Lecture 1: Compact K¨ahler manifolds

Lecture 1: Compact K¨ ahler manifolds

Vincent Guedj

Institut de Math´ematiques de Toulouse

PhD course,

Rome, April 2021

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 1 / 19

Quasi-plurisubharmonic functions on K¨ ahler manifolds

The aim of today’s lectures is to

explain the definition and provide examples of K¨ ahler metrics; motivate the search for canonical K¨ ahler metrics;

discuss basic properties of quasi-plurisubharmonic functions. The precise plan is as follows:

Lecture 1: a panoramic view of compact K¨ ahler manifolds.

Lecture 2: uniform integrability properties of quasi-psh functions.

Quasi-plurisubharmonic functions on K¨ ahler manifolds

The aim of today’s lectures is to

explain the definition and provide examples of K¨ ahler metrics;

motivate the search for canonical K¨ ahler metrics;

discuss basic properties of quasi-plurisubharmonic functions. The precise plan is as follows:

Lecture 1: a panoramic view of compact K¨ ahler manifolds. Lecture 2: uniform integrability properties of quasi-psh functions.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 2 / 19

Quasi-plurisubharmonic functions on K¨ ahler manifolds

The aim of today’s lectures is to

explain the definition and provide examples of K¨ ahler metrics;

motivate the search for canonical K¨ ahler metrics;

discuss basic properties of quasi-plurisubharmonic functions. The precise plan is as follows:

Lecture 1: a panoramic view of compact K¨ ahler manifolds.

Lecture 2: uniform integrability properties of quasi-psh functions.

Quasi-plurisubharmonic functions on K¨ ahler manifolds

The aim of today’s lectures is to

explain the definition and provide examples of K¨ ahler metrics;

motivate the search for canonical K¨ ahler metrics;

discuss basic properties of quasi-plurisubharmonic functions.

The precise plan is as follows:

Lecture 1: a panoramic view of compact K¨ ahler manifolds. Lecture 2: uniform integrability properties of quasi-psh functions.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 2 / 19

Quasi-plurisubharmonic functions on K¨ ahler manifolds

The aim of today’s lectures is to

explain the definition and provide examples of K¨ ahler metrics;

motivate the search for canonical K¨ ahler metrics;

discuss basic properties of quasi-plurisubharmonic functions.

The precise plan is as follows:

Lecture 1: a panoramic view of compact K¨ ahler manifolds.

Lecture 2: uniform integrability properties of quasi-psh functions.

Quasi-plurisubharmonic functions on K¨ ahler manifolds

The aim of today’s lectures is to

explain the definition and provide examples of K¨ ahler metrics;

motivate the search for canonical K¨ ahler metrics;

discuss basic properties of quasi-plurisubharmonic functions.

The precise plan is as follows:

Lecture 1: a panoramic view of compact K¨ ahler manifolds.

Lecture 2: uniform integrability properties of quasi-psh functions.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 2 / 19

Quasi-plurisubharmonic functions on K¨ ahler manifolds

The aim of today’s lectures is to

explain the definition and provide examples of K¨ ahler metrics;

motivate the search for canonical K¨ ahler metrics;

discuss basic properties of quasi-plurisubharmonic functions.

The precise plan is as follows:

Lecture 1: a panoramic view of compact K¨ ahler manifolds.

Lecture 2: uniform integrability properties of quasi-psh functions.

K¨ahler manifolds

K¨ ahler metrics

Let X be a compact complex manifold of dimension n ∈ N

.

Using local charts and partition of unity, one can find plenty of hermitian forms,

ω

= P

i ,j =1

ω

idz

∧ d z

. where

(z

) are local holomorphic coordinates;

ω

are smooth functions such that (ω

) is hermitian at all points. Definition

The hermitian form ω is K¨ ahler if its is closed d ω = 0. A manifold is K¨ ahler if it admits a K¨ ahler form.

Euclidean metric in local chart=local example of K¨ ahler metric. Using cut-off functions destroy the condition d ω = 0.

Such a form is associated to a Riemannian metric on TX .

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 3 / 19

K¨ahler manifolds

K¨ ahler metrics

Let X be a compact complex manifold of dimension n ∈ N

. Using local charts and partition of unity, one can find plenty of hermitian forms,

ω

= P

i ,j =1

ω

idz

∧ d z

. where

(z

) are local holomorphic coordinates;

ω

are smooth functions such that (ω

) is hermitian at all points. Definition

The hermitian form ω is K¨ ahler if its is closed d ω = 0. A manifold is K¨ ahler if it admits a K¨ ahler form.

Euclidean metric in local chart=local example of K¨ ahler metric. Using cut-off functions destroy the condition d ω = 0.

Such a form is associated to a Riemannian metric on TX .

K¨ahler manifolds

K¨ ahler metrics

Let X be a compact complex manifold of dimension n ∈ N

. Using local charts and partition of unity, one can find plenty of hermitian forms,

ω

= P

i ,j =1

ω

idz

∧ d z

.

where

(z

) are local holomorphic coordinates;

ω

are smooth functions such that (ω

) is hermitian at all points. Definition

The hermitian form ω is K¨ ahler if its is closed d ω = 0. A manifold is K¨ ahler if it admits a K¨ ahler form.

Euclidean metric in local chart=local example of K¨ ahler metric. Using cut-off functions destroy the condition d ω = 0.

