Analysis on simply stratified complex manifolds II. 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
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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.
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