Analysis on simply stratified complex manifoldsII. Differential topology of stratified manifolds

Gerardo Mendoza

Temple University

Partial Differential Equations in Complex Geometry

and Singular Spaces,

American University of Beirut

Center for Advanced Mathematical Sciences

November 2014

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 1 / 13

Analysis on simply stratified complex manifolds

..............................................

and noncompact

A number of problems in partial differential equations on stratified or

noncompact manifolds become amenable to treatment after a process

of (real) “resolution” of singularities or of compactification. The lectures

will describe the differential topology of the resulting spaces, the various

kinds of differential operators that arise, and give an overview of some

results of the theory.

In this second lecture I will describe a variety of structures that arise

for which related differential operators can be studied with ideas more

or less uniform across the various situations.

A fuller dicussion of stratified manifolds was ommitted due to time constraints.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 2 / 13

Analysis on simply stratified complex manifolds.................................

......

.......and noncompact

A number of problems in partial differential equations on stratified or

noncompact manifolds become amenable to treatment after a process

of (real) “resolution” of singularities or of compactification. The lectures

will describe the differential topology of the resulting spaces, the various

kinds of differential operators that arise, and give an overview of some

results of the theory.

In this second lecture I will describe a variety of structures that arise

for which related differential operators can be studied with ideas more

or less uniform across the various situations.

A fuller dicussion of stratified manifolds was ommitted due to time constraints.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 2 / 13

Analysis on simply stratified complex manifolds.................................

......

.......and noncompact

A number of problems in partial differential equations on stratified or

noncompact manifolds become amenable to treatment after a process

of (real) “resolution” of singularities or of compactification. The lectures

will describe the differential topology of the resulting spaces, the various

kinds of differential operators that arise, and give an overview of some

results of the theory.

In this second lecture I will describe a variety of structures that arise

for which related differential operators can be studied with ideas more

or less uniform across the various situations.

A fuller dicussion of stratified manifolds was ommitted due to time constraints.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 2 / 13

Analysis on simply stratified complex manifolds.................................

......

.......and noncompact

A number of problems in partial differential equations on stratified or

noncompact manifolds become amenable to treatment after a process

of (real) “resolution” of singularities or of compactification. The lectures

will describe the differential topology of the resulting spaces, the various

kinds of differential operators that arise, and give an overview of some

results of the theory.

In this second lecture I will describe a variety of structures that arise

for which related differential operators can be studied with ideas more

or less uniform across the various situations.

A fuller dicussion of stratified manifolds was ommitted due to time constraints.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 2 / 13

Motivation: b-manifolds

Let M be a smooth compact manifold with boundary. The b-structure

of M is the space Vb of all smooth vector fields on M which over the

boundary are tangential to the boundary.

For example, on the interval [−1, 1], every such vector field is of the form

X = (1− x2)d

dx.

∂M = N

M x←−−

x :M→ R,

x > 0 inM

dx 6= 0 near N

In general, if M is a manifold with boundary N then V ∈ Vb if and only

if in any coordinate system x , y1, . . . , yn−1

near the boundary (x > 0 inM),

V = a0x∂

∂x+

m∑j=1

aj∂

∂yjThese vector fields can be realized as the

image by a suitable map, of the space of sections of another vector bundle.

This is essentially based on a theorem of Serre in the analytic category,

adapted by Swan to the continuous category.

Specifically...

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 3 / 13

Motivation: b-manifoldsLet M be a smooth compact manifold with boundary. The b-structure

of M is the space Vb of all smooth vector fields on M which over the

boundary are tangential to the boundary.

For example, on the interval [−1, 1], every such vector field is of the form

X = (1− x2)d

dx.

∂M = N

M x←−−

x :M→ R,

x > 0 inM

dx 6= 0 near N

In general, if M is a manifold with boundary N then V ∈ Vb if and only

if in any coordinate system x , y1, . . . , yn−1

near the boundary (x > 0 inM),

V = a0x∂

∂x+

m∑j=1

aj∂

∂yjThese vector fields can be realized as the

image by a suitable map, of the space of sections of another vector bundle.

This is essentially based on a theorem of Serre in the analytic category,

adapted by Swan to the continuous category.

Specifically...

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 3 / 13

Motivation: b-manifoldsLet M be a smooth compact manifold with boundary. The b-structure

of M is the space Vb of all smooth vector fields on M which over the

boundary are tangential to the boundary.

For example, on the interval [−1, 1], every such vector field is of the form

X = (1− x2)d

dx.

∂M = N

M x←−−

x :M→ R,

x > 0 inM

dx 6= 0 near N

In general, if M is a manifold with boundary N then V ∈ Vb if and only

if in any coordinate system x , y1, . . . , yn−1

near the boundary (x > 0 inM),

V = a0x∂

∂x+

m∑j=1

aj∂

∂yjThese vector fields can be realized as the

image by a suitable map, of the space of sections of another vector bundle.

This is essentially based on a theorem of Serre in the analytic category,

adapted by Swan to the continuous category.

Specifically...

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 3 / 13

Motivation: b-manifoldsLet M be a smooth compact manifold with boundary. The b-structure

of M is the space Vb of all smooth vector fields on M which over the

boundary are tangential to the boundary.

For example, on the interval [−1, 1], every such vector field is of the form

X = (1− x2)d

dx.

∂M = N

M x←−−

x :M→ R,

x > 0 inM

dx 6= 0 near N

In general, if M is a manifold with boundary N then V ∈ Vb if and only

if in any coordinate system x , y1, . . . , yn−1

near the boundary (x > 0 inM),

V = a0x∂

∂x+

m∑j=1

aj∂

∂yjThese vector fields can be realized as the

image by a suitable map, of the space of sections of another vector bundle.

This is essentially based on a theorem of Serre in the analytic category,

adapted by Swan to the continuous category.

Specifically...

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 3 / 13

Motivation: b-manifoldsLet M be a smooth compact manifold with boundary. The b-structure

of M is the space Vb of all smooth vector fields on M which over the

boundary are tangential to the boundary.

For example, on the interval [−1, 1], every such vector field is of the form

X = (1− x2)d

dx.

∂M = N

M x←−−

x :M→ R,

x > 0 inM

dx 6= 0 near N

In general, if M is a manifold with boundary N then V ∈ Vb if and only

if in any coordinate system x , y1, . . . , yn−1

near the boundary (x > 0 inM),

V = a0x∂

∂x+

m∑j=1

aj∂

∂yjThese vector fields can be realized as the

image by a suitable map, of the space of sections of another vector bundle.

This is essentially based on a theorem of Serre in the analytic category,

adapted by Swan to the continuous category.

Specifically...Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 3 / 13

Theorem. Let M be a connected paracompact smooth manifold, let E

be a module over C∞(M). Suppose E is locally free finitely generated.

Then there is a vector bundle E →M such that E is isomorphic to

C∞(M;E ).