Such a form is associated to a Riemannian metric on TX .

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 3 / 19

K¨ahler manifolds

K¨ ahler metrics

Let X be a compact complex manifold of dimension n ∈ N

. Using local charts and partition of unity, one can find plenty of hermitian forms,

ω

= P

i ,j =1

ω

idz

∧ d z

. where

(z

) are local holomorphic coordinates;

ω

are smooth functions such that (ω

) is hermitian at all points. Definition

The hermitian form ω is K¨ ahler if its is closed d ω = 0. A manifold is K¨ ahler if it admits a K¨ ahler form.

Euclidean metric in local chart=local example of K¨ ahler metric. Using cut-off functions destroy the condition d ω = 0.

Such a form is associated to a Riemannian metric on TX .

K¨ahler manifolds

K¨ ahler metrics

Let X be a compact complex manifold of dimension n ∈ N

. Using local charts and partition of unity, one can find plenty of hermitian forms,

ω

= P

i ,j =1

ω

idz

∧ d z

. where

(z

) are local holomorphic coordinates;

ω

are smooth functions such that (ω

) is hermitian at all points.

Definition

The hermitian form ω is K¨ ahler if its is closed d ω = 0. A manifold is K¨ ahler if it admits a K¨ ahler form.

Euclidean metric in local chart=local example of K¨ ahler metric. Using cut-off functions destroy the condition d ω = 0.

Such a form is associated to a Riemannian metric on TX .

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 3 / 19

K¨ahler manifolds

K¨ ahler metrics

Let X be a compact complex manifold of dimension n ∈ N

. Using local charts and partition of unity, one can find plenty of hermitian forms,

ω

= P

i ,j =1

ω

idz

∧ d z

. where

(z

) are local holomorphic coordinates;

ω

are smooth functions such that (ω

) is hermitian at all points.

Definition

The hermitian form ω is K¨ ahler if its is closed d ω = 0.

A manifold is K¨ ahler if it admits a K¨ ahler form.

Euclidean metric in local chart=local example of K¨ ahler metric. Using cut-off functions destroy the condition d ω = 0.

Such a form is associated to a Riemannian metric on TX .

K¨ahler manifolds

K¨ ahler metrics

Let X be a compact complex manifold of dimension n ∈ N

. Using local charts and partition of unity, one can find plenty of hermitian forms,

ω

= P

i ,j =1

ω

idz

∧ d z

. where

(z

) are local holomorphic coordinates;

ω

are smooth functions such that (ω

) is hermitian at all points.

Definition

The hermitian form ω is K¨ ahler if its is closed d ω = 0.

A manifold is K¨ ahler if it admits a K¨ ahler form.

Euclidean metric in local chart=local example of K¨ ahler metric. Using cut-off functions destroy the condition d ω = 0.

Such a form is associated to a Riemannian metric on TX .

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 3 / 19

K¨ahler manifolds

K¨ ahler metrics

Let X be a compact complex manifold of dimension n ∈ N

. Using local charts and partition of unity, one can find plenty of hermitian forms,

ω

= P

i ,j =1

ω

idz

∧ d z

. where

(z

) are local holomorphic coordinates;

ω

are smooth functions such that (ω

) is hermitian at all points.

Definition

The hermitian form ω is K¨ ahler if its is closed d ω = 0.

A manifold is K¨ ahler if it admits a K¨ ahler form.

Using cut-off functions destroy the condition d ω = 0.

Such a form is associated to a Riemannian metric on TX .

K¨ahler manifolds

K¨ ahler metrics

Let X be a compact complex manifold of dimension n ∈ N

. Using local charts and partition of unity, one can find plenty of hermitian forms,

ω

= P

i ,j =1

ω

idz

∧ d z

. where

(z

) are local holomorphic coordinates;

ω

are smooth functions such that (ω

) is hermitian at all points.

Definition

The hermitian form ω is K¨ ahler if its is closed d ω = 0.

A manifold is K¨ ahler if it admits a K¨ ahler form.

Euclidean metric in local chart=local example of K¨ ahler metric.

Using cut-off functions destroy the condition d ω = 0.

Such a form is associated to a Riemannian metric on TX .

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 3 / 19

K¨ahler manifolds

K¨ ahler metrics

Let X be a compact complex manifold of dimension n ∈ N

. Using local charts and partition of unity, one can find plenty of hermitian forms,

ω

= P

i ,j =1

ω

idz

∧ d z

. where

(z

) are local holomorphic coordinates;

ω

are smooth functions such that (ω

) is hermitian at all points.

Definition

The hermitian form ω is K¨ ahler if its is closed d ω = 0.

A manifold is K¨ ahler if it admits a K¨ ahler form.

K¨ahler manifolds

Examples

Example (Tori)

Euclidean form ω = P

α=1

idz

∧ d z

closed

and invariant by translations

⇒ K¨ ahler form on compact tori X = C

/Λ, where Λ ⊂ R

is a lattice. Example (Projective space)

The cplx projective space CP

= set of complex lines through 0 ∈ C

. Homogeneous coordinates [z] = [z

: · · · : z

] = [λz

: · · · : λz

], λ ∈ C

. Charts x ∈ C

7→ [x

, . . . , 1, . . . , x

] ∈ U

= {[z] ∈ CP

, z

6= 0} ∼ C

. The K¨ ahler form ω =

∂∂ log[1 + ||x ||

] defines a K¨ ahler form on CP

This is the Fubini-Study K¨ ahler form. Exercise: check that R

CPⁿ

ω

= 1.

