# Structure theorems for compact Kähler manifolds

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1 Structure theorems for compact Kähler manifolds Jean-Pierre D ly joint work with Frédéric Campana & Thomas Peternell Institut Fourier, Université de Grenoble I, France & Académie des Sciences de Paris KSCV10 Conference, August 7 11, 2014 Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 1/17 [1] Goals Analyze the geometric structure of projective or compact Kähler 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 CX. L X 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 [for X projective: c 1 (L) Eff ]. L X is positive (semipositive) if h = e ϕ smooth s.t. Θ L,h = dd c logh > 0 ( 0) on X. L is nef if ε > 0, h ε = e ϕ ε smooth such that Θ L,hε = dd c logh ε εω on X [for X projective: L C 0, C alg. curve]. Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 2/17 [1]

2 Complex curves (n = 1) : genus and curvature K X = Λ n T X, deg(k X) = 2g 2 Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 3/17 [2] 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 Ĉ the strict transform of C. Then thus K X = µ K P 2 O( E i ) K X = µ O P 2(3) O( E i ), One has K X = µ O P 2(C) O( E i ) = O X (Ĉ). K X Γ = Ĉ Γ 0 if Γ Ĉ, K X Ĉ = ( K X) 2 = (Ĉ)2 = C 2 9 = 0 K X nef. Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 4/17 [2]

3 Rationally connected manifolds In fact G := ( K X ) Ĉ O P 2 C(3) O C ( p i ) Pic 0 (C) 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. Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 5/17 [3] Ex. of compact Kähler manifolds with K X 0 (Recall: By Yau, K X 0 ω Kähler with Ricci(ω) 0.) Ricci flat manifolds Complex tori T = C q /Λ Holomorphic symplectic manifolds S (also called hyperkähler): σ H 0 (S,Ω 2 S ) symplectic Calabi-Yau manifolds Y: π 1 (Y) finite and some multiple of K Y is trivial (may assume π 1 (Y) = 1 and K Y trivial by passing to some finite étale cover) the rather large class of rationally connected manifolds Z with K Z 0 all products T S j Y k Z l. Main result. Essentially, this is a complete list! Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 6/17 [3]

4 Structure theorem for manifolds with K X 0 Theorem [Campana, D, Peternell, 2012] Let X be a compact Kähler manifold with K X 0. Then (a) holomorphic and isometric splitting X C q Y j S k Z l where Y j = Calabi-Yau (holonomy SU(n j )), S k = holomorphic symplectic (holonomy Sp(n k /2)), and Z l = RC with K Zl 0 (holonomy U(n l )). (b) There exists a finite étale Galois cover X X such that the Albanese map α : X Alb( X) is an (isometrically) locally trivial holomorphic fiber bundle whose fibers are products Yj S k Z l, as described in (a). (c) π 1 ( X) Z 2q Γ, Γ finite ( almost abelian group). Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 7/17 [4] Criterion for rational connectedness Criterion Let X be a projective algebraic n-dimensional manifold. The following properties are equivalent. (a) X is rationally connected. (b) For every invertible subsheaf F Ω p X := O(Λp T X ), 1 p n, F is not psef. (c) For every invertible subsheaf F O((T X ) p ), p 1, F is not psef. (d) For some (resp. for any) ample line bundle A on X, there exists a constant C A > 0 such that H 0 (X,(T X ) m A k ) = 0 m, k N with m C A k. Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 8/17 [4]

5 Proof of the RC criterion Proof (essentially from Peternell 2006) (a) (d) is easy (RC implies there are many rational curves on which T X, so TX < 0), (d) (c) and (c) (b) are trivial. Thus the only thing left to complete the proof is (b) (a). Consider the MRC quotient π : X Y, given by the equivalence relation x y if x and y can be joined by a chain of rational curves. Then (by definition) the fibers are RC, maximal, and a result of Graber-Harris-Starr (2002) implies that Y is not uniruled. By BDPP (2004), Y not uniruled K Y psef. Then π K Y Ω p X where p = dimy, which is a contradiction unless p = 0, and therefore X is RC. Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 9/17 [5] Generalized holonomy principle Generalized holonomy principle Let (E,h) X be a hermitian holomorphic vector bundle of rank r over X compact/c. Assume that Θ E,h ωn 1 (n 1)! = B ωn n!, B Herm(E,E), B 0 on X. Let H the restricted holonomy group of (E,h). Then (a) If there exists a psef invertible sheaf L O((E ) m ), then L is flat and invariant under parallel transport by the connection of (E ) m induced by the Chern connection of (E,h); moreover, H acts trivially on L. (b) If H satisfies H = U(r), then none of the invertible sheaves L O((E ) m ) can be psef for m 1. Proof. L O((E ) m ) which has trace of curvature 0 while Θ L 0, use Bochner formula. Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 10/17 [5]