E locally free finitely generated means:

for every p ∈M there is r ∈ N, a neighborhood U of p, η1, . . . , ηr ∈ E ,

and χ ∈ C∞c (U) with χ(p) 6= 0 such that for any φ ∈ E :

a) there are f 1, . . . , f r ∈ C∞(M) such that χφ =∑r

j=1 fjηj ;

b) if∑r

j=1 fjηj = χφ =

∑rj=1 f

jηj , then fj = fj on suppχ for

each j = 1, . . . , r .

Theorem. Let F →M be a smooth vector bundle and E ⊂ C∞(M;F )

a submodule. If E is locally free finitely generated and E →M is a

vector bundle such that E ≈ C∞(M;E ), then there is a unique bundle

homomorphism ι : E → F such that ι∗ : C∞(E ;M)→ C∞(M;F ) is an

isomorphism onto E .

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 4 / 13

Theorem. Let M be a connected paracompact smooth manifold, let E

be a module over C∞(M). Suppose E is locally free finitely generated.

Then there is a vector bundle E →M such that E is isomorphic to

C∞(M;E ).

E locally free finitely generated means:

for every p ∈M there is r ∈ N, a neighborhood U of p, η1, . . . , ηr ∈ E ,

and χ ∈ C∞c (U) with χ(p) 6= 0 such that for any φ ∈ E :

a) there are f 1, . . . , f r ∈ C∞(M) such that χφ =∑r

j=1 fjηj ;

b) if∑r

j=1 fjηj = χφ =

∑rj=1 f

jηj , then fj = fj on suppχ for

each j = 1, . . . , r .

Theorem. Let F →M be a smooth vector bundle and E ⊂ C∞(M;F )

a submodule. If E is locally free finitely generated and E →M is a

vector bundle such that E ≈ C∞(M;E ), then there is a unique bundle

homomorphism ι : E → F such that ι∗ : C∞(E ;M)→ C∞(M;F ) is an

isomorphism onto E .

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 4 / 13

Theorem. Let M be a connected paracompact smooth manifold, let E

be a module over C∞(M). Suppose E is locally free finitely generated.

Then there is a vector bundle E →M such that E is isomorphic to

C∞(M;E ).

E locally free finitely generated means:

for every p ∈M there is r ∈ N, a neighborhood U of p, η1, . . . , ηr ∈ E ,

and χ ∈ C∞c (U) with χ(p) 6= 0 such that for any φ ∈ E :

a) there are f 1, . . . , f r ∈ C∞(M) such that χφ =∑r

j=1 fjηj ;

b) if∑r

j=1 fjηj = χφ =

∑rj=1 f

jηj , then fj = fj on suppχ for

each j = 1, . . . , r .

Theorem. Let F →M be a smooth vector bundle and E ⊂ C∞(M;F )

a submodule. If E is locally free finitely generated and E →M is a

vector bundle such that E ≈ C∞(M;E ), then there is a unique bundle

homomorphism ι : E → F such that ι∗ : C∞(E ;M)→ C∞(M;F ) is an

isomorphism onto E .

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 4 / 13

b-manifoldsLet M be a compact manifold with boundary N , let Vb ⊂ C∞(M;TM)

consist of all vector fields of M tangent to N . Then Vb is a module over

C∞(M).

If p ∈M and w1, . . . ,wn are coordinates about p, then every V ∈ Vb

is uniquely of the form∑

bµ∂wµ near p with smooth bj .

If p ∈ N and x , y1, . . . , yn−1 are coordinates on M near p with x a

defining function for N , then again any V ∈ Vb is uniquely of the form

V = a0x∂x +∑

j aj∂yj near p with smooth aµ.

This implies that Vb is the space of sections of a vector bundle, denotedbTM. There is a map bev : bTM→ TM. which is an isomorphism overM. The bundle bTM is the structure bundle of M, and M with bTM is

a b-manifold.

b-manifolds model manifolds with cylindrical ends.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 5 / 13

b-manifoldsLet M be a compact manifold with boundary N , let Vb ⊂ C∞(M;TM)

consist of all vector fields of M tangent to N . Then Vb is a module over

C∞(M).

If p ∈M and w1, . . . ,wn are coordinates about p, then every V ∈ Vb

is uniquely of the form∑

bµ∂wµ near p with smooth bj .

If p ∈ N and x , y1, . . . , yn−1 are coordinates on M near p with x a

defining function for N , then again any V ∈ Vb is uniquely of the form

V = a0x∂x +∑

j aj∂yj near p with smooth aµ.

This implies that Vb is the space of sections of a vector bundle, denotedbTM. There is a map bev : bTM→ TM. which is an isomorphism overM. The bundle bTM is the structure bundle of M, and M with bTM is

a b-manifold.

b-manifolds model manifolds with cylindrical ends.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 5 / 13

b-manifoldsLet M be a compact manifold with boundary N , let Vb ⊂ C∞(M;TM)

consist of all vector fields of M tangent to N . Then Vb is a module over

C∞(M).

If p ∈M and w1, . . . ,wn are coordinates about p, then every V ∈ Vb

is uniquely of the form∑

bµ∂wµ near p with smooth bj .

If p ∈ N and x , y1, . . . , yn−1 are coordinates on M near p with x a

defining function for N , then again any V ∈ Vb is uniquely of the form

V = a0x∂x +∑

j aj∂yj near p with smooth aµ.

This implies that Vb is the space of sections of a vector bundle, denotedbTM. There is a map bev : bTM→ TM. which is an isomorphism overM. The bundle bTM is the structure bundle of M, and M with bTM is

a b-manifold.

b-manifolds model manifolds with cylindrical ends.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 5 / 13

b-manifoldsLet M be a compact manifold with boundary N , let Vb ⊂ C∞(M;TM)

consist of all vector fields of M tangent to N . Then Vb is a module over

C∞(M).

If p ∈M and w1, . . . ,wn are coordinates about p, then every V ∈ Vb

is uniquely of the form∑

bµ∂wµ near p with smooth bj .

If p ∈ N and x , y1, . . . , yn−1 are coordinates on M near p with x a

defining function for N , then again any V ∈ Vb is uniquely of the form

V = a0x∂x +∑

j aj∂yj near p with smooth aµ.

This implies that Vb is the space of sections of a vector bundle, denotedbTM. There is a map bev : bTM→ TM. which is an isomorphism overM.

The bundle bTM is the structure bundle of M, and M with bTM is

a b-manifold.

b-manifolds model manifolds with cylindrical ends.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 5 / 13

b-manifoldsLet M be a compact manifold with boundary N , let Vb ⊂ C∞(M;TM)

consist of all vector fields of M tangent to N . Then Vb is a module over

C∞(M).

If p ∈M and w1, . . . ,wn are coordinates about p, then every V ∈ Vb

is uniquely of the form∑

bµ∂wµ near p with smooth bj .

If p ∈ N and x , y1, . . . , yn−1 are coordinates on M near p with x a

defining function for N , then again any V ∈ Vb is uniquely of the form

V = a0x∂x +∑

j aj∂yj near p with smooth aµ.