Example (Hopf surface)

The surface X = C

/hz 7→ 2zi ∼ S

× S

does not admit any K¨ ahler form.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 4 / 19

K¨ahler manifolds

Examples

Example (Tori)

Euclidean form ω = P

α=1

idz

∧ d z

closed and invariant by translations

⇒ K¨ ahler form on compact tori X = C

/Λ, where Λ ⊂ R

is a lattice. Example (Projective space)

The cplx projective space CP

= set of complex lines through 0 ∈ C

. Homogeneous coordinates [z] = [z

: · · · : z

] = [λz

: · · · : λz

], λ ∈ C

. Charts x ∈ C

7→ [x

, . . . , 1, . . . , x

] ∈ U

= {[z] ∈ CP

, z

6= 0} ∼ C

. The K¨ ahler form ω =

∂∂ log[1 + ||x ||

] defines a K¨ ahler form on CP

This is the Fubini-Study K¨ ahler form. Exercise: check that R

CPⁿ

ω

= 1. Example (Hopf surface)

The surface X = C

/hz 7→ 2zi ∼ S

× S

does not admit any K¨ ahler form.

K¨ahler manifolds

Examples

Example (Tori)

Euclidean form ω = P

α=1

idz

∧ d z

closed and invariant by translations

⇒ K¨ ahler form on compact tori X = C

/Λ, where Λ ⊂ R

is a lattice.

Example (Projective space)

The cplx projective space CP

= set of complex lines through 0 ∈ C

. Homogeneous coordinates [z] = [z

: · · · : z

] = [λz

: · · · : λz

], λ ∈ C

. Charts x ∈ C

7→ [x

, . . . , 1, . . . , x

] ∈ U

= {[z] ∈ CP

, z

6= 0} ∼ C

. The K¨ ahler form ω =

∂∂ log[1 + ||x ||

] defines a K¨ ahler form on CP

This is the Fubini-Study K¨ ahler form. Exercise: check that R

CPⁿ

ω

= 1.

Example (Hopf surface)

The surface X = C

/hz 7→ 2zi ∼ S

× S

does not admit any K¨ ahler form.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 4 / 19

K¨ahler manifolds

Examples

Example (Tori)

Euclidean form ω = P

α=1

idz

∧ d z

closed and invariant by translations

⇒ K¨ ahler form on compact tori X = C

/Λ, where Λ ⊂ R

is a lattice.

Example (Projective space)

The cplx projective space CP

= set of complex lines through 0 ∈ C

.

Homogeneous coordinates [z] = [z

: · · · : z

] = [λz

: · · · : λz

], λ ∈ C

. Charts x ∈ C

7→ [x

, . . . , 1, . . . , x

] ∈ U

= {[z] ∈ CP

, z

6= 0} ∼ C

. The K¨ ahler form ω =

∂∂ log[1 + ||x ||

] defines a K¨ ahler form on CP

This is the Fubini-Study K¨ ahler form. Exercise: check that R

CPⁿ

ω

= 1. Example (Hopf surface)

The surface X = C

/hz 7→ 2zi ∼ S

× S

does not admit any K¨ ahler form.

K¨ahler manifolds

Examples

Example (Tori)

Euclidean form ω = P

α=1

idz

∧ d z

closed and invariant by translations

⇒ K¨ ahler form on compact tori X = C

/Λ, where Λ ⊂ R

is a lattice.

Example (Projective space)

The cplx projective space CP

= set of complex lines through 0 ∈ C

. Homogeneous coordinates [z] = [z

: · · · : z

] = [λz

: · · · : λz

], λ ∈ C

.

Charts x ∈ C

7→ [x

, . . . , 1, . . . , x

] ∈ U

= {[z] ∈ CP

, z

6= 0} ∼ C

. The K¨ ahler form ω =

∂∂ log[1 + ||x ||

] defines a K¨ ahler form on CP

This is the Fubini-Study K¨ ahler form. Exercise: check that R

CPⁿ

ω

= 1.

Example (Hopf surface)

The surface X = C

/hz 7→ 2zi ∼ S

× S

does not admit any K¨ ahler form.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 4 / 19

K¨ahler manifolds

Examples

Example (Tori)

Euclidean form ω = P

α=1

idz

∧ d z

closed and invariant by translations

⇒ K¨ ahler form on compact tori X = C

/Λ, where Λ ⊂ R

is a lattice.

Example (Projective space)

The cplx projective space CP

= set of complex lines through 0 ∈ C

. Homogeneous coordinates [z] = [z

: · · · : z

] = [λz

: · · · : λz

], λ ∈ C

. Charts x ∈ C

7→ [x

, . . . , 1, . . . , x

] ∈ U

= {[z] ∈ CP

, z

6= 0} ∼ C

.

The K¨ ahler form ω =

∂∂ log[1 + ||x ||

] defines a K¨ ahler form on CP

This is the Fubini-Study K¨ ahler form. Exercise: check that R

CPⁿ

ω

= 1. Example (Hopf surface)

The surface X = C

/hz 7→ 2zi ∼ S

× S

does not admit any K¨ ahler form.

