Structure theorems for compact K¨ ahler manifolds Jean-Pierre Demailly joint work with Fr´ ed´ eric Campana & Thomas Peternell Institut Fourier, Universit´ e de Grenoble I, France & Acad´ emie des Sciences de Paris KSCV10 Conference, August 7–11, 2014 1/17 [1:1] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Goals Analyze the geometric structure of projective or compact K¨ ahler manifolds 2/17 [1:2] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Goals Analyze the geometric structure of projective or compact K¨ ahler manifolds As is well known since the beginning of the XX th century at least, the geometry depends on the sign of the curvature of the canonical line bundle K X = Λ n T ∗ X , n = dim C X . 2/17 [2:3] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Goals Analyze the geometric structure of projective or compact K¨ ahler manifolds As is well known since the beginning of the XX th century at least, the geometry depends on the sign of the curvature of the canonical line bundle K X = Λ n T ∗ X , n = dim C X . L → X is pseudoeffective (psef) if ∃ h = e − ϕ , ϕ ∈ L 1 loc , (possibly singular) such that Θ L , h = − dd c log h ≥ 0 on X , in the sense of currents [for X projective: c 1 ( L ) ∈ Eff ]. 2/17 [3:4] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Goals Analyze the geometric structure of projective or compact K¨ ahler manifolds As is well known since the beginning of the XX th century at least, the geometry depends on the sign of the curvature of the canonical line bundle K X = Λ n T ∗ X , n = dim C X . L → X is pseudoeffective (psef) if ∃ h = e − ϕ , ϕ ∈ L 1 loc , (possibly singular) such that Θ L , h = − dd c log h ≥ 0 on X , in the sense of currents [for X projective: c 1 ( L ) ∈ Eff ]. L → X is positive (semipositive) if ∃ h = e − ϕ smooth s.t. Θ L , h = − dd c log h > 0 ( ≥ 0) on X . 2/17 [4:5] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Goals Analyze the geometric structure of projective or compact K¨ ahler manifolds As is well known since the beginning of the XX th century at least, the geometry depends on the sign of the curvature of the canonical line bundle K X = Λ n T ∗ X , n = dim C X . L → X is pseudoeffective (psef) if ∃ h = e − ϕ , ϕ ∈ L 1 loc , (possibly singular) such that Θ L , h = − dd c log h ≥ 0 on X , in the sense of currents [for X projective: c 1 ( L ) ∈ Eff ]. L → X is positive (semipositive) if ∃ h = e − ϕ smooth s.t. Θ L , h = − dd c log h > 0 ( ≥ 0) on X . L is nef if ∀ ε > 0, ∃ h ε = e − ϕ ε smooth such that Θ L , h ε = − dd c log h ε ≥ − εω on X [for X projective: L · C ≥ 0 , ∀ C alg. curve]. 2/17 [5:6] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Complex curves ( n = 1) : genus and curvature K X = Λ n T ∗ X , deg( K X ) = 2 g − 2 3/17 [1:7] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Comparison of positivity concepts Recall that for a line bundle > 0 ⇔ ample ⇒ semiample ⇒ semipositive ⇒ nef ⇒ psef but none of the reverse implications in red holds true. 4/17 [1:8] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Comparison of positivity concepts Recall that for a line bundle > 0 ⇔ ample ⇒ semiample ⇒ semipositive ⇒ nef ⇒ psef but none of the reverse implications in red holds true. Example Let X be the rational surface obtained by blowing up P 2 in 9 distinct points { p i } on a smooth (cubic) elliptic curve C ⊂ P 2 , µ : X → P 2 and ˆ C the strict transform of C . 4/17 [2:9] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Comparison of positivity concepts Recall that for a line bundle > 0 ⇔ ample ⇒ semiample ⇒ semipositive ⇒ nef ⇒ psef but none of the reverse implications in red holds true. Example Let X be the rational surface obtained by blowing up P 2 in 9 distinct points { p i } on a smooth (cubic) elliptic curve C ⊂ P 2 , µ : X → P 2 and ˆ C the strict transform of C . Then K X = µ ∗ K P 2 ⊗ O ( � E i ) ⇒ − K X = µ ∗ O P 2 (3) ⊗ O ( − � E i ) , thus − K X = µ ∗ O P 2 ( C ) ⊗ O ( − � E i ) = O X (ˆ C ) . 