This implies that Vb is the space of sections of a vector bundle, denotedbTM. There is a map bev : bTM→ TM. which is an isomorphism overM. The bundle bTM is the structure bundle of M, and M with bTM is

a b-manifold.

b-manifolds model manifolds with cylindrical ends.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 5 / 13

b-manifoldsLet M be a compact manifold with boundary N , let Vb ⊂ C∞(M;TM)

consist of all vector fields of M tangent to N . Then Vb is a module over

C∞(M).

If p ∈M and w1, . . . ,wn are coordinates about p, then every V ∈ Vb

is uniquely of the form∑

bµ∂wµ near p with smooth bj .

If p ∈ N and x , y1, . . . , yn−1 are coordinates on M near p with x a

defining function for N , then again any V ∈ Vb is uniquely of the form

V = a0x∂x +∑

j aj∂yj near p with smooth aµ.

This implies that Vb is the space of sections of a vector bundle, denotedbTM. There is a map bev : bTM→ TM. which is an isomorphism overM. The bundle bTM is the structure bundle of M, and M with bTM is

a b-manifold.

b-manifolds model manifolds with cylindrical ends.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 5 / 13

e-manifoldsAgain M is a compact manifold with boundary N , but in addition there

is a fibration

Z ⊂ N

Y?

Let Ve be the subspace of C∞(M;TM) whose element are, over the

boundary, tangent to the fibers of ℘. This is a locally free finitely

generated module over C∞(M). The associated vector bundle is eTM.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 6 / 13

Θ-structuresLet Ω ⊂ Cn be an open bounded domain with smooth boundary, let

K (z , ζ) be the Schwartz kernel of the Bergman projection:

Π : L2(Ω)→ H2(Ω), Π(f ) =

∫ΩK (z , ζ)f (ζ)dλ(ζ).

The Bergman metric is

g =∑

i ,j

∂2 logK∆(z)

∂zi∂z jdzi ⊗ dz j , K∆(z) = K (z , z)

Let ρ be a defining function for ∂Ω, positive in Ω. By a theorem of

Fefferman there are smooth functions φ, ψ near Ω with φ > 0 such that

K∆(z) = φρ−n−1 + ψ log ρ.

Ignoring ψ,

g =n + 1

ρ2

(∂ρ⊗ ∂ρ−

∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)+O(1) as ρ→ 0.

Strict pseudoconvexity: −∑ ∂2ρ

∂zi∂z jwiw j > 0 if

∑wj∂ρ

∂zi= 0 & w 6= 0

We look for vector fields Vj s.t [g(Vi ,V j)] has the least degeneration.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 7 / 13

Θ-structuresLet Ω ⊂ Cn be an open bounded domain with smooth boundary, let

K (z , ζ) be the Schwartz kernel of the Bergman projection:

Π : L2(Ω)→ H2(Ω), Π(f ) =

∫ΩK (z , ζ)f (ζ)dλ(ζ).

The Bergman metric is

g =∑

i ,j

∂2 logK∆(z)

∂zi∂z jdzi ⊗ dz j , K∆(z) = K (z , z)

Let ρ be a defining function for ∂Ω, positive in Ω. By a theorem of

Fefferman there are smooth functions φ, ψ near Ω with φ > 0 such that

K∆(z) = φρ−n−1 + ψ log ρ.

Ignoring ψ,

g =n + 1

ρ2

(∂ρ⊗ ∂ρ−

∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)+O(1) as ρ→ 0.

Strict pseudoconvexity: −∑ ∂2ρ

∂zi∂z jwiw j > 0 if

∑wj∂ρ

∂zi= 0 & w 6= 0

We look for vector fields Vj s.t [g(Vi ,V j)] has the least degeneration.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 7 / 13

Θ-structuresLet Ω ⊂ Cn be an open bounded domain with smooth boundary, let

K (z , ζ) be the Schwartz kernel of the Bergman projection:

Π : L2(Ω)→ H2(Ω), Π(f ) =

∫ΩK (z , ζ)f (ζ)dλ(ζ).

The Bergman metric is

g =∑

i ,j

∂2 logK∆(z)

∂zi∂z jdzi ⊗ dz j , K∆(z) = K (z , z)

Let ρ be a defining function for ∂Ω, positive in Ω. By a theorem of

Fefferman there are smooth functions φ, ψ near Ω with φ > 0 such that

K∆(z) = φρ−n−1 + ψ log ρ.

Ignoring ψ,

g =n + 1

ρ2

(∂ρ⊗ ∂ρ−

∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)+O(1) as ρ→ 0.

Strict pseudoconvexity: −∑ ∂2ρ

∂zi∂z jwiw j > 0 if

∑wj∂ρ

∂zi= 0 & w 6= 0

We look for vector fields Vj s.t [g(Vi ,V j)] has the least degeneration.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 7 / 13

g =n + 1

ρ2

(∂ρ⊗ ∂ρ−

∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)+O(1) as ρ→ 0.

Strict pseudoconvexity: −∑ ∂2ρ

∂zi∂z jwiw j > 0 if

∑wj∂ρ

∂zi= 0 & w 6= 0

We look for vector fields Vj s.t [g(Vi ,V j)] has the least degeneration.

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω. So 〈∂ρ, Lj〉 = 0 and (in ρ ≥ 0)

g(ρ1/2Li , ρ1/2Lj) =

n + 1

ρ2

(−∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)(ρ1/2Li , ρ

1/2Lj)

while

g(ρH, ρH) =n + 1

ρ2

(ρ∑j

∣∣∣ ∂ρ∂zj

∣∣∣2)2+O(ρ) as ρ→ 0

Take Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.Must change C∞ struc-

ture to make ρ1/2 smooth.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 8 / 13

g =n + 1

ρ2

(∂ρ⊗ ∂ρ−

∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)+O(1) as ρ→ 0.

Strict pseudoconvexity: −∑ ∂2ρ

∂zi∂z jwiw j > 0 if

∑wj∂ρ

∂zi= 0 & w 6= 0

We look for vector fields Vj s.t [g(Vi ,V j)] has the least degeneration.

Let H =1

2

(∇ρ− iJ∇ρ

).

Then H =∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω. So 〈∂ρ, Lj〉 = 0 and (in ρ ≥ 0)

g(ρ1/2Li , ρ1/2Lj) =

n + 1

ρ2

(−∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)(ρ1/2Li , ρ

1/2Lj)

while

g(ρH, ρH) =n + 1

ρ2

(ρ∑j

∣∣∣ ∂ρ∂zj

∣∣∣2)2+O(ρ) as ρ→ 0

Take Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.Must change C∞ struc-

ture to make ρ1/2 smooth.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 8 / 13

g =n + 1

ρ2

(∂ρ⊗ ∂ρ−

∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)+O(1) as ρ→ 0.

Strict pseudoconvexity: −∑ ∂2ρ

∂zi∂z jwiw j > 0 if

∑wj∂ρ

∂zi= 0 & w 6= 0

We look for vector fields Vj s.t [g(Vi ,V j)] has the least degeneration.

