(X, KX ⊗ 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]
K¨ ahler case. Assume c1(L) nef for simplicity. Then c1(L) + εω K¨
ere equation: ∃hε on L, (ΘL,hε + εω)n = Cεωn. Here Cε ≥ n
p
L,h · (εω)n−p ∼ Cεn−p, p = nd(L, h).
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 18/21[1:57]
K¨ ahler case. Assume c1(L) nef for simplicity. Then c1(L) + εω K¨
ere equation: ∃hε on L, (ΘL,hε + εω)n = Cεωn. Here Cε ≥ n
p
L,h · (εω)n−p ∼ Cεn−p, p = nd(L, h).
the eigenvalues of ΘL,h + εω w.r.to ω. Then λ1 . . . λn = Cε ≥ Const εn−p and
X
λq+1 . . . λn ωn =
Θn−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]
K¨ ahler case. Assume c1(L) nef for simplicity. Then c1(L) + εω K¨
ere equation: ∃hε on L, (ΘL,hε + εω)n = Cεωn. Here Cε ≥ n
p
L,h · (εω)n−p ∼ Cεn−p, p = nd(L, h).
the eigenvalues of ΘL,h + εω w.r.to ω. Then λ1 . . . λn = Cε ≥ Const εn−p and
X
λq+1 . . . λn ωn =
Θn−q
L,h ∧ ωq ≤ Const,
∀q ≥ 1, so λq+1 . . . λn ≤ C on a large open set U ⊂ X and λq
q ≥ λ1 . . . λq ≥ cεn−p
⇒ λq ≥ cε(n−p)/q on U, q
j=1(λj − ε) ≥ λq − 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]
λj = eigenvalues of (ΘL,hε+εω) ⇒ (eigenvalues of ΘL,hε) = λj − ε.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 19/21[1:60]
λj = eigenvalues of (ΘL,hε+εω) ⇒ (eigenvalues of ΘL,hε) = λj − ε. Bochner-Kodaira formula yields ∂u2
ε + ∂∗u2 ε ≥
(λj − ε)
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 19/21[2:61]
λj = eigenvalues of (ΘL,hε+εω) ⇒ (eigenvalues of ΘL,hε) = λj − ε. Bochner-Kodaira formula yields ∂u2
ε + ∂∗u2 ε ≥
(λj − ε)
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]
Definition (Campana) A compact K¨ ahler manifold is said to be simple if there are no positive dimensional analytic sets Ax ⊂ X through a very generic point x ∈ X.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 20/21[1:63]
Definition (Campana) A compact K¨ ahler manifold is said to be simple if there are no positive dimensional analytic sets Ax ⊂ X through a very generic point x ∈ X. Well-known fact A complex torus X = Cn/Λ defined by a sufficiently generic lattice Λ ⊂ Cn is simple, and in fact has no positive dimensional analytic subset A X at all.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 20/21[2:64]
Definition (Campana) A compact K¨ ahler manifold is said to be simple if there are no positive dimensional analytic sets Ax ⊂ X through a very generic point x ∈ X. Well-known fact A complex torus X = Cn/Λ defined by a sufficiently generic lattice Λ ⊂ Cn 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]
Definition (Campana) A compact K¨ ahler manifold is said to be simple if there are no positive dimensional analytic sets Ax ⊂ X through a very generic point x ∈ X. Well-known fact A complex torus X = Cn/Λ defined by a sufficiently generic lattice Λ ⊂ Cn 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 K¨ ahler manifolds are either generic complex tori, generic hyperk¨ ahler 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]
Theorem (Campana - D - Verbitsky, 2013) Let X be a compact K¨ ahler 3-fold without any positive dimensional analytic subset A X. Then X is a complex 3-dimensional torus.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21[1:67]
Theorem (Campana - D - Verbitsky, 2013) Let X be a compact K¨ ahler 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 Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21[2:68]
Theorem (Campana - D - Verbitsky, 2013) Let X be a compact K¨ ahler 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) KX 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]
Theorem (Campana - D - Verbitsky, 2013) Let X be a compact K¨ ahler 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) KX is pseudoeffective: otherwise X would be covered by rational curves (Brunella 2008), hence in fact nef. All multiplier ideal sheaves I(h) are trivial H0(X, Ωn−q
X
⊗ K ⊗m−1
X
) → Hq(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]
Theorem (Campana - D - Verbitsky, 2013) Let X be a compact K¨ ahler 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) KX is pseudoeffective: otherwise X would be covered by rational curves (Brunella 2008), hence in fact nef. All multiplier ideal sheaves I(h) are trivial H0(X, Ωn−q
X
⊗ K ⊗m−1
X
) → Hq(X, K ⊗m
X
) is surjective Hilbert polynomial P(m) = χ(X, K ⊗m
X
) is bounded, hence χ(X, OX) = 0.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles 21/21[5:71]
Theorem (Campana - D - Verbitsky, 2013) Let X be a compact K¨ ahler 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) KX is pseudoeffective: otherwise X would be covered by rational curves (Brunella 2008), hence in fact nef. All multiplier ideal sheaves I(h) are trivial H0(X, Ωn−q
X
⊗ K ⊗m−1
X
) → Hq(X, K ⊗m
X
) is surjective Hilbert polynomial P(m) = χ(X, K ⊗m
X
) is bounded, hence χ(X, OX) = 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]
[BCHM10] Birkar, C., Cascini, P., Hacon, C., McKernan, J.: Existence of minimal models for varieties of log general type, arXiv: 0610203 [math.AG], J. Amer. Math. Soc. 23 (2010) 405–468 [Bou02] Boucksom, S.: Cˆ
et´ es complexes compactes, Thesis, Grenoble, 2002. [BDPP13] Boucksom, S., Demailly, J.-P., Paun M., Peternell, Th.: The pseudo-effective cone of a compact K¨ ahler manifold and varieties of negative Kodaira dimension, arXiv: 0405285 [math.AG] ; J. Algebraic Geom. 22 (2013) 201–248 [CDV13] Campana, F., Demailly, J.-P., Verbitsky, M.: Compact K¨ ahler 3-manifolds without non-trivial subvarieties, hal-00819044, arXiv: 1304.7891 [math.CV].
