The decomposition theorem: the smooth case Arnaud Beauville - PowerPoint PPT Presentation

The decomposition theorem: the smooth case Arnaud Beauville Universit´ e Cˆ ote d’Azur CIRM, April 2018 (virtual) Arnaud Beauville The decomposition theorem: the smooth case

The decomposition theorem This introductory talk is devoted to the history of the following theorem: Decomposition theorem ahler manifold with c 1 p M q “ 0 in H 2 p M , R q . Let M be a compact K¨ etale with M 1 “ T ˆ ś i X i ˆ ś There exists M 1 Ñ M finite ´ j Y j T “ complex torus; X i “ X simply connected projective, dim ě 3, H 0 p X , Ω ˚ X q “ C ‘ C ω , where ω is a generator of K X p Calabi-Yau manifolds q . Y j “ Y compact simply connected, H 0 p Y , Ω ˚ Y q “ C r σ s , where σ P H 0 p Y , Ω 2 Y q is everywhere non-degenerate p irreducible symplectic manifolds q . Arnaud Beauville The decomposition theorem: the smooth case

Splitting the Theorem in two To describe the history, it is convenient to split it in two theorems: Theorem A ahler manifold with c 1 p M q “ 0 in H 2 p M , R q . Let M be a compact K¨ There exists T ˆ X Ñ M finite ´ etale, T complex torus, X compact simply connected with K X – O X . This has highly nontrivial consequences: Corollary 1 q K b n M – O M for some n . 2 q π 1 p M q is virtually abelian. Theorem B M compact simply connected K¨ ahler manifold with K M – O M ñ M – ś i X i ˆ ś ù j Y j as in the Theorem. Arnaud Beauville The decomposition theorem: the smooth case

The Calabi conjecture At the ICM 1954, Calabi announced (as a theorem) his now famous conjecture. In our case: Calabi’s conjecture c R 1 p M q “ 0 ù ñ M admits a Ricci-flat K¨ ahler metric. In a 1957 paper, he restates it as a conjecture, and gives as its main application a weak version of Theorem A: Proposition (Calabi) ahler metric ñ Theorem A’ : M admits a Ricci-flat K¨ etale, T complex torus, H 0 p X , Ω 1 D T ˆ X Ñ M finite ´ X q “ 0. By studying the automorphism group, Matsushima proved: Proposition (Matsushima, 1969) Theorem A’ holds for M projective (with c R 1 p M q “ 0). Arnaud Beauville The decomposition theorem: the smooth case

Bogomolov 1974 In 1974 appear 2 papers by Bogomolov: 1 K¨ ahler manifolds with trivial canonical class ; 2 On the decomposition of K¨ ahler manifolds with trivial canonical class . In 1 he reproves Theorem A’ in the projective case, and proves (?) K b n M – O M in the K¨ ahler case. 2 he announces Theorem B (a slightly weaker form): In K M – O M and π 1 p M q “ 0 ñ M – X ˆ ś j Y j , with H 0 p X , Ω 2 X q “ 0, Y j symplectic. Arnaud Beauville The decomposition theorem: the smooth case

The attempted proof of Theorem B Sketch of proof : The heart of the proof is the following statement: If T M “ E ‘ F with E , F integrable and det p E q “ det p F q “ O M , M – X ˆ Y with E – T X , F – T Y . Without the condition det p E q “ det p F q “ O M , this is an open problem – there are partial results by Druel, H¨ oring, Brunella- Pereira-Touzet. It is hard to see how the extra condition on det could help. What the paper says: “There exists a linear connection on M for which E and F are parallel. Hence the result” . The connection cannot be holomorphic (this would imply c i p M q “ 0 for all i ). There certainly exists such a C 8 connection on M (just take one on E and one on F ), but then?? Arnaud Beauville The decomposition theorem: the smooth case

After Yau’s theorem In 1977 Yau announces his proof of the Calabi conjecture (the proof appears in 1978). As we will see below, the decomposition theorem is a direct consequence of Yau’s theorem, plus some basic results in differential geometry. I believe that this became soon common knowledge among differential geometers, but for some reason nobody bothered to write it down explicitely. Here is why I did it 5 years later. In 1978 Bogomolov published another paper Hamiltonian K¨ ahler manifolds where he claims that no holomorphic symplectic mani- fold exists in dimension ą 2. The error lies in an algebraic manipulation, where I do not understand how he moves from one line to the next. Arnaud Beauville The decomposition theorem: the smooth case