6 Generically nef vector bundles Definition Let X compact Kähler manifold, E X torsion free sheaf. (a) E is generically nef with respect to the Kähler class ω if (b) Let µ ω (S) 0 for all torsion free quotients E S 0. If E is ω-generically nef for all ω, we simply say that E is generically nef. 0 = E 0 E 1... E s = E be a filtration of E by torsion free coherent subsheaves such that the quotients E i+1 /E i are ω-stable subsheaves of E/E i of maximal rank. We call such a sequence a refined Harder-Narasimhan (HN) filtration w.r.t. ω. Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 11/17 [6] Characterization of generically nef vector bundles It is a standard fact that refined HN-filtrations always exist, moreover µ ω (E i /E i 1 ) ν ω (E i+1 /E i ) for all i. Proposition Let (X,ω) be a compact Kähler manifold and E a torsion free sehaf on X. Then E is ω-generically nef if and only if for some refined HN-filtration. µ ω (E i+1 /E i ) 0 Proof. Easy arguments on filtrations. Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 12/17 [6]

7 A result of J. Cao about manifolds with K X nef Theorem (Junyan Cao, 2013) Let X be a compact Kähler manifold with K X nef. Then the tangent bundle T X is ω-generically nef for all Kähler classes ω. Proof. use the fact that ε > 0, Kähler metric with Ricci(ω ε ) εω ε (Yau, DPS 1995). From this, one can deduce Theorem Let X be a compact Kähler manifold with nef anticanonical bundle. Then the bundles T m X are ω-generically nef for all Kähler classes ω and all positive integers m. In particular, the bundles S k T X and p T X are ω-generically nef. Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 13/17 [7] A lemma on sections of contravariant tensors Lemma Let (X,ω) be a compact Kähler manifold with K X nef and η H 0 (X,(Ω 1 X ) m L) where L is a numerically trivial line bundle on X. Then the filtered parts of η w.r.t. the refined HN filtration are parallel w.r.t. the Bando-Siu metric in the 0 slope parts, and the < 0 slope parts vanish. Proof. By Cao s theorem there exists a refined HN-filtration 0 = E 0 E 1... E s = T m X with ω-stable quotients E i+1 /E i such that µ ω (E i+1 /E i ) 0 for all i. Then we use the fact that any section in a (semi-)negative slope reflexive sheaf E i+1 /E i L is parallel w.r.t. its Bando-Siu metric (resp. vanishes). Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 14/17 [7]

8 Smoothness of the Albanese morphism (after Cao) Theorem (J.Cao 2013, D-Peternell, 2014) Non-zero holomorphic p-forms on a compact Kähler manifold X with K X nef vanish only on the singular locus of the refined HN filtration of T X. This implies the following result essentially due to J.Cao. Corollary Let X be a compact Kähler manifold with nef anticanonical bundle. Then the Albanese map α : X Alb(X) is a submersion on the complement of the HN filtration singular locus in X [ α surjects onto Alb(X) (Paun 2012)]. Proof. The differential dα is given by (dη 1,...,dη q ) where (η 1,...,η q ) is a basis of 1-forms, q = dimh 0 (X,Ω 1 X ). Cao s thm rank of (dη 1,...,dη q ) is = q generically. Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 15/17 [8] Emerging general picture Conjecture (known for X projective!) Let X be compact Kähler, and let X Y be the MRC fibration (after taking suitable blow-ups to make it a genuine morphism). Then K Y is psef. Proof? Take the part of slope > 0 in the HN filtration of T X w.r.t. to classes in the dual of the psef cone, and apply duality. Remaining problems Develop the theory of singular Calabi-Yau and singular holomorphic symplectic manifolds. Show that the slope 0 part corresponds to blown-up tori, singular Calabi-Yau and singular holomorphic symplectic manifolds (as fibers and targets). The rest of T X (slope < 0) yields a general type quotient. Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 16/17 [8]

9 The end Thank you for your attention! Jean-Pierre D ly KSCV10, Gyeong-Ju, August 11, 2014 Structure theorems for compact Kähler manifolds 17/17 [9]

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