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zj

Assume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω. So 〈∂ρ, Lj〉 = 0 and (in ρ ≥ 0)

g(ρ1/2Li , ρ1/2Lj) =

n + 1

ρ2

(−∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)(ρ1/2Li , ρ

1/2Lj)

while

g(ρH, ρH) =n + 1

ρ2

(ρ∑j

∣∣∣ ∂ρ∂zj

∣∣∣2)2+O(ρ) as ρ→ 0

Take Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.Must change C∞ struc-

ture to make ρ1/2 smooth.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 8 / 13

g =n + 1

ρ2

(∂ρ⊗ ∂ρ−

∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)+O(1) as ρ→ 0.

Strict pseudoconvexity: −∑ ∂2ρ

∂zi∂z jwiw j > 0 if

∑wj∂ρ

∂zi= 0 & w 6= 0

We look for vector fields Vj s.t [g(Vi ,V j)] has the least degeneration.

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω.

The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω. So 〈∂ρ, Lj〉 = 0 and (in ρ ≥ 0)

g(ρ1/2Li , ρ1/2Lj) =

n + 1

ρ2

(−∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)(ρ1/2Li , ρ

1/2Lj)

while

g(ρH, ρH) =n + 1

ρ2

(ρ∑j

∣∣∣ ∂ρ∂zj

∣∣∣2)2+O(ρ) as ρ→ 0

Take Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.Must change C∞ struc-

ture to make ρ1/2 smooth.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 8 / 13

g =n + 1

ρ2

(∂ρ⊗ ∂ρ−

∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)+O(1) as ρ→ 0.

Strict pseudoconvexity: −∑ ∂2ρ

∂zi∂z jwiw j > 0 if

∑wj∂ρ

∂zi= 0 & w 6= 0

We look for vector fields Vj s.t [g(Vi ,V j)] has the least degeneration.

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω.

So 〈∂ρ, Lj〉 = 0 and (in ρ ≥ 0)

g(ρ1/2Li , ρ1/2Lj) =

n + 1

ρ2

(−∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)(ρ1/2Li , ρ

1/2Lj)

while

g(ρH, ρH) =n + 1

ρ2

(ρ∑j

∣∣∣ ∂ρ∂zj

∣∣∣2)2+O(ρ) as ρ→ 0

Take Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.Must change C∞ struc-

ture to make ρ1/2 smooth.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 8 / 13

g =n + 1

ρ2

(∂ρ⊗ ∂ρ−

∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)+O(1) as ρ→ 0.

Strict pseudoconvexity: −∑ ∂2ρ

∂zi∂z jwiw j > 0 if

∑wj∂ρ

∂zi= 0 & w 6= 0

We look for vector fields Vj s.t [g(Vi ,V j)] has the least degeneration.

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω. So 〈∂ρ, Lj〉 = 0 and (in ρ ≥ 0)

g(ρ1/2Li , ρ1/2Lj) =

n + 1

ρ2

(−∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)(ρ1/2Li , ρ

1/2Lj)

while

g(ρH, ρH) =n + 1

ρ2

(ρ∑j

∣∣∣ ∂ρ∂zj

∣∣∣2)2+O(ρ) as ρ→ 0

Take Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.Must change C∞ struc-

ture to make ρ1/2 smooth.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 8 / 13

g =n + 1

ρ2

(∂ρ⊗ ∂ρ−

∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)+O(1) as ρ→ 0.

Strict pseudoconvexity: −∑ ∂2ρ

∂zi∂z jwiw j > 0 if

∑wj∂ρ

∂zi= 0 & w 6= 0

We look for vector fields Vj s.t [g(Vi ,V j)] has the least degeneration.

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω. So 〈∂ρ, Lj〉 = 0 and (in ρ ≥ 0)

g(ρ1/2Li , ρ1/2Lj) =

n + 1

ρ2

(−∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)(ρ1/2Li , ρ

1/2Lj)

while

g(ρH, ρH) =n + 1

ρ2

(ρ∑j

∣∣∣ ∂ρ∂zj

∣∣∣2)2+O(ρ) as ρ→ 0

Take Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.Must change C∞ struc-

ture to make ρ1/2 smooth.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 8 / 13

g =n + 1

ρ2

(∂ρ⊗ ∂ρ−

∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)+O(1) as ρ→ 0.

Strict pseudoconvexity: −∑ ∂2ρ

∂zi∂z jwiw j > 0 if

∑wj∂ρ

∂zi= 0 & w 6= 0

We look for vector fields Vj s.t [g(Vi ,V j)] has the least degeneration.

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω. So 〈∂ρ, Lj〉 = 0 and (in ρ ≥ 0)

g(ρ1/2Li , ρ1/2Lj) =

n + 1

ρ2

(−∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)(ρ1/2Li , ρ

1/2Lj)

while

g(ρH, ρH) =n + 1

ρ2

(ρ∑j

∣∣∣ ∂ρ∂zj

∣∣∣2)2+O(ρ) as ρ→ 0

Take Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.

Must change C∞ struc-

ture to make ρ1/2 smooth.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 8 / 13

g =n + 1

ρ2

(∂ρ⊗ ∂ρ−

∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)+O(1) as ρ→ 0.

Strict pseudoconvexity: −∑ ∂2ρ

∂zi∂z jwiw j > 0 if

∑wj∂ρ

∂zi= 0 & w 6= 0

We look for vector fields Vj s.t [g(Vi ,V j)] has the least degeneration.

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω. So 〈∂ρ, Lj〉 = 0 and (in ρ ≥ 0)

g(ρ1/2Li , ρ1/2Lj) =

n + 1

ρ2

(−∑i ,j

ρ∂2ρ

∂zi∂z jdzi ⊗ dz j

)(ρ1/2Li , ρ

1/2Lj)

while

g(ρH, ρH) =n + 1

ρ2

(ρ∑j

∣∣∣ ∂ρ∂zj

∣∣∣2)2+O(ρ) as ρ→ 0

Take Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.Must change C∞ struc-

ture to make ρ1/2 smooth.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 8 / 13

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω.

Let Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.

With zj = xj + ixn+j , change variables (assume ρx2n 6= 0):

xj = yj , j = 1, . . . , 2n − 1, x2n = ρ, set ζj = yj + iyj+n, j < n.

GetLj =

∂

∂ζj−

ρzj∣∣∂ρ∣∣2n−1∑`=1

ρz`∂

∂ζj+ρzn2

∂

∂yn

H =n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn+∣∣∂ρ∣∣2 ∂

∂ρ

Now put ρ = r2:

∂

∂ρ=

1

2r

∂

∂r

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 9 / 13

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω. Let Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.

With zj = xj + ixn+j , change variables (assume ρx2n 6= 0):

xj = yj , j = 1, . . . , 2n − 1, x2n = ρ, set ζj = yj + iyj+n, j < n.

GetLj =

∂

∂ζj−

ρzj∣∣∂ρ∣∣2n−1∑`=1

ρz`∂

∂ζj+ρzn2

∂

∂yn

H =n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn+∣∣∂ρ∣∣2 ∂

∂ρ

Now put ρ = r2:

∂

∂ρ=

1

2r

∂

∂r

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 9 / 13

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω. Let Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.

With zj = xj + ixn+j , change variables (assume ρx2n 6= 0):

xj = yj , j = 1, . . . , 2n − 1, x2n = ρ, set ζj = yj + iyj+n, j < n.