K¨ahler manifolds

Examples

Example (Tori)

Euclidean form ω = P

α=1

idz

∧ d z

closed and invariant by translations

⇒ K¨ ahler form on compact tori X = C

/Λ, where Λ ⊂ R

is a lattice.

Example (Projective space)

The cplx projective space CP

= set of complex lines through 0 ∈ C

. Homogeneous coordinates [z] = [z

: · · · : z

] = [λz

: · · · : λz

], λ ∈ C

. Charts x ∈ C

7→ [x

, . . . , 1, . . . , x

] ∈ U

= {[z] ∈ CP

, z

6= 0} ∼ C

. The K¨ ahler form ω =

∂∂ log[1 + ||x ||

] defines a K¨ ahler form on CP

This is the Fubini-Study K¨ ahler form. Exercise: check that R

CPⁿ

ω

= 1.

Example (Hopf surface)

The surface X = C

/hz 7→ 2zi ∼ S

× S

does not admit any K¨ ahler form.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 4 / 19

K¨ahler manifolds

Examples

Example (Tori)

Euclidean form ω = P

α=1

idz

∧ d z

closed and invariant by translations

⇒ K¨ ahler form on compact tori X = C

/Λ, where Λ ⊂ R

is a lattice.

Example (Projective space)

The cplx projective space CP

= set of complex lines through 0 ∈ C

. Homogeneous coordinates [z] = [z

: · · · : z

] = [λz

: · · · : λz

], λ ∈ C

. Charts x ∈ C

7→ [x

, . . . , 1, . . . , x

] ∈ U

= {[z] ∈ CP

, z

6= 0} ∼ C

. The K¨ ahler form ω =

∂∂ log[1 + ||x ||

] defines a K¨ ahler form on CP

This is the Fubini-Study K¨ ahler form.

Exercise: check that R

CPⁿ

ω

= 1. Example (Hopf surface)

The surface X = C

/hz 7→ 2zi ∼ S

× S

does not admit any K¨ ahler form.

K¨ahler manifolds

Examples

Example (Tori)

Euclidean form ω = P

α=1

idz

∧ d z

closed and invariant by translations

⇒ K¨ ahler form on compact tori X = C

/Λ, where Λ ⊂ R

is a lattice.

Example (Projective space)

The cplx projective space CP

= set of complex lines through 0 ∈ C

. Homogeneous coordinates [z] = [z

: · · · : z

] = [λz

: · · · : λz

], λ ∈ C

. Charts x ∈ C

7→ [x

, . . . , 1, . . . , x

] ∈ U

= {[z] ∈ CP

, z

6= 0} ∼ C

. The K¨ ahler form ω =

∂∂ log[1 + ||x ||

] defines a K¨ ahler form on CP

This is the Fubini-Study K¨ ahler form. Exercise: check that R

CPⁿ

ω

= 1.

Example (Hopf surface)

The surface X = C

/hz 7→ 2zi ∼ S

× S

does not admit any K¨ ahler form.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 4 / 19

K¨ahler manifolds

Examples

Example (Tori)

Euclidean form ω = P

α=1

idz

∧ d z

closed and invariant by translations

⇒ K¨ ahler form on compact tori X = C

/Λ, where Λ ⊂ R

is a lattice.

Example (Projective space)

The cplx projective space CP

= set of complex lines through 0 ∈ C

. Homogeneous coordinates [z] = [z

: · · · : z

] = [λz

: · · · : λz

], λ ∈ C

. Charts x ∈ C

7→ [x

, . . . , 1, . . . , x

] ∈ U

= {[z] ∈ CP

, z

6= 0} ∼ C

. The K¨ ahler form ω =

∂∂ log[1 + ||x ||

] defines a K¨ ahler form on CP

This is the Fubini-Study K¨ ahler form. Exercise: check that R

CPⁿ

ω

= 1.

K¨ahler manifolds

Basic constructions

If ω is K¨ ahler, then so is ω + i ∂∂ϕ if ϕ ∈ C

(X , R) is C

-small.

A product of compact K¨ ahler manifolds is K¨ ahler, cf ω(x , y ) = ω

(x ) + ω

(y ) = K¨ ahler form on X × Y . If f : X → Y is a holomorphic embedding and ω

K¨ ahler, then

ω

= f

ω

is a K¨ ahler form on X .

⇒ a submanifold of a compact K¨ ahler manifold is K¨ ahler.

⇒ any projective algebraic manifold is K¨ ahler.

⇒ any compact Riemann surface is K¨ ahler.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 5 / 19

K¨ahler manifolds

Basic constructions

If ω is K¨ ahler, then so is ω + i ∂∂ϕ if ϕ ∈ C

(X , R) is C

-small.

A product of compact K¨ ahler manifolds is K¨ ahler,

cf ω(x , y ) = ω

(x ) + ω

(y ) = K¨ ahler form on X × Y . If f : X → Y is a holomorphic embedding and ω

K¨ ahler, then

ω

= f

ω

is a K¨ ahler form on X .

⇒ a submanifold of a compact K¨ ahler manifold is K¨ ahler.

⇒ any projective algebraic manifold is K¨ ahler.

⇒ any compact Riemann surface is K¨ ahler.

K¨ahler manifolds

Basic constructions

If ω is K¨ ahler, then so is ω + i ∂∂ϕ if ϕ ∈ C

(X , R) is C

-small.