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
[Cao12] Cao, J.: Numerical dimension and a Kawamata-Viehweg-Nadel type vanishing theorem on compact K¨ ahler manifolds, arXiv: 1210.5692 [math.AG] [Dem91] Demailly, J.-P. : Transcendental proof of a generalized Kawamata-Viehweg vanishing theorem, C. R.
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] Demailly, J.-P. : Regularization of closed positive currents and Intersection Theory, J. Alg. Geom. 1 (1992) 361–409 [DHP10] Demailly, J.-P., Hacon, C., P˘ aun, M.: Extension theorems, Non-vanishing and the existence of good minimal models, arXiv: 1012.0493 [math.AG], to appear in Acta Math. 2013.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
[DP04] Demailly, J.-P., Paun, M., Numerical characterization
ahler cone of a compact K¨ ahler manifold, arXiv: 0105176 [math.AG], Annals of Mathematics, 159 (2004), 1247–1274. [DPS01] Demailly, J.-P., Peternell, Th., Schneider M.: Pseudo-effective line bundles on compact K¨ ahler manifolds, International Journal of Math. 6 (2001), 689–741. [Mou95] Mourougane, Ch.: Versions k¨ ahl´ eriennes du th´ eor` eme d’annulation de Bogomolov-Sommese, C. R. Acad.
[Mou97] Mourougane, Ch.: Notions de positivit´ e et d’amplitude des fibr´ es vectoriels, Th´ eor` emes d’annulation sur les vari´ et´ es k¨ ahleriennes, Th` ese Universit´ e de Grenoble I, 1997. [Mou98] Mourougane, Ch.: Versions k¨ ahl´ eriennes du th´ eor` eme d’annulation de Bogomolov, Dedicated to the memory of Fernando Serrano, Collect. Math. 49 (1998)
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
433–445. [Nad90] Nadel, A.: Multiplier ideal sheaves and K¨ ahler-Einstein metrics of positive scalar curvature, Ann. of
[OT87] Ohsawa, T., Takegoshi, K.: On the extension of L2 holomorphic functions, Math. Zeitschrift, 195 (1987) 197–204. [Pau07] P˘ aun, M.: Siu’s invariance of plurigenera: a
[Pau12] P˘ aun, M.: Relative critical exponents, non-vanishing and metrics with minimal singularities, arXiv: 0807.3109 [math.CV], Inventiones Math. 187 (2012) 195–258. [Sho85] Shokurov, V.: The non-vanishing theorem, Izv. Akad. Nauk SSSR 49 (1985) & Math. USSR Izvestiya 26 (1986) 591–604 [Siu98] Siu, Y.-T.: Invariance of plurigenera, Invent. Math. 134 (1998) 631–639.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles
[Siu00] Siu, Y.-T.: Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily
Springer, Berlin, 2002, 223–277. [Siu08] Siu, Y.-T.: Finite Generation of Canonical Ring by Analytic Method, arXiv: 0803.2454 [math.AG]. [Siu09] Siu, Y.-T.: Abundance conjecture, arXiv: 0912.0576 [math.AG]. [Sko72] Skoda, H.: Application des techniques L2 ` a la th´ eorie des id´ eaux d’une alg` ebre de fonctions holomorphes avec poids,
[Sko78] Skoda, H.: Morphismes surjectifs de fibr´ es vectoriels semi-positifs, Ann. Sci. Ecole Norm. Sup. 11 (1978) 577–611.
Jean-Pierre Demailly – Abel Symposium, July 5, 2013 On the cohomology of pseudoeffective line bundles