GetLj =

∂

∂ζj−

ρzj∣∣∂ρ∣∣2n−1∑`=1

ρz`∂

∂ζj+ρzn2

∂

∂yn

H =n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn+∣∣∂ρ∣∣2 ∂

∂ρ

Now put ρ = r2:

∂

∂ρ=

1

2r

∂

∂r

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 9 / 13

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω. Let Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.

With zj = xj + ixn+j , change variables (assume ρx2n 6= 0):

xj = yj , j = 1, . . . , 2n − 1, x2n = ρ, set ζj = yj + iyj+n, j < n.

GetLj =

∂

∂ζj−

ρzj∣∣∂ρ∣∣2n−1∑`=1

ρz`∂

∂ζj+ρzn2

∂

∂yn

H =n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn+∣∣∂ρ∣∣2 ∂

∂ρ

Now put ρ = r2:

∂

∂ρ=

1

2r

∂

∂r

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 9 / 13

Let H =1

2

(∇ρ− iJ∇ρ

). Then H =

∑j

∂ρ

∂z j

∂

∂zjAssume ρzn 6= 0 near

p0 ∈ ∂Ω. The vector fields

Lj = ∂zj −∂ρ/∂zjHρ

H, j = 1, . . . , n − 1

are tangent to ∂Ω. Let Vj = ρ1/2Lj , j = 1, . . . , n − 1, Vn = ρH.

With zj = xj + ixn+j , change variables (assume ρx2n 6= 0):

xj = yj , j = 1, . . . , 2n − 1, x2n = ρ, set ζj = yj + iyj+n, j < n.

GetLj =

∂

∂ζj−

ρzj∣∣∂ρ∣∣2n−1∑`=1

ρz`∂

∂ζj+ρzn2

∂

∂yn

H =n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn+∣∣∂ρ∣∣2 ∂

∂ρ

Now put ρ = r2:

∂

∂ρ=

1

2r

∂

∂r

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 9 / 13

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Making ρ1/2 smooth corresponds to:

Let

Ω1/2 = (z , r) ∈ Ω× R : ρ(z)− r2 = 0, r ≥ 0,define π : Ω1/2 → Ω by π(z , r) = z . Then π is a homeomorphism,

separately a diffeomorphism from the region

Ω1/2 onto Ω, and

of ∂Ω1/2 onto ∂Ω.

The differential of π along ∂Ω1/2 has one dimensional kernel (spanned

by ∂r ). Let ι∗∂Ωθ = i∂ρ, as a form on ∂Ω1/2. Define Θ by

〈Θ, v〉 =

〈θ, v〉 if v ∈ T∂Ω1/2

0 if v ∈ ker dπ

Define VΘ ⊂ C∞(Ω1/2;TΩ1/2) by specifying V ∈ VΘ if V vanishes on

∂Ω1/2 and 〈Θ,V 〉 = O(r2).

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 10 / 13

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Making ρ1/2 smooth corresponds to: Let

Ω1/2 = (z , r) ∈ Ω× R : ρ(z)− r2 = 0, r ≥ 0,define π : Ω1/2 → Ω by π(z , r) = z .

Then π is a homeomorphism,

separately a diffeomorphism from the region

Ω1/2 onto Ω, and

of ∂Ω1/2 onto ∂Ω.

The differential of π along ∂Ω1/2 has one dimensional kernel (spanned

by ∂r ). Let ι∗∂Ωθ = i∂ρ, as a form on ∂Ω1/2. Define Θ by

〈Θ, v〉 =

〈θ, v〉 if v ∈ T∂Ω1/2

0 if v ∈ ker dπ

Define VΘ ⊂ C∞(Ω1/2;TΩ1/2) by specifying V ∈ VΘ if V vanishes on

∂Ω1/2 and 〈Θ,V 〉 = O(r2).

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 10 / 13

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Making ρ1/2 smooth corresponds to: Let

Ω1/2 = (z , r) ∈ Ω× R : ρ(z)− r2 = 0, r ≥ 0,define π : Ω1/2 → Ω by π(z , r) = z . Then π is a homeomorphism,

separately a diffeomorphism from the region

Ω1/2 onto Ω, and

of ∂Ω1/2 onto ∂Ω.

The differential of π along ∂Ω1/2 has one dimensional kernel (spanned

by ∂r ). Let ι∗∂Ωθ = i∂ρ, as a form on ∂Ω1/2. Define Θ by

〈Θ, v〉 =

〈θ, v〉 if v ∈ T∂Ω1/2

0 if v ∈ ker dπ

Define VΘ ⊂ C∞(Ω1/2;TΩ1/2) by specifying V ∈ VΘ if V vanishes on

∂Ω1/2 and 〈Θ,V 〉 = O(r2).

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 10 / 13

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Making ρ1/2 smooth corresponds to: Let

Ω1/2 = (z , r) ∈ Ω× R : ρ(z)− r2 = 0, r ≥ 0,define π : Ω1/2 → Ω by π(z , r) = z . Then π is a homeomorphism,

separately a diffeomorphism from the region

Ω1/2 onto Ω,

and

of ∂Ω1/2 onto ∂Ω.

The differential of π along ∂Ω1/2 has one dimensional kernel (spanned

by ∂r ). Let ι∗∂Ωθ = i∂ρ, as a form on ∂Ω1/2. Define Θ by

〈Θ, v〉 =

〈θ, v〉 if v ∈ T∂Ω1/2

0 if v ∈ ker dπ

Define VΘ ⊂ C∞(Ω1/2;TΩ1/2) by specifying V ∈ VΘ if V vanishes on

∂Ω1/2 and 〈Θ,V 〉 = O(r2).

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 10 / 13

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Making ρ1/2 smooth corresponds to: Let

Ω1/2 = (z , r) ∈ Ω× R : ρ(z)− r2 = 0, r ≥ 0,define π : Ω1/2 → Ω by π(z , r) = z . Then π is a homeomorphism,

separately a diffeomorphism from the region

Ω1/2 onto Ω, and

of ∂Ω1/2 onto ∂Ω.

The differential of π along ∂Ω1/2 has one dimensional kernel (spanned

by ∂r ). Let ι∗∂Ωθ = i∂ρ, as a form on ∂Ω1/2. Define Θ by

〈Θ, v〉 =

〈θ, v〉 if v ∈ T∂Ω1/2

0 if v ∈ ker dπ

Define VΘ ⊂ C∞(Ω1/2;TΩ1/2) by specifying V ∈ VΘ if V vanishes on

∂Ω1/2 and 〈Θ,V 〉 = O(r2).

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 10 / 13

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Making ρ1/2 smooth corresponds to: Let

Ω1/2 = (z , r) ∈ Ω× R : ρ(z)− r2 = 0, r ≥ 0,define π : Ω1/2 → Ω by π(z , r) = z . Then π is a homeomorphism,

separately a diffeomorphism from the region

Ω1/2 onto Ω, and

of ∂Ω1/2 onto ∂Ω.