A product of compact K¨ ahler manifolds is K¨ ahler, cf ω(x , y ) = ω

(x ) + ω

(y ) = K¨ ahler form on X × Y .

If f : X → Y is a holomorphic embedding and ω

K¨ ahler, then ω

= f

ω

is a K¨ ahler form on X .

⇒ a submanifold of a compact K¨ ahler manifold is K¨ ahler.

⇒ any projective algebraic manifold is K¨ ahler.

⇒ any compact Riemann surface is K¨ ahler.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 5 / 19

K¨ahler manifolds

Basic constructions

If ω is K¨ ahler, then so is ω + i ∂∂ϕ if ϕ ∈ C

(X , R) is C

-small.

A product of compact K¨ ahler manifolds is K¨ ahler, cf ω(x , y ) = ω

(x ) + ω

(y ) = K¨ ahler form on X × Y . If f : X → Y is a holomorphic embedding and ω

K¨ ahler,

then ω

= f

ω

is a K¨ ahler form on X .

⇒ a submanifold of a compact K¨ ahler manifold is K¨ ahler.

⇒ any projective algebraic manifold is K¨ ahler.

⇒ any compact Riemann surface is K¨ ahler.

K¨ahler manifolds

Basic constructions

If ω is K¨ ahler, then so is ω + i ∂∂ϕ if ϕ ∈ C

(X , R) is C

-small.

A product of compact K¨ ahler manifolds is K¨ ahler, cf ω(x , y ) = ω

(x ) + ω

(y ) = K¨ ahler form on X × Y . If f : X → Y is a holomorphic embedding and ω

K¨ ahler, then

ω

= f

ω

is a K¨ ahler form on X .

⇒ a submanifold of a compact K¨ ahler manifold is K¨ ahler.

⇒ any projective algebraic manifold is K¨ ahler.

⇒ any compact Riemann surface is K¨ ahler.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 5 / 19

K¨ahler manifolds

Basic constructions

If ω is K¨ ahler, then so is ω + i ∂∂ϕ if ϕ ∈ C

(X , R) is C

-small.

A product of compact K¨ ahler manifolds is K¨ ahler, cf ω(x , y ) = ω

(x ) + ω

(y ) = K¨ ahler form on X × Y . If f : X → Y is a holomorphic embedding and ω

K¨ ahler, then

ω

= f

ω

is a K¨ ahler form on X .

⇒ a submanifold of a compact K¨ ahler manifold is K¨ ahler.

⇒ any projective algebraic manifold is K¨ ahler.

⇒ any compact Riemann surface is K¨ ahler.

K¨ahler manifolds

Basic constructions

If ω is K¨ ahler, then so is ω + i ∂∂ϕ if ϕ ∈ C

(X , R) is C

-small.

A product of compact K¨ ahler manifolds is K¨ ahler, cf ω(x , y ) = ω

(x ) + ω

(y ) = K¨ ahler form on X × Y . If f : X → Y is a holomorphic embedding and ω

K¨ ahler, then

ω

= f

ω

is a K¨ ahler form on X .

⇒ a submanifold of a compact K¨ ahler manifold is K¨ ahler.

⇒ any projective algebraic manifold is K¨ ahler.

⇒ any compact Riemann surface is K¨ ahler.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 5 / 19

K¨ahler manifolds

Basic constructions

If ω is K¨ ahler, then so is ω + i ∂∂ϕ if ϕ ∈ C

(X , R) is C

-small.

A product of compact K¨ ahler manifolds is K¨ ahler, cf ω(x , y ) = ω

(x ) + ω

(y ) = K¨ ahler form on X × Y . If f : X → Y is a holomorphic embedding and ω

K¨ ahler, then

ω

= f

ω

is a K¨ ahler form on X .

⇒ a submanifold of a compact K¨ ahler manifold is K¨ ahler.

⇒ any projective algebraic manifold is K¨ ahler.

⇒ any compact Riemann surface is K¨ ahler.

K¨ahler manifolds

Blow-ups

Example (Local blow up of a point) Let B be a ball centered at zero in C

and consider B = {(z, `) ∈ B × P ˜

, z ∈ `}.

This is a complex manifold of C

× P

of dimension n such that the projection π : (z, `) ∈ ˜ B 7→ z ∈ B satisfies E := π

(0) ∼ P

; π is a biholomorphism from ˜ B \ E onto B \ {0};

ω = π

dd

(log |z| + |z|

) − [E ] is a K¨ ahler form on ˜ B

Can globalize this construction, blowing up an arbitrary point in X . One can also blow up any submanifold Y ⊂ X of codimension ≥ 2. The blow up of a K¨ ahler manifold is a K¨ ahler manifold.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 6 / 19

K¨ahler manifolds

Blow-ups

Example (Local blow up of a point)

Let B be a ball centered at zero in C

and consider B = {(z, `) ∈ B × P ˜

, z ∈ `}.

This is a complex manifold of C

× P

of dimension n such that the projection π : (z, `) ∈ ˜ B 7→ z ∈ B satisfies E := π

(0) ∼ P

; π is a biholomorphism from ˜ B \ E onto B \ {0};

ω = π

dd

(log |z| + |z|

) − [E ] is a K¨ ahler form on ˜ B

Can globalize this construction, blowing up an arbitrary point in X .

One can also blow up any submanifold Y ⊂ X of codimension ≥ 2.