My personal involvement In 1982 Fujiki constructed a counter-example in dimension 4. I soon realized how to extend his construction in any dimension, then I started to study these manifolds and found a number of interesting features. I gave a talk at Harvard beginning of 83; Phil Griffiths, who was an influential editor of the JDG at the time, suggested that I submit my paper there. He added that the JDG was looking for papers with a survey aspect, so that general remarks on manifolds with c 1 “ 0 would be welcome. This is why I wrote a detailed proof of the decomposition theorem. Now let me sketch how the theorem indeed follows from the Calabi conjecture. Arnaud Beauville The decomposition theorem: the smooth case

� � Basics on holonomy p M , g q Riemannian manifold ù parallel transport: v 0 v 1 γ p ù ϕ γ : T p p M q Ý „ Ñ T q p M q q with ϕ γ ˝ ϕ δ “ ϕ δγ . In particular, ϕ : t loops at p u Ý Ñ O p T p p M qq ; Im ϕ : “ H p “ holonomy (sub-)group at p , closed in O p T p p M qq . ` ˘ A tensor field τ is parallel if ϕ γ τ p p q “ τ p q q for every γ . Holonomy principle Evaluation at p gives a bijective correspondence between: parallel tensor fields; tensors on T p p M q invariant under H p . Arnaud Beauville The decomposition theorem: the smooth case

Examples p M , g q with complex structure J P End p T M q , J 2 “ ´ I . 1 p g , J q K¨ ahler ð ñ J parallel ð ñ H p Ă U p T p p M qq . 2 g Ricci-flat ð ñ p K M , g q flat ð ñ H p Ă SU p T p p M qq . 3 The symplectic group: Sp p r q “ U p 2 r q X Sp p 2 r , C q Ă GL p C 2 r q “ U p r , H q Ă GL p H r q . H p Ă Sp p T p p M qq ð ñ D σ 2- form holomorphic symplectic parallel ð ñ D I , J , K parallel complex structures defining H Ñ End p T M q ( M is hyperk¨ ahler ). It is a remarkable fact that there are very few possibilities for the holonomy representation: Arnaud Beauville The decomposition theorem: the smooth case

The de Rham and Berger theorems From now on we assume that M is compact and simply connected . Theorem (de Rham) T p p M q “ À ñ M – ś V i stable under H p ù M i , with V i “ T p i p M i q and H p – ś i i i H p i . Thus we are reduced to irreducible manifolds, i.e. with irreducible holonomy representation. In his thesis (1955), Berger gave a complete list of these representations. In the special case of K¨ ahler manifolds: Theorem (Berger) p M , g q K¨ ahler non symmetric, H p irreducible ñ H p “ U , SU or Sp . Arnaud Beauville The decomposition theorem: the smooth case

Sketch of proof of Theorem B Theorem B : M compact K¨ ahler with π 1 p M q “ 0, K M “ O M . By Yau’s theorem M carries a K¨ ahler metric which is Ricci-flat, that is, with holonomy contained in SU. By the de Rham and Berger theorems, M – ś i X i ˆ ś j Y j , where the X ’s have holonomy SU p n q and the Y ’s Sp p r q (we view SU p 2 q as Sp p 1 q ). To compute H 0 p Ω ˚ q we use the holonomy principle, plus the Bochner principle On a compact K¨ ahler Ricci-flat manifold, a holomorphic tensor field is parallel. ‚ For H “ SU p n q , the only invariant tensor is the determinant. X q “ C ‘ C ω . Then h 2 , 0 “ 0 ñ X projective. Thus H 0 p X , Ω ˚ ‚ For H “ Sp p r q , the only invariant tensors are the powers of the symplectic form, hence H 0 p Y , Ω ˚ Y q “ C r σ s . Arnaud Beauville The decomposition theorem: the smooth case

Sketch of proof of Theorem A M compact K¨ ahler Ricci-flat. Ñ C k ˆ X , Cheeger-Gromoll (1971): isometric isomorphism r Ý „ M with X compact simply connected. Thus M “ p C k ˆ X q{ Γ, with Γ Ă Aut p C k q ˆ Aut p X q . Aut p X q finite ñ D Γ 1 Ă Γ of finite index acting trivially on X . Bieberbach’s theorem ñ D Γ 2 Ă Γ 1 of finite index acting on C k by translations. Then p C k ˆ X q{ Γ 2 – T ˆ X Ñ M finite ´ etale. THE END Arnaud Beauville The decomposition theorem: the smooth case