Klt varieties with trivial canonical class – Holonomy, differential forms, and fundamental groups II Analytic aspects Henri Guenancia joint work with Daniel Greb and Stefan Kebekus April 2020 H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 1 / 13
Background The singular Ricci-flat metric X projective variety of dimension n , H ample line bundle. Main assumption X has canonical singularities and K X ∼ O X . Theorem (Eyssidieux-Guedj-Zeriahi ’06) There exists a unique smooth K¨ ahler form ω on X reg such that 1 ω ∈ c 1 ( H ) | X reg , 2 Ric ω = 0, � X reg ω n = c 1 ( H ) n . 3 H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 2 / 13
Background Decomposition of the tangent sheaf Theorem (Greb-Kebekus-Peternell ’12, Guenancia ’15) There exists a finite quasi-´ etale cover f : A × Z → X and a decomposition � T Z = E i i ∈ I such that q ( Z ) = 0, i.e. ∀ Z ′ → Z quasi-´ etale finite, h 0 ( Z ′ , Ω [1] � Z ′ ) = 0. 1 2 Each E i has vanishing c 1 , is strongly stable wrt any polarization. 3 The vector bundle E i | Z reg is invariant under parallel transport by ω Z defined by f ∗ ω = ω A ⊕ ω Z . H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 3 / 13
Background Objectives q ( X ) = 0 and decompose T X = � Assume � i ∈ I E i into strongly stable pieces. Road map 1 Classify/compute Hol ω ( X reg , E i ). 2 Relate the geometry of E i (e.g. global sections of tensor bundles) to its holonomy group Hol ω ( X reg , E i ). 3 Classify varieties X with T X strongly stable (i.e. | I | = 1). H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 4 / 13
Holonomy: definitions and basic properties Quick recap on riemannian holonomy (1) ( M , g ) Riemannian manifold, Levi-Civita connection ∇ g on T M , x ∈ M . Parallel transport Given a loop γ : [0 , 1] → M with γ (0) = γ (1) = x and v ∈ T M , x , ∃ ! smooth section v ( t ) ∈ T M ,γ ( t ) such that 1 v (0) = v . 2 ∇ g γ ′ ( t ) v ( t ) = 0. Define τ γ ( v ) := v (1) ∈ T M , x � τ γ ∈ O ( T M , x , g x ). H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 5 / 13
Holonomy: definitions and basic properties Quick recap on riemannian holonomy (2) Holonomy group We define G := Hol ( M , g ) := { τ γ , γ loop at x } ⊂ O ( T M , x ) and the connected component of the identity G ◦ := Hol ◦ ( M , g ) := { τ γ , γ loop at x homotopic to 0 } ⊂ O ( T M , x ) Link with fundamental group G ◦ ⊳ G is normal and ∃ canonical surjection π 1 ( M ) ։ G / G ◦ . H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 6 / 13
Holonomy: definitions and basic properties Quick recap on riemannian holonomy (3) Main example ( M , ω ) K¨ ahler Ricci-flat � G ⊂ SU ( n ). Subbundles If E ⊂ T M is a subbundle invariant by parallel transport, then the holonomy G decomposes as G 1 × G 2 where G 1 � E x and G 2 � E ⊥ x and one sets Hol ( M , E , g ) = G 1 . If M is complex and E a holomorphic subbundle, then the C ∞ splitting T M = E ⊕ E ⊥ is actually holomorphic. H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 7 / 13
Holonomy of singular Ricci-flat metrics Elementary results q ( X ) = 0 and decompose T X = � Assume � i ∈ I E i into strongly stable pieces of rank r i , set G = Hol ( X reg , ω ) for some fixed x ∈ X reg Splitting G decomposes as � G = G i i ∈ I with G i ⊂ SU ( r i ). Irreducibility vs Stability The action G i � C r i (resp G ◦ i � C r i ) is irreducible iff E i is stable (resp. strongly stable) wrt H . H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 8 / 13
Holonomy of singular Ricci-flat metrics Classification of the restricted holonomy i = Hol ( X reg , E i , ω ) ◦ is irreducible. � The standard action of G ◦ Berger-Simons classification One of the following cases holds 1 G ◦ i = { 1 } . 2 G ◦ i = Sp ( r i / 2). 3 G ◦ i = SU ( r i ). Case 1. E i | X reg is a flat vector bundle: impossible since by Druel’s result, it would mean that an abelian variety splits off X (after a finite quasi-´ etale cover maybe). Case 2. In the last two cases, G i / G ◦ i injects in U (1); in particular, it is abelian � it is finite! H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 9 / 13
Holonomy of singular Ricci-flat metrics Proof of finiteness claim By Greb-Kebekus-Peternell, can assume that any linear representation of π 1 ( X reg ) extends to π 1 ( X ). G i / G ◦ π 1 ( X reg ) i π 1 ( X ) H 1 ( X , Z ) but the first homology group is finite as rank ( H 1 ( X , Z )) = dim C H 1 ( X , C ) = 2 dim C H 0 ( X , Ω [1] X ) = 0 since q ( X ) ≤ � q ( X ) = 0. H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 10 / 13
Holonomy of singular Ricci-flat metrics Consequence of the classification result Consequence of the previous results: The holonomy group of ( X reg , ω ) is known Up to passing to a further quasi-´ etale finite cover, one can assume that G i is either SU ( r i ) or Sp ( r i / 2) and G is the product of these groups. In particular, one can compute explicitely the algebra of G i -invariant (resp. G -invariant) vectors under the standard or tensor representations of E i , x (resp. T X , x ). H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 11 / 13
The Bochner principle The result Theorem: the Bochner principle (Greb-G-Kebekus ’17) Given a linear representation ρ of GL ( n , C ), the evaluation map at x X ) and T ρ ( G ) induces a bijection between H 0 ( X reg , T ρ X , x . Idea of proof In the smooth case, if σ is a holomorphic tensor, then Bochner-Weitzenb¨ ock formula reads ∆ ω | σ | 2 ω = |∇ ω σ | 2 ω since Ric ω = 0. As the integral of a Laplacian is zero, we get ∇ ω σ = 0, i.e. σ is parallel, i.e. σ x is G -invariant. H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 12 / 13
The Bochner principle The result Theorem: the Bochner principle (Greb-G-Kebekus ’17) Given a linear representation ρ of GL ( n , C ), the evaluation map at x X ) and T ρ ( G ) induces a bijection between H 0 ( X reg , T ρ X , x . Calabi-Yau and Irreducible Holomorphic Symplectic varieties If G = SU ( n ) or G = Sp ( n / 2), then � n n � � C [Ω] , Ω = triv. of K X . (Λ p ( C n ) ∗ ) G = H 0 ( X , Ω [ p ] X ) = C [ σ ] , σ = symplectic 2-form . p =0 p =0 and the same holds for any finite, quasi-´ etale cover Y → X . H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 12 / 13
The Bochner principle Two corollaries Strongly stable varieties are CY or IHS Assume n ≥ 2, and T X is strongly stable wrt H . Then, there is a finite, quasi-´ etale cover Y → X such either 1 Y is CY variety (i.e. G = SU ( n )) or 2 Y is an IHS variety (i.e. G = Sp ( n / 2)). The next corollary plays a key role in H¨ oring-Peternell’s proof of the decomposition theorem. Symmetric power of the tangent sheaf. Assume n ≥ 2, and T X is strongly stable wrt H . Then, for any r ≥ 1, the sheaf Sym [ r ] T X is strongly stable wrt H . H. Guenancia Holonomy of singular Ricci-flat metrics April 2020 13 / 13
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