The blow up of a K¨ ahler manifold is a K¨ ahler manifold.

K¨ahler manifolds

Blow-ups

Example (Local blow up of a point)

Let B be a ball centered at zero in C

and consider B = {(z, `) ∈ B × P ˜

, z ∈ `}.

This is a complex manifold of C

× P

of dimension n such that

the projection π : (z, `) ∈ ˜ B 7→ z ∈ B satisfies E := π

(0) ∼ P

; π is a biholomorphism from ˜ B \ E onto B \ {0};

ω = π

dd

(log |z| + |z|

) − [E ] is a K¨ ahler form on ˜ B

Can globalize this construction, blowing up an arbitrary point in X . One can also blow up any submanifold Y ⊂ X of codimension ≥ 2. The blow up of a K¨ ahler manifold is a K¨ ahler manifold.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 6 / 19

K¨ahler manifolds

Blow-ups

Example (Local blow up of a point)

Let B be a ball centered at zero in C

and consider B = {(z, `) ∈ B × P ˜

, z ∈ `}.

This is a complex manifold of C

× P

of dimension n such that the projection π : (z, `) ∈ ˜ B 7→ z ∈ B satisfies E := π

(0) ∼ P

;

π is a biholomorphism from ˜ B \ E onto B \ {0}; ω = π

dd

(log |z| + |z|

) − [E ] is a K¨ ahler form on ˜ B

Can globalize this construction, blowing up an arbitrary point in X .

One can also blow up any submanifold Y ⊂ X of codimension ≥ 2.

The blow up of a K¨ ahler manifold is a K¨ ahler manifold.

K¨ahler manifolds

Blow-ups

Example (Local blow up of a point)

Let B be a ball centered at zero in C

and consider B = {(z, `) ∈ B × P ˜

, z ∈ `}.

This is a complex manifold of C

× P

of dimension n such that the projection π : (z, `) ∈ ˜ B 7→ z ∈ B satisfies E := π

(0) ∼ P

; π is a biholomorphism from ˜ B \ E onto B \ {0};

ω = π

dd

(log |z| + |z|

) − [E ] is a K¨ ahler form on ˜ B

Can globalize this construction, blowing up an arbitrary point in X . One can also blow up any submanifold Y ⊂ X of codimension ≥ 2. The blow up of a K¨ ahler manifold is a K¨ ahler manifold.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 6 / 19

K¨ahler manifolds

Blow-ups

Example (Local blow up of a point)

Let B be a ball centered at zero in C

and consider B = {(z, `) ∈ B × P ˜

, z ∈ `}.

This is a complex manifold of C

× P

of dimension n such that the projection π : (z, `) ∈ ˜ B 7→ z ∈ B satisfies E := π

(0) ∼ P

; π is a biholomorphism from ˜ B \ E onto B \ {0};

ω = π

dd

(log |z| + |z|

) − [E ] is a K¨ ahler form on ˜ B

Can globalize this construction, blowing up an arbitrary point in X .

One can also blow up any submanifold Y ⊂ X of codimension ≥ 2.

The blow up of a K¨ ahler manifold is a K¨ ahler manifold.

K¨ahler manifolds

Blow-ups

Example (Local blow up of a point)

Let B be a ball centered at zero in C

and consider B = {(z, `) ∈ B × P ˜

, z ∈ `}.

This is a complex manifold of C

× P

of dimension n such that the projection π : (z, `) ∈ ˜ B 7→ z ∈ B satisfies E := π

(0) ∼ P

; π is a biholomorphism from ˜ B \ E onto B \ {0};

ω = π

dd

(log |z| + |z|

) − [E ] is a K¨ ahler form on ˜ B

Can globalize this construction, blowing up an arbitrary point in X .

One can also blow up any submanifold Y ⊂ X of codimension ≥ 2. The blow up of a K¨ ahler manifold is a K¨ ahler manifold.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 6 / 19

K¨ahler manifolds

Blow-ups

Example (Local blow up of a point)

Let B be a ball centered at zero in C

and consider B = {(z, `) ∈ B × P ˜

, z ∈ `}.

This is a complex manifold of C

× P

of dimension n such that the projection π : (z, `) ∈ ˜ B 7→ z ∈ B satisfies E := π

(0) ∼ P

; π is a biholomorphism from ˜ B \ E onto B \ {0};

ω = π

dd

(log |z| + |z|

) − [E ] is a K¨ ahler form on ˜ B

Can globalize this construction, blowing up an arbitrary point in X .

The blow up of a K¨ ahler manifold is a K¨ ahler manifold.

K¨ahler manifolds

Blow-ups

Example (Local blow up of a point)

Let B be a ball centered at zero in C

and consider B = {(z, `) ∈ B × P ˜

, z ∈ `}.

This is a complex manifold of C

× P

of dimension n such that the projection π : (z, `) ∈ ˜ B 7→ z ∈ B satisfies E := π

(0) ∼ P

; π is a biholomorphism from ˜ B \ E onto B \ {0};

ω = π

dd

(log |z| + |z|

) − [E ] is a K¨ ahler form on ˜ B

Can globalize this construction, blowing up an arbitrary point in X . One can also blow up any submanifold Y ⊂ X of codimension ≥ 2.