4/17 [3:10] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Comparison of positivity concepts Recall that for a line bundle > 0 ⇔ ample ⇒ semiample ⇒ semipositive ⇒ nef ⇒ psef but none of the reverse implications in red holds true. Example Let X be the rational surface obtained by blowing up P 2 in 9 distinct points { p i } on a smooth (cubic) elliptic curve C ⊂ P 2 , µ : X → P 2 and ˆ C the strict transform of C . Then K X = µ ∗ K P 2 ⊗ O ( � E i ) ⇒ − K X = µ ∗ O P 2 (3) ⊗ O ( − � E i ) , thus − K X = µ ∗ O P 2 ( C ) ⊗ O ( − � E i ) = O X (ˆ C ) . One has − K X · Γ = ˆ if Γ � = ˆ C · Γ ≥ 0 C , C = ( − K X ) 2 = (ˆ C ) 2 = C 2 − 9 = 0 − K X · ˆ ⇒ − K X nef. 4/17 [4:11] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Rationally connected manifolds In fact C ≃ O P 2 | C (3) ⊗ O C ( − � p i ) ∈ Pic 0 ( C ) G := ( − K X ) | ˆ 5/17 [1:12] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Rationally connected manifolds In fact C ≃ O P 2 | C (3) ⊗ O C ( − � p i ) ∈ Pic 0 ( C ) G := ( − K X ) | ˆ If G is a torsion point in Pic 0 ( C ), then one can show that − K X is semi-ample, but otherwise it is not semi-ample. 5/17 [2:13] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Rationally connected manifolds In fact C ≃ O P 2 | C (3) ⊗ O C ( − � p i ) ∈ Pic 0 ( C ) G := ( − K X ) | ˆ If G is a torsion point in Pic 0 ( C ), then one can show that − K X is semi-ample, but otherwise it is not semi-ample. Brunella has shown that − K X is C ∞ semipositive if c 1 ( G ) satisfies a diophantine condition found by T. Ueda, but that otherwise it may not be semipositive (although nef). 5/17 [3:14] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Rationally connected manifolds In fact C ≃ O P 2 | C (3) ⊗ O C ( − � p i ) ∈ Pic 0 ( C ) G := ( − K X ) | ˆ If G is a torsion point in Pic 0 ( C ), then one can show that − K X is semi-ample, but otherwise it is not semi-ample. Brunella has shown that − K X is C ∞ semipositive if c 1 ( G ) satisfies a diophantine condition found by T. Ueda, but that otherwise it may not be semipositive (although nef). P 2 # 9 points is an example of rationally connected manifold: 5/17 [4:15] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Rationally connected manifolds In fact C ≃ O P 2 | C (3) ⊗ O C ( − � p i ) ∈ Pic 0 ( C ) G := ( − K X ) | ˆ If G is a torsion point in Pic 0 ( C ), then one can show that − K X is semi-ample, but otherwise it is not semi-ample. Brunella has shown that − K X is C ∞ semipositive if c 1 ( G ) satisfies a diophantine condition found by T. Ueda, but that otherwise it may not be semipositive (although nef). P 2 # 9 points is an example of rationally connected manifold: Definition Recall that a compact complex manifold is said to be rationally connected (or RC for short) if any 2 points can be joined by a chain of rational curves 5/17 [5:16] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Rationally connected manifolds In fact C ≃ O P 2 | C (3) ⊗ O C ( − � p i ) ∈ Pic 0 ( C ) G := ( − K X ) | ˆ If G is a torsion point in Pic 0 ( C ), then one can show that − K X is semi-ample, but otherwise it is not semi-ample. Brunella has shown that − K X is C ∞ semipositive if c 1 ( G ) satisfies a diophantine condition found by T. Ueda, but that otherwise it may not be semipositive (although nef). P 2 # 9 points is an example of rationally connected manifold: Definition Recall that a compact complex manifold is said to be rationally connected (or RC for short) if any 2 points can be joined by a chain of rational curves Remark. X = P 2 blown-up in ≥ 10 points is RC but − K X not nef. 5/17 [6:17] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Ex. of compact K¨ ahler manifolds with − K X ≥ 0 (Recall: By Yau, − K X ≥ 0 ⇔ ∃ ω K¨ ahler with Ricci( ω ) ≥ 0.) 6/17 [1:18] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
Ex. of compact K¨ ahler manifolds with − K X ≥ 0 (Recall: By Yau, − K X ≥ 0 ⇔ ∃ ω K¨ ahler with Ricci( ω ) ≥ 0.) Ricci flat manifolds – Complex tori T = C q / Λ 6/17 [2:19] Jean-Pierre Demailly – KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact K¨ ahler manifolds
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