The differential of π along ∂Ω1/2 has one dimensional kernel (spanned

by ∂r ).

Let ι∗∂Ωθ = i∂ρ, as a form on ∂Ω1/2. Define Θ by

〈Θ, v〉 =

〈θ, v〉 if v ∈ T∂Ω1/2

0 if v ∈ ker dπ

Define VΘ ⊂ C∞(Ω1/2;TΩ1/2) by specifying V ∈ VΘ if V vanishes on

∂Ω1/2 and 〈Θ,V 〉 = O(r2).

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 10 / 13

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Making ρ1/2 smooth corresponds to: Let

Ω1/2 = (z , r) ∈ Ω× R : ρ(z)− r2 = 0, r ≥ 0,define π : Ω1/2 → Ω by π(z , r) = z . Then π is a homeomorphism,

separately a diffeomorphism from the region

Ω1/2 onto Ω, and

of ∂Ω1/2 onto ∂Ω.

The differential of π along ∂Ω1/2 has one dimensional kernel (spanned

by ∂r ). Let ι∗∂Ωθ = i∂ρ, as a form on ∂Ω1/2. Define Θ by

〈Θ, v〉 =

〈θ, v〉 if v ∈ T∂Ω1/2

0 if v ∈ ker dπ

Define VΘ ⊂ C∞(Ω1/2;TΩ1/2) by specifying V ∈ VΘ if V vanishes on

∂Ω1/2 and 〈Θ,V 〉 = O(r2).

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 10 / 13

Vj = rLj , j < n, Vn = r2( n−1∑`=1

∂ρ

∂z j

∂

∂ζj+ρzn2

∂

∂yn

)+ r∣∣∂ρ∣∣2 ∂

∂r

Making ρ1/2 smooth corresponds to: Let

Ω1/2 = (z , r) ∈ Ω× R : ρ(z)− r2 = 0, r ≥ 0,define π : Ω1/2 → Ω by π(z , r) = z . Then π is a homeomorphism,

separately a diffeomorphism from the region

Ω1/2 onto Ω, and

of ∂Ω1/2 onto ∂Ω.

The differential of π along ∂Ω1/2 has one dimensional kernel (spanned

by ∂r ). Let ι∗∂Ωθ = i∂ρ, as a form on ∂Ω1/2. Define Θ by

〈Θ, v〉 =

〈θ, v〉 if v ∈ T∂Ω1/2

0 if v ∈ ker dπ

Define VΘ ⊂ C∞(Ω1/2;TΩ1/2) by specifying V ∈ VΘ if V vanishes on

∂Ω1/2 and 〈Θ,V 〉 = O(r2).

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 10 / 13

Scattering structuresThe model for these is radial compactification of Rn.

This means to

replace Rn by a ball and then attach the boundary in a specific way:

View Rn as P = (x , 1) ∈ Rn × R : x ∈ Rn, map it onto the open

hemisphere (w , z) ∈ Rn × R : |w |2 + z2 = 1, z > 0 via

(x , 1) 7→( x√

1 + |x |2,

1√1 + |x |2

)then attach the boundary of the hemisphere

in the standard way. Write Sn+ for the closed

upper hemisphere; Vsc is generated over

C∞(Sn+) by the push-forward of the vector fields

∂xj of Rn. In spherical coordinates x = ρω, ρ > 0, ω ∈ Sn, if ωn 6= 0,

∂xj = ωj∂ρ +1

ρ

n−1∑`=1

(δj` − ωjω`)∂ω`

with r = 1/ρ,

∂xj = ωj r2∂r + r

n−1∑`=1

(δj` − ωjω`)∂ω`

Vsc locally spanned

by r2∂r and r∂ω`

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 11 / 13

Scattering structuresThe model for these is radial compactification of Rn. This means to

replace Rn by a ball and then attach the boundary in a specific way:

View Rn as P = (x , 1) ∈ Rn × R : x ∈ Rn, map it onto the open

hemisphere (w , z) ∈ Rn × R : |w |2 + z2 = 1, z > 0 via

(x , 1) 7→( x√

1 + |x |2,

1√1 + |x |2

)then attach the boundary of the hemisphere

in the standard way. Write Sn+ for the closed

upper hemisphere; Vsc is generated over

C∞(Sn+) by the push-forward of the vector fields

∂xj of Rn. In spherical coordinates x = ρω, ρ > 0, ω ∈ Sn, if ωn 6= 0,

∂xj = ωj∂ρ +1

ρ

n−1∑`=1

(δj` − ωjω`)∂ω`

with r = 1/ρ,

∂xj = ωj r2∂r + r

n−1∑`=1

(δj` − ωjω`)∂ω`

Vsc locally spanned

by r2∂r and r∂ω`

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 11 / 13

Scattering structuresThe model for these is radial compactification of Rn. This means to

replace Rn by a ball and then attach the boundary in a specific way:

View Rn as P = (x , 1) ∈ Rn × R : x ∈ Rn,

map it onto the open

hemisphere (w , z) ∈ Rn × R : |w |2 + z2 = 1, z > 0 via

(x , 1) 7→( x√

1 + |x |2,

1√1 + |x |2

)then attach the boundary of the hemisphere

in the standard way. Write Sn+ for the closed

upper hemisphere; Vsc is generated over

C∞(Sn+) by the push-forward of the vector fields

∂xj of Rn. In spherical coordinates x = ρω, ρ > 0, ω ∈ Sn, if ωn 6= 0,

∂xj = ωj∂ρ +1

ρ

n−1∑`=1

(δj` − ωjω`)∂ω`

with r = 1/ρ,

∂xj = ωj r2∂r + r

n−1∑`=1

(δj` − ωjω`)∂ω`

Vsc locally spanned

by r2∂r and r∂ω`

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 11 / 13

Scattering structuresThe model for these is radial compactification of Rn. This means to

replace Rn by a ball and then attach the boundary in a specific way:

View Rn as P = (x , 1) ∈ Rn × R : x ∈ Rn, map it onto the open

hemisphere (w , z) ∈ Rn × R : |w |2 + z2 = 1, z > 0 via

(x , 1) 7→( x√

1 + |x |2,

1√1 + |x |2

)

then attach the boundary of the hemisphere

in the standard way. Write Sn+ for the closed

upper hemisphere; Vsc is generated over

C∞(Sn+) by the push-forward of the vector fields

∂xj of Rn. In spherical coordinates x = ρω, ρ > 0, ω ∈ Sn, if ωn 6= 0,

∂xj = ωj∂ρ +1

ρ

n−1∑`=1

(δj` − ωjω`)∂ω`

with r = 1/ρ,

∂xj = ωj r2∂r + r

n−1∑`=1

(δj` − ωjω`)∂ω`

Vsc locally spanned

by r2∂r and r∂ω`

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 11 / 13

Scattering structuresThe model for these is radial compactification of Rn. This means to

replace Rn by a ball and then attach the boundary in a specific way:

View Rn as P = (x , 1) ∈ Rn × R : x ∈ Rn, map it onto the open

hemisphere (w , z) ∈ Rn × R : |w |2 + z2 = 1, z > 0 via

(x , 1) 7→( x√

1 + |x |2,

1√1 + |x |2

)then attach the boundary of the hemisphere

in the standard way. Write Sn+ for the closed

upper hemisphere; Vsc is generated over

C∞(Sn+) by the push-forward of the vector fields

∂xj of Rn.