The blow up of a K¨ ahler manifold is a K¨ ahler manifold.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 6 / 19

K¨ahler manifolds

Local characterizations

Proposition (Normal coordinates) Let (X , ω) be a cplx hermitian manifold.

The form ω is K¨ ahler iff for each p ∈ X there exists local holomorphic coordinates centered at p such that

ω =

n

X

i ,j =1

ω

idz

∧ d z

with ω

= δ

+ O(||z||

).

Proposition (Local ∂∂-lemma)

Let (X , ω) be a cplx hermitian manifold. The form ω is K¨ ahler iff locally ω = i ∂∂ϕ,

where ϕ is smooth and strictly plurisubharmonic.

,→ Not usually possible globally (max principle), but...

K¨ahler manifolds

Local characterizations

Proposition (Normal coordinates)

Let (X , ω) be a cplx hermitian manifold. The form ω is K¨ ahler iff for each p ∈ X

there exists local holomorphic coordinates centered at p such that ω =

n

X

i ,j =1

ω

idz

∧ d z

with ω

= δ

+ O(||z||

).

Proposition (Local ∂∂-lemma)

Let (X , ω) be a cplx hermitian manifold. The form ω is K¨ ahler iff locally ω = i ∂∂ϕ,

where ϕ is smooth and strictly plurisubharmonic.

,→ Not usually possible globally (max principle), but...

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 7 / 19

K¨ahler manifolds

Local characterizations

Proposition (Normal coordinates)

Let (X , ω) be a cplx hermitian manifold. The form ω is K¨ ahler iff for each p ∈ X there exists local holomorphic coordinates centered at p such that

ω =

n

X

i ,j =1

ω

idz

∧ d z

with ω

= δ

+ O(||z||

).

Proposition (Local ∂∂-lemma)

Let (X , ω) be a cplx hermitian manifold. The form ω is K¨ ahler iff locally ω = i ∂∂ϕ,

where ϕ is smooth and strictly plurisubharmonic.

,→ Not usually possible globally (max principle), but...

K¨ahler manifolds

Local characterizations

Proposition (Normal coordinates)

Let (X , ω) be a cplx hermitian manifold. The form ω is K¨ ahler iff for each p ∈ X there exists local holomorphic coordinates centered at p such that

ω =

n

X

i ,j =1

ω

idz

∧ d z

with ω

= δ

+ O(||z||

).

Proposition (Local ∂∂-lemma)

Let (X , ω) be a cplx hermitian manifold.

The form ω is K¨ ahler iff locally ω = i ∂∂ϕ,

where ϕ is smooth and strictly plurisubharmonic.

,→ Not usually possible globally (max principle), but...

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 7 / 19

K¨ahler manifolds

Local characterizations

Proposition (Normal coordinates)

Let (X , ω) be a cplx hermitian manifold. The form ω is K¨ ahler iff for each p ∈ X there exists local holomorphic coordinates centered at p such that

ω =

n

X

i ,j =1

ω

idz

∧ d z

with ω

= δ

+ O(||z||

).

Proposition (Local ∂∂-lemma)

Let (X , ω) be a cplx hermitian manifold. The form ω is K¨ ahler iff locally ω = i ∂∂ϕ,

where ϕ is smooth and strictly plurisubharmonic.

,→ Not usually possible globally (max principle), but...

K¨ahler manifolds

Local characterizations

Proposition (Normal coordinates)

Let (X , ω) be a cplx hermitian manifold. The form ω is K¨ ahler iff for each p ∈ X there exists local holomorphic coordinates centered at p such that

ω =

n

X

i ,j =1

ω

idz

∧ d z

with ω

= δ

+ O(||z||

).

Proposition (Local ∂∂-lemma)

Let (X , ω) be a cplx hermitian manifold. The form ω is K¨ ahler iff locally ω = i ∂∂ϕ,

where ϕ is smooth and strictly plurisubharmonic.

,→ Not usually possible globally (max principle), but...

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 7 / 19

K¨ahler manifolds

Local characterizations

Proposition (Normal coordinates)

Let (X , ω) be a cplx hermitian manifold. The form ω is K¨ ahler iff for each p ∈ X there exists local holomorphic coordinates centered at p such that

ω =

n

X

i ,j =1

ω

idz

∧ d z

with ω

= δ

+ O(||z||

).

Proposition (Local ∂∂-lemma)

Let (X , ω) be a cplx hermitian manifold. The form ω is K¨ ahler iff locally

ω = i ∂∂ϕ,

K¨ahler manifolds

The (global) ∂∂-lemma

A K¨ ahler form ω defines a deRham class {ω} ∈ H

(X , R).

Theorem (∂∂-lemma)

If two K¨ ahler forms ω, ω

define the same cohomology class, then there exists a (essentially unique) ϕ ∈ C

(X , R) such that ω

= ω + i ∂∂ϕ.

Let K

denote set of smooth functions ϕ s.t. ω + i ∂∂ϕ > 0 is K¨ ahler. PSH(X , ω) =the closure of K

in L

will be the hero of Lecture 2. Theorem (Hodge decomposition theorem)

If X is compact K¨ ahler then H

(X , C) is isomorphic to H

(X , C) and H

(X , C) = ⊕

H

(X , C).

,→ In particular b

(X ) = 2h

(X ) even, while b

(S

× S

) = 1 so no Hopf.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 8 / 19

K¨ahler manifolds

The (global) ∂∂-lemma

A K¨ ahler form ω defines a deRham class {ω} ∈ H

(X , R).

