Canonical metrics on holomorphic Courant algebroids Roberto Rubio - PowerPoint PPT Presentation

Canonical metrics on holomorphic Courant algebroids Roberto Rubio Universitat Aut` onoma de Barcelona Encuentro REAG 2019 Murcia, 10th April 2019 Joint work with Garcia-Fernandez, Shahbazi, and Tipler, arXiv:1803.01873.

Notation For an almost complex structure J on a smooth manifold X , we have the induced decomposition of complex differential forms Ω p , q ( X ) ⊂ Ω • C ( X ).

Notation For an almost complex structure J on a smooth manifold X , we have the induced decomposition of complex differential forms Ω p , q ( X ) ⊂ Ω • C ( X ). For ( X , J ) an almost complex manifold, a hermitian metric g is a riemannian metric for which J is orthogonal, which is equivalently given by ω = g ( J · , · ) ∈ Ω 1 , 1 ( X ) . An ω ∈ Ω 1 , 1 ( X ) with g = ω ( · , J · ) riemannian is called positive.

Notation For an almost complex structure J on a smooth manifold X , we have the induced decomposition of complex differential forms Ω p , q ( X ) ⊂ Ω • C ( X ). For ( X , J ) an almost complex manifold, a hermitian metric g is a riemannian metric for which J is orthogonal, which is equivalently given by ω = g ( J · , · ) ∈ Ω 1 , 1 ( X ) . An ω ∈ Ω 1 , 1 ( X ) with g = ω ( · , J · ) riemannian is called positive. We say that ( X , J ) is K¨ ahler if J is integrable and there exists a hermitian metric ω ∈ Ω 1 , 1 that is closed, d ω = 0. K¨ ahler class: [ ω ] ∈ H 2 ( M , R ). Alternatively, J integrable and hol ( ω ) ⊂ U ( n ), where n = dim C X .

Notation For an almost complex structure J on a smooth manifold X , we have the induced decomposition of complex differential forms Ω p , q ( X ) ⊂ Ω • C ( X ). For ( X , J ) an almost complex manifold, a hermitian metric g is a riemannian metric for which J is orthogonal, which is equivalently given by ω = g ( J · , · ) ∈ Ω 1 , 1 ( X ) . An ω ∈ Ω 1 , 1 ( X ) with g = ω ( · , J · ) riemannian is called positive. We say that ( X , J ) is K¨ ahler if J is integrable and there exists a hermitian metric ω ∈ Ω 1 , 1 that is closed, d ω = 0. K¨ ahler class: [ ω ] ∈ H 2 ( M , R ). Alternatively, J integrable and hol ( ω ) ⊂ U ( n ), where n = dim C X . For integrable J , we have d = ∂ + ¯ ∂ .

Calabi’s conjecture (1954/1957)

Calabi’s conjecture (1954/1957)

Calabi’s conjecture (1954/1957)

20 years later

Yau’s solution ahler with ω n = n ! µ ? For X compact K¨ ahler with volume µ , is there ω K¨

Yau’s solution ahler with ω n = n ! µ ? For X compact K¨ ahler with volume µ , is there ω K¨ ahler class [ ω 0 ] ∈ H 2 ( M , R ), the Calabi Problem For metrics on a fixed K¨ with smooth volume form µ reduces to solve the Complex Monge-Amp` ere equation ∂ϕ ) n = n ! µ ( ω 0 + 2 i ∂ ¯ for a smooth function ϕ on X .

Yau’s solution ahler with ω n = n ! µ ? For X compact K¨ ahler with volume µ , is there ω K¨ ahler class [ ω 0 ] ∈ H 2 ( M , R ), the Calabi Problem For metrics on a fixed K¨ with smooth volume form µ reduces to solve the Complex Monge-Amp` ere equation ∂ϕ ) n = n ! µ ( ω 0 + 2 i ∂ ¯ for a smooth function ϕ on X . Theorem (Yau ’77) Let X be a compact K¨ ahler manifold with smooth volume µ . Then there ahler metric with ω n = n ! µ in any K¨ exists a unique K¨ ahler class.