In spherical coordinates x = ρω, ρ > 0, ω ∈ Sn, if ωn 6= 0,

∂xj = ωj∂ρ +1

ρ

n−1∑`=1

(δj` − ωjω`)∂ω`

with r = 1/ρ,

∂xj = ωj r2∂r + r

n−1∑`=1

(δj` − ωjω`)∂ω`

Vsc locally spanned

by r2∂r and r∂ω`

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 11 / 13

Scattering structuresThe model for these is radial compactification of Rn. This means to

replace Rn by a ball and then attach the boundary in a specific way:

View Rn as P = (x , 1) ∈ Rn × R : x ∈ Rn, map it onto the open

hemisphere (w , z) ∈ Rn × R : |w |2 + z2 = 1, z > 0 via

(x , 1) 7→( x√

1 + |x |2,

1√1 + |x |2

)then attach the boundary of the hemisphere

in the standard way. Write Sn+ for the closed

upper hemisphere; Vsc is generated over

C∞(Sn+) by the push-forward of the vector fields

∂xj of Rn. In spherical coordinates x = ρω, ρ > 0, ω ∈ Sn, if ωn 6= 0,

∂xj = ωj∂ρ +1

ρ

n−1∑`=1

(δj` − ωjω`)∂ω`

with r = 1/ρ,

∂xj = ωj r2∂r + r

n−1∑`=1

(δj` − ωjω`)∂ω`

Vsc locally spanned

by r2∂r and r∂ω`

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 11 / 13

Scattering structuresThe model for these is radial compactification of Rn. This means to

replace Rn by a ball and then attach the boundary in a specific way:

View Rn as P = (x , 1) ∈ Rn × R : x ∈ Rn, map it onto the open

hemisphere (w , z) ∈ Rn × R : |w |2 + z2 = 1, z > 0 via

(x , 1) 7→( x√

1 + |x |2,

1√1 + |x |2

)then attach the boundary of the hemisphere

in the standard way. Write Sn+ for the closed

upper hemisphere; Vsc is generated over

C∞(Sn+) by the push-forward of the vector fields

∂xj of Rn. In spherical coordinates x = ρω, ρ > 0, ω ∈ Sn, if ωn 6= 0,

∂xj = ωj∂ρ +1

ρ

n−1∑`=1

(δj` − ωjω`)∂ω`

with r = 1/ρ,

∂xj = ωj r2∂r + r

n−1∑`=1

(δj` − ωjω`)∂ω`

Vsc locally spanned

by r2∂r and r∂ω`

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 11 / 13

Scattering structuresThe model for these is radial compactification of Rn. This means to

replace Rn by a ball and then attach the boundary in a specific way:

View Rn as P = (x , 1) ∈ Rn × R : x ∈ Rn, map it onto the open

hemisphere (w , z) ∈ Rn × R : |w |2 + z2 = 1, z > 0 via

(x , 1) 7→( x√

1 + |x |2,

1√1 + |x |2

)then attach the boundary of the hemisphere

in the standard way. Write Sn+ for the closed

upper hemisphere; Vsc is generated over

C∞(Sn+) by the push-forward of the vector fields

∂xj of Rn. In spherical coordinates x = ρω, ρ > 0, ω ∈ Sn, if ωn 6= 0,

∂xj = ωj∂ρ +1

ρ

n−1∑`=1

(δj` − ωjω`)∂ω`

with r = 1/ρ,

∂xj = ωj r2∂r + r

n−1∑`=1

(δj` − ωjω`)∂ω`

Vsc locally spanned

by r2∂r and r∂ω`

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 11 / 13

All these examples share the property that the spaces V are closed

under Lie bracket.

This allows for the definition of complexes, for example

· · ·C∞(M;∧qbT ∗M)→ C∞(M;

∧q+1bT ∗M)→ · · ·by way of Cartan’s formula for the differential

(q + 1)bdφ(V0, . . . ,Vq) =∑j

(−1)jVjφ(V0, . . . , Vj , . . . ,Vq)

+∑j<k

(−1)j+kφ([Vj ,Vk ], φ1, . . . , Vj , . . . , Vk , . . . ,Vq).

We take advantage of this also to define complex structures.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 12 / 13

All these examples share the property that the spaces V are closed

under Lie bracket. This allows for the definition of complexes, for example

· · ·C∞(M;∧qbT ∗M)→ C∞(M;

∧q+1bT ∗M)→ · · ·

by way of Cartan’s formula for the differential

(q + 1)bdφ(V0, . . . ,Vq) =∑j

(−1)jVjφ(V0, . . . , Vj , . . . ,Vq)

+∑j<k

(−1)j+kφ([Vj ,Vk ], φ1, . . . , Vj , . . . , Vk , . . . ,Vq).

We take advantage of this also to define complex structures.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 12 / 13

All these examples share the property that the spaces V are closed

under Lie bracket. This allows for the definition of complexes, for example

· · ·C∞(M;∧qbT ∗M)→ C∞(M;

∧q+1bT ∗M)→ · · ·by way of Cartan’s formula for the differential

(q + 1)bdφ(V0, . . . ,Vq) =∑j

(−1)jVjφ(V0, . . . , Vj , . . . ,Vq)

+∑j<k

(−1)j+kφ([Vj ,Vk ], φ1, . . . , Vj , . . . , Vk , . . . ,Vq).

We take advantage of this also to define complex structures.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 12 / 13

All these examples share the property that the spaces V are closed

under Lie bracket. This allows for the definition of complexes, for example

· · ·C∞(M;∧qbT ∗M)→ C∞(M;

∧q+1bT ∗M)→ · · ·by way of Cartan’s formula for the differential

(q + 1)bdφ(V0, . . . ,Vq) =∑j

(−1)jVjφ(V0, . . . , Vj , . . . ,Vq)

+∑j<k

(−1)j+kφ([Vj ,Vk ], φ1, . . . , Vj , . . . , Vk , . . . ,Vq).

We take advantage of this also to define complex structures.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 12 / 13

All these examples share the property that the spaces V are closed

under Lie bracket. This allows for the definition of complexes, for example

· · ·C∞(M;∧qbT ∗M)→ C∞(M;

∧q+1bT ∗M)→ · · ·by way of Cartan’s formula for the differential

(q + 1)bdφ(V0, . . . ,Vq) =∑j

(−1)jVjφ(V0, . . . , Vj , . . . ,Vq)

+∑j<k

(−1)j+kφ([Vj ,Vk ], φ1, . . . , Vj , . . . , Vk , . . . ,Vq).

We take advantage of this also to define complex structures.

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 12 / 13

A complex b-manifold is a manifold M with boundary

together with aninvolutive sub-bundle

bT 0,1M⊂ C bTM

of the complexification of its b-tangent bundle, such that

bT 0,1M+ bT 0,1M = C bTM

as a direct sum.

