365
Mass in K¨ ahler Geometry Claude LeBrun Stony Brook University New Horizons in Twistor Theory Oxford, January 5, 2017
Mass in
Kahler Geometry
Claude LeBrunStony Brook University
New Horizons in Twistor TheoryOxford, January 5, 2017
1
Joint work with
2
Joint work with
Hans-Joachim HeinUniversity of Maryland
3
Joint work with
Hans-Joachim HeinFordham University
4
Joint work with
Hans-Joachim HeinFordham University
Comm. Math. Phys. 347 (2016) 621–653.
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn
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n ≥ 3
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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n ≥ 3
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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n ≥ 3
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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10
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
12
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
13
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
14
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), scalar curvature ∈ L1
15
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
16
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
17
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
18
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
19
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
20
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
21
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
22
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each “end” component of is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
23
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(R
n −Dn)/Γi, where Γi ⊂ O(n), such that
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24
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(R
n −Dn)/Γi, where Γi ⊂ O(n), such that
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25
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(R
n −Dn)/Γi, where Γi ⊂ O(n), such that
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26
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
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27
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
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28
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
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29
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
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30
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
31
Why consider ALE spaces?
32
Key examples:
33
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
34
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
35
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
By contrast, any Ricci-flat AE manifold must beflat, by the Bishop-Gromov inequality. . .
36
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
37
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
38
vv
vvv
Data: ` points in R3. =⇒ V with ∆V = 0
39
vv
vvv
Data: ` points in R3. =⇒ V with ∆V = 0
V =∑j=1
1
2%j
40
vv
vvv
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Data: ` points in R3. =⇒ V with ∆V = 0
V =∑j=1
1
2%j
F = ?dV curvature θ on P → R3 − pts.
41
vv
vvv
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Data: ` points in R3. =⇒ V with ∆V = 0
g = V h + V −1θ2
F = ?dV curvature θ on P → R3 − pts.
42
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Data: ` points in R3. =⇒ V with ∆V = 0
g = V h + V −1θ2
on P . Then take M4 = Riemannian completion.
43
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
44
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
The G-H metrics are hyper-Kahler, and were soonrediscovered independently by Hitchin.
45
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
46
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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47
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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48
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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49
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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50
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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51
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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52
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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53
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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54
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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55
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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56
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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57
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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58
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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59
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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60
Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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61
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
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62
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
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Sp(1) = SU(2)
63
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
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Sp(1) = SU(2)
⇐⇒ Λ+ flat and trivial.
64
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
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Sp(1) = SU(2)
⇐⇒ Λ+ flat and trivial.
Locally, ⇐⇒ s = 0, r = 0, W+ = 0.
65
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
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Sp(1) = SU(2)
Ricci-flat and Kahler,
for many different complex structures!
66
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ U(2)
s
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Sp(1) ⊂ U(2)
Ricci-flat and Kahler,
for many different complex structures!
67
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
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Sp(1) = SU(2)
Ricci-flat and Kahler,
for many different complex structures!
68
All these complex structures can be repackaged as
Penrose Twistor Space (Z6, J),
which is a complex 3-manifold
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69
All these complex structures can be repackaged as
Penrose Twistor Space (Z, J),
which is a complex 3-manifold
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70
All these complex structures can be repackaged as
Penrose Twistor Space (Z, J),
which is a complex 3-manifold.
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71
All these complex structures can be repackaged as
Penrose Twistor Space (Z, J),
which is a complex 3-manifold.
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Riemannian non-linear graviton construction.
72
Hitchin’s Twistor Spaces:
73
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3⊃ R3.
74
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
75
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
76
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)
77
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)
xy = (z − P1) · · · (z − P`)
78
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)
xy = (z − P1) · · · (z − P`)
79
Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)
xy = (z − P1) · · · (z − P`)is the twistor space of a Gibbons-Hawking metric.
80
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
The G-H metrics are hyper-Kahler, and were soonrediscovered independently by Hitchin.
81
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
The G-H metrics are hyper-Kahler, and were soonrediscovered independently by Hitchin.
Hitchin conjectured that similar metrics would existfor each finite Γ ⊂ SU(2).
82
Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
The G-H metrics are hyper-Kahler, and were soonrediscovered independently by Hitchin.