Theorem (∂∂-lemma)

If two K¨ ahler forms ω, ω

define the same cohomology class,

then there exists a (essentially unique) ϕ ∈ C

(X , R) such that ω

= ω + i ∂∂ϕ.

Let K

denote set of smooth functions ϕ s.t. ω + i ∂∂ϕ > 0 is K¨ ahler. PSH(X , ω) =the closure of K

in L

will be the hero of Lecture 2. Theorem (Hodge decomposition theorem)

If X is compact K¨ ahler then H

(X , C) is isomorphic to H

(X , C) and H

(X , C) = ⊕

H

(X , C).

,→ In particular b

(X ) = 2h

(X ) even, while b

(S

× S

) = 1 so no Hopf.

K¨ahler manifolds

The (global) ∂∂-lemma

A K¨ ahler form ω defines a deRham class {ω} ∈ H

(X , R).

Theorem (∂∂-lemma)

If two K¨ ahler forms ω, ω

define the same cohomology class, then there exists a (essentially unique) ϕ ∈ C

(X , R) such that ω

= ω + i ∂∂ϕ.

Let K

denote set of smooth functions ϕ s.t. ω + i ∂∂ϕ > 0 is K¨ ahler. PSH(X , ω) =the closure of K

in L

will be the hero of Lecture 2. Theorem (Hodge decomposition theorem)

If X is compact K¨ ahler then H

(X , C) is isomorphic to H

(X , C) and H

(X , C) = ⊕

H

(X , C).

,→ In particular b

(X ) = 2h

(X ) even, while b

(S

× S

) = 1 so no Hopf.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 8 / 19

K¨ahler manifolds

The (global) ∂∂-lemma

A K¨ ahler form ω defines a deRham class {ω} ∈ H

(X , R).

Theorem (∂∂-lemma)

If two K¨ ahler forms ω, ω

define the same cohomology class, then there exists a (essentially unique) ϕ ∈ C

(X , R) such that ω

= ω + i ∂∂ϕ.

Let K

denote set of smooth functions ϕ s.t. ω + i ∂∂ϕ > 0 is K¨ ahler.

PSH(X , ω) =the closure of K

in L

will be the hero of Lecture 2. Theorem (Hodge decomposition theorem)

If X is compact K¨ ahler then H

(X , C) is isomorphic to H

(X , C) and H

(X , C) = ⊕

H

(X , C).

,→ In particular b

(X ) = 2h

(X ) even, while b

(S

× S

) = 1 so no Hopf.

K¨ahler manifolds

The (global) ∂∂-lemma

A K¨ ahler form ω defines a deRham class {ω} ∈ H

(X , R).

Theorem (∂∂-lemma)

If two K¨ ahler forms ω, ω

define the same cohomology class, then there exists a (essentially unique) ϕ ∈ C

(X , R) such that ω

= ω + i ∂∂ϕ.

Let K

denote set of smooth functions ϕ s.t. ω + i ∂∂ϕ > 0 is K¨ ahler.

PSH(X , ω) =the closure of K

in L

will be the hero of Lecture 2.

Theorem (Hodge decomposition theorem)

If X is compact K¨ ahler then H

(X , C) is isomorphic to H

(X , C) and H

(X , C) = ⊕

H

(X , C).

,→ In particular b

(X ) = 2h

(X ) even, while b

(S

× S

) = 1 so no Hopf.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 8 / 19

K¨ahler manifolds

The (global) ∂∂-lemma

A K¨ ahler form ω defines a deRham class {ω} ∈ H

(X , R).

Theorem (∂∂-lemma)

If two K¨ ahler forms ω, ω

define the same cohomology class, then there exists a (essentially unique) ϕ ∈ C

(X , R) such that ω

= ω + i ∂∂ϕ.

Let K

denote set of smooth functions ϕ s.t. ω + i ∂∂ϕ > 0 is K¨ ahler.

PSH(X , ω) =the closure of K

in L

will be the hero of Lecture 2.

Theorem (Hodge decomposition theorem)

If X is compact K¨ ahler then H

(X , C) is isomorphic to H

(X , C)

and H

(X , C) = ⊕

H

(X , C).

,→ In particular b

(X ) = 2h

(X ) even, while b

(S

× S

) = 1 so no Hopf.

K¨ahler manifolds

The (global) ∂∂-lemma

A K¨ ahler form ω defines a deRham class {ω} ∈ H

(X , R).

Theorem (∂∂-lemma)

If two K¨ ahler forms ω, ω

define the same cohomology class, then there exists a (essentially unique) ϕ ∈ C

(X , R) such that ω

= ω + i ∂∂ϕ.

Let K

denote set of smooth functions ϕ s.t. ω + i ∂∂ϕ > 0 is K¨ ahler.

PSH(X , ω) =the closure of K

in L

will be the hero of Lecture 2.

Theorem (Hodge decomposition theorem)

If X is compact K¨ ahler then H

(X , C) is isomorphic to H

(X , C) and H

(X , C) = ⊕

H

(X , C).

,→ In particular b

(X ) = 2h

(X ) even, while b

(S

× S

) = 1 so no Hopf.

Vincent Guedj (IMT) Lecture 1: Compact K¨ahler manifolds April 2021 8 / 19