The boundary of such a manifold inherits an interesting structure which inthe compact case resembles that of a circle bundle of a holomorphic linebundle over a complex manifold.

I’ll state classification theorems for such structures generalizing theclassification of complex line bundles by their Chern class and ofholomorphic line bundles by the Picard group. These classificationtheorems permit the construction of new complex b-manifolds out of agiven one. I’ll give details on how the proofs go about

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 13 / 13

A complex b-manifold is a manifold M with boundary together with aninvolutive sub-bundle

bT 0,1M⊂ C bTM

of the complexification of its b-tangent bundle,

such that

bT 0,1M+ bT 0,1M = C bTM

as a direct sum.

The boundary of such a manifold inherits an interesting structure which inthe compact case resembles that of a circle bundle of a holomorphic linebundle over a complex manifold.

I’ll state classification theorems for such structures generalizing theclassification of complex line bundles by their Chern class and ofholomorphic line bundles by the Picard group. These classificationtheorems permit the construction of new complex b-manifolds out of agiven one. I’ll give details on how the proofs go about

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 13 / 13

A complex b-manifold is a manifold M with boundary together with aninvolutive sub-bundle

bT 0,1M⊂ C bTM

of the complexification of its b-tangent bundle, such that

bT 0,1M+ bT 0,1M = C bTM

as a direct sum.

The boundary of such a manifold inherits an interesting structure which inthe compact case resembles that of a circle bundle of a holomorphic linebundle over a complex manifold.

I’ll state classification theorems for such structures generalizing theclassification of complex line bundles by their Chern class and ofholomorphic line bundles by the Picard group. These classificationtheorems permit the construction of new complex b-manifolds out of agiven one. I’ll give details on how the proofs go about

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 13 / 13

A complex b-manifold is a manifold M with boundary together with aninvolutive sub-bundle

bT 0,1M⊂ C bTM

of the complexification of its b-tangent bundle, such that

bT 0,1M+ bT 0,1M = C bTM

as a direct sum.

The boundary of such a manifold inherits an interesting structure which inthe compact case resembles that of a circle bundle of a holomorphic linebundle over a complex manifold.

I’ll state classification theorems for such structures generalizing theclassification of complex line bundles by their Chern class and ofholomorphic line bundles by the Picard group. These classificationtheorems permit the construction of new complex b-manifolds out of agiven one. I’ll give details on how the proofs go about

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 13 / 13

A complex b-manifold is a manifold M with boundary together with aninvolutive sub-bundle

bT 0,1M⊂ C bTM

of the complexification of its b-tangent bundle, such that

bT 0,1M+ bT 0,1M = C bTM

as a direct sum.

The boundary of such a manifold inherits an interesting structure which inthe compact case resembles that of a circle bundle of a holomorphic linebundle over a complex manifold.

I’ll state classification theorems for such structures generalizing theclassification of complex line bundles by their Chern class and ofholomorphic line bundles by the Picard group. These classificationtheorems permit the construction of new complex b-manifolds out of agiven one. I’ll give details on how the proofs go about

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 13 / 13

A complex b-manifold is a manifold M with boundary together with aninvolutive sub-bundle

bT 0,1M⊂ C bTM

of the complexification of its b-tangent bundle, such that

bT 0,1M+ bT 0,1M = C bTM

as a direct sum.

The boundary of such a manifold inherits an interesting structure which inthe compact case resembles that of a circle bundle of a holomorphic linebundle over a complex manifold.

I’ll state classification theorems for such structures generalizing theclassification of complex line bundles by their Chern class and ofholomorphic line bundles by the Picard group. These classificationtheorems permit the construction of new complex b-manifolds out of agiven one. I’ll give details on how the proofs go about

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 13 / 13

A complex b-manifold is a manifold M with boundary together with aninvolutive sub-bundle

bT 0,1M⊂ C bTM

of the complexification of its b-tangent bundle, such that

bT 0,1M+ bT 0,1M = C bTM

as a direct sum.

The boundary of such a manifold inherits an interesting structure which inthe compact case resembles that of a circle bundle of a holomorphic linebundle over a complex manifold.

I’ll state classification theorems for such structures generalizing theclassification of complex line bundles by their Chern class and ofholomorphic line bundles by the Picard group. These classificationtheorems permit the construction of new complex b-manifolds out of agiven one. I’ll give details on how the proofs go about

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 13 / 13

A complex b-manifold is a manifold M with boundary together with aninvolutive sub-bundle

bT 0,1M⊂ C bTM

of the complexification of its b-tangent bundle, such that

bT 0,1M+ bT 0,1M = C bTM

as a direct sum.

The boundary of such a manifold inherits an interesting structure which inthe compact case resembles that of a circle bundle of a holomorphic linebundle over a complex manifold.

I’ll state classification theorems for such structures generalizing theclassification of complex line bundles by their Chern class and ofholomorphic line bundles by the Picard group.

These classificationtheorems permit the construction of new complex b-manifolds out of agiven one. I’ll give details on how the proofs go about

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 13 / 13

A complex b-manifold is a manifold M with boundary together with aninvolutive sub-bundle

bT 0,1M⊂ C bTM

of the complexification of its b-tangent bundle, such that

bT 0,1M+ bT 0,1M = C bTM

as a direct sum.

The boundary of such a manifold inherits an interesting structure which inthe compact case resembles that of a circle bundle of a holomorphic linebundle over a complex manifold.

I’ll state classification theorems for such structures generalizing theclassification of complex line bundles by their Chern class and ofholomorphic line bundles by the Picard group. These classificationtheorems permit the construction of new complex b-manifolds out of agiven one.

I’ll give details on how the proofs go about

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 13 / 13

A complex b-manifold is a manifold M with boundary together with aninvolutive sub-bundle

bT 0,1M⊂ C bTM

of the complexification of its b-tangent bundle, such that

bT 0,1M+ bT 0,1M = C bTM

as a direct sum.

The boundary of such a manifold inherits an interesting structure which inthe compact case resembles that of a circle bundle of a holomorphic linebundle over a complex manifold.

I’ll state classification theorems for such structures generalizing theclassification of complex line bundles by their Chern class and ofholomorphic line bundles by the Picard group. These classificationtheorems permit the construction of new complex b-manifolds out of agiven one. I’ll give details on how the proofs go about

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 13 / 13

A complex b-manifold is a manifold M with boundary together with aninvolutive sub-bundle

bT 0,1M⊂ C bTM

of the complexification of its b-tangent bundle, such that

bT 0,1M+ bT 0,1M = C bTM

as a direct sum.

The boundary of such a manifold inherits an interesting structure which inthe compact case resembles that of a circle bundle of a holomorphic linebundle over a complex manifold.

I’ll state classification theorems for such structures generalizing theclassification of complex line bundles by their Chern class and ofholomorphic line bundles by the Picard group. These classificationtheorems permit the construction of new complex b-manifolds out of agiven one. I’ll give details on how the proofs go about

Gerardo Mendoza (Temple University) Differential topology of stratified manifolds Beirut, November 2014 13 / 13