Calabi-Yau metrics For X admitting a holomorphic volume form Ω (Calabi-Yau manifold), K X := Λ n T ∗ X ∼ = Ω O X , we can use a multiple of Ω ∧ Ω as µ , say, ω n = ( − 1) n ( n − 1) i n Ω ∧ Ω , 2 and the holonomy of the metric is further reduced to SU ( n ) (Calabi-Yau metric). In particular, it is K¨ ahler and Ricci flat.

Calabi-Yau metrics For X admitting a holomorphic volume form Ω (Calabi-Yau manifold), K X := Λ n T ∗ X ∼ = Ω O X , we can use a multiple of Ω ∧ Ω as µ , say, ω n = ( − 1) n ( n − 1) i n Ω ∧ Ω , 2 and the holonomy of the metric is further reduced to SU ( n ) (Calabi-Yau metric). In particular, it is K¨ ahler and Ricci flat. Theorem (Yau ’77) Let ( X , Ω) be a Calabi-Yau manifold. In each K¨ ahler class there exists a unique K¨ ahler metric with holonomy SU ( n ).

Can we extend Yau’s Theorem to complex non-K¨ ahler manifolds?

Can we extend Yau’s Theorem to complex non-K¨ ahler manifolds? We say that a hermitian metric given by a form ω is: K¨ ahler if d ω = 0, ahler with torsion if ∂ ¯ pluriclosed or strong K¨ ∂ω = 0, balanced if d ω n − 1 = 0, Gauduchon if ∂ ¯ ∂ ( ω n − 1 ) = 0.

Can we extend Yau’s Theorem to complex non-K¨ ahler manifolds? We say that a hermitian metric given by a form ω is: K¨ ahler if d ω = 0, ahler with torsion if ∂ ¯ pluriclosed or strong K¨ ∂ω = 0, balanced if d ω n − 1 = 0, Gauduchon if ∂ ¯ ∂ ( ω n − 1 ) = 0. Theorem (Gauduchon ’77): A compact complex manifold X admits a Gauduchon metric on each hermitian conformal class, unique up to scaling when n > 1.

Can we extend Yau’s Theorem to complex non-K¨ ahler manifolds? We say that a hermitian metric given by a form ω is: K¨ ahler if d ω = 0, ahler with torsion if ∂ ¯ pluriclosed or strong K¨ ∂ω = 0, balanced if d ω n − 1 = 0, Gauduchon if ∂ ¯ ∂ ( ω n − 1 ) = 0. Theorem (Gauduchon ’77): A compact complex manifold X admits a Gauduchon metric on each hermitian conformal class, unique up to scaling when n > 1. But this does not relate to any cohomological quantity.

X compact complex manifold of dimension n , with c 1 ( X ) = 0 ∈ H 2 ( X , Z ).

X compact complex manifold of dimension n , with c 1 ( X ) = 0 ∈ H 2 ( X , Z ). Definition: An SU ( n )-structure on X is a pair (Ψ , ω ) such that ω ∈ Ω 1 , 1 ( X ) that is positive (i.e., g is riemannian), Ψ is a non-vanishing complex ( n , 0)-form on X , normalized such that n ( n − 1) � Ψ � ω = 1 (that is, ω n = ( − 1) i n Ψ ∧ Ψ) 2

X compact complex manifold of dimension n , with c 1 ( X ) = 0 ∈ H 2 ( X , Z ). Definition: An SU ( n )-structure on X is a pair (Ψ , ω ) such that ω ∈ Ω 1 , 1 ( X ) that is positive (i.e., g is riemannian), Ψ is a non-vanishing complex ( n , 0)-form on X , normalized such that n ( n − 1) � Ψ � ω = 1 (that is, ω n = ( − 1) i n Ψ ∧ Ψ) 2 Lee form : only θ ω ∈ Ω 1 ( X ) such that d ω n − 1 = θ ω ∧ ω n − 1 (or θ ω = Jd ∗ ω ).

X compact complex manifold of dimension n , with c 1 ( X ) = 0 ∈ H 2 ( X , Z ). Definition: An SU ( n )-structure on X is a pair (Ψ , ω ) such that ω ∈ Ω 1 , 1 ( X ) that is positive (i.e., g is riemannian), Ψ is a non-vanishing complex ( n , 0)-form on X , normalized such that n ( n − 1) � Ψ � ω = 1 (that is, ω n = ( − 1) i n Ψ ∧ Ψ) 2 Lee form : only θ ω ∈ Ω 1 ( X ) such that d ω n − 1 = θ ω ∧ ω n − 1 (or θ ω = Jd ∗ ω ). Definition: An SU ( n )-structure (Ψ , ω ) is a solution to the twisted Calabi-Yau system on X if: (3) ∂ ¯ (1) d Ψ − θ ω ∧ Ψ = 0 , (2) d θ ω = 0 , ∂ω = 0 .

(3) ∂ ¯ (1) d Ψ − θ ω ∧ Ψ = 0 , (2) d θ ω = 0 , ∂ω = 0 ,

(3) ∂ ¯ (1) d Ψ − θ ω ∧ Ψ = 0 , (2) d θ ω = 0 , ∂ω = 0 , (1) + (2) ⇒ the Bismut connection ∇ + = ∇ g − d c ω/ 2 satisfies hol ( ∇ + ) ⊂ SU ( n ) (Calabi-Yau with torsion, recall d c = − J ◦ d ◦ J ). (3) ⇒ ω strong K¨ ahler with torsion, or pluriclosed.

(3) ∂ ¯ (1) d Ψ − θ ω ∧ Ψ = 0 , (2) d θ ω = 0 , ∂ω = 0 , (1) + (2) ⇒ the Bismut connection ∇ + = ∇ g − d c ω/ 2 satisfies hol ( ∇ + ) ⊂ SU ( n ) (Calabi-Yau with torsion, recall d c = − J ◦ d ◦ J ). (3) ⇒ ω strong K¨ ahler with torsion, or pluriclosed. Moreover, the class [ θ ω ] is an invariant of the solutions: for fixed J , all solutions ω give the same class. when [ θ ω ] = 0 ∈ H 1 ( X , R ), X admits a holomorphic volume form Ω and the equations are equivalent to the Calabi-Yau condition: n ( n − 1) ω n = ( − 1) i n Ω ∧ Ω . d ω = 0 , 2

The twisted Calabi-Yau system admits solutions for both K¨ ahler and non-K¨ ahler surfaces.

The twisted Calabi-Yau system admits solutions for both K¨ ahler and non-K¨ ahler surfaces. Proposition (Garcia-Fernandez–R–Shahbazi–Tipler) A compact complex surface X admits a solution of the twisted Calabi-Yau system if and only if X ∼ = K 3 or T 4 , when [ θ ω ] = 0, X = C 2 \{ 0 } / Γ is a quaternionic Hopf surface, when [ θ ω ] � = 0.

The twisted Calabi-Yau system admits solutions for both K¨ ahler and non-K¨ ahler surfaces. Proposition (Garcia-Fernandez–R–Shahbazi–Tipler) A compact complex surface X admits a solution of the twisted Calabi-Yau system if and only if X ∼ = K 3 or T 4 , when [ θ ω ] = 0, X = C 2 \{ 0 } / Γ is a quaternionic Hopf surface, when [ θ ω ] � = 0. Observe: if X Hopf surface, then H 2 ( X , R ) = 0. What is the analogue of K¨ ahler cone in Yau’s Theorem?

� � � � � � Cohomologies in complex geometry H • , • BC ( X ) H • , • H • , • H k ∂ ( X ) dR ( X , C ) ∂ ( X ) ¯ H • , • A ( X ) A ( X ) = Ker ∂ ¯ BC ( X ) = Ker d ∂ H p , q H p , q ∂ , Im ∂ ¯ Im ∂ ⊕ ¯ ∂ Notation: H • , • p + q = k H p , q ∗ ( X ) = � ( X ) ∗