Hitchin conjectured that similar metrics would existfor each finite Γ ⊂ SU(2).
This conjecture was proved by Kronheimer, 1986.
83
Felix Klein, 1884: C2/Γ → C3
Zk+1 ←→ xy + zk+1 = 0
Dih∗k−2 ←→ x2 + z(y2 + zk−2) = 0
T ∗ ←→ x2 + y3 + z4 = 0
O∗ ←→ x2 + y3 + yz3 = 0
I∗ ←→ x2 + y3 + z5 = 0
84
Zk+1←→Ak • • • •.......................................................................................................................................................
Dih∗k−2←→Dk • • • ••.................................................................................................................
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T ∗←→E6 • • • •••................................................................................................................................................................................................................
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O∗←→E7 • • • ••• •..................................................................................................................................................................................................................................................................
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I∗←→E8 • • • ••• • •....................................................................................................................................................................................................................................................................................................................
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85
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ U(2)
s
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86
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ U(2)
s
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In real dimension 4, any Kahler manifold satisfies
|W+|2 =s2
24
87
Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ U(2)
s
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In real dimension 4, any Kahler manifold satisfies
|W+|2 =s2
24
so that W+ = 0 ⇐⇒ s = 0.
88
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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89
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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Kahler case: holomorphic section of KZ−1/2
vanishing at D := (M,J) ∩ (M,−J).
90
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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Kahler case: holomorphic section of KZ−1/2
vanishing at D := (M,J) ∩ (M,−J).
=⇒ Normal bundle of (M,J) in Z is KM−1.
91
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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0→ T 1,0M → T 1,0Z|M → KM−1→ 0
=⇒ Normal bundle of (M,J) in Z is KM−1.
92
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Extension class: [ω] ∈ H1(M,O(KM⊗T 1,0M)).
93
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Extension class: [ω] ∈ H1(M,Ω1).
94
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Extension class: [ω] ∈ H1(M,Ω1).
95
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Kahler class: [ω] ∈ H1(M,Ω1).
96
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Kahler form: ω = g(J ·, ·).
97
Some AE/ALE Scalar-Flat Kahler Surfaces:
98
Some AE/ALE Scalar-Flat Kahler Surfaces:
(L ’91)
99
Some AE/ALE Scalar-Flat Kahler Surfaces:
100
Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
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.............
Data: k + 1 points in H3. =⇒ V with ∆V = 0
101
Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
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Data: k + 1 points in H3. =⇒ V with ∆V = 0
V = 1 +`
e2%0 − 1+
k∑j=1
1
e2%j − 1
102
Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
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Data: k + 1 points in H3. =⇒ V with ∆V = 0
F = ?dV curvature θ on P → H3 − pts.
103
Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
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.............
Data: k + 1 points in H3. =⇒ V with ∆V = 0
g =1
4 sinh2 %0
(V h + V −1θ2
)
104
Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
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.............
Riemannian completion is ALE scalar-flat Kahler.
g =1
4 sinh2 %0
(V h + V −1θ2
)
105
Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
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.............
Riemannian completion is AE ⇐⇒ ` = 1:
V = 1 +`
e2%0 − 1+
k∑j=1
1
e2%j − 1
106
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4⊃ R1,3 ⊃ H3
107
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3⊃ H3
108
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
109
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
f
vv
vvv
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.............
So k + 1 points in H3 give rise to
110
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
f
vv
vvv
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So k + 1 points in H3 give rise to
P0, P1, . . . , Pk ∈ H0(CP1 × CP1,O(1, 1)).
111
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
112
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
113
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
114
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
Then twistor space Z obtained from Z by
115
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
Then twistor space Z obtained from Z by
• removing curve in zero section cut out by P0,
116
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
Then twistor space Z obtained from Z by
• removing curve in zero section cut out by P0,
• adding two rational curves at infinity, and
117
Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
Then twistor space Z obtained from Z by
• removing curve in zero section cut out by P0,
• adding two rational curves at infinity, and
•making small resolutions of isolated singularities.
118
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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119
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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rr Z
M 4
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Lots more ALE scalar-flat Kahler surfaces now known:
120
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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rr Z
M 4
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Lots more ALE scalar-flat Kahler surfaces now known:
Joyce, Calderbank-Singer, Lock-Viaclovsky. . .
121
Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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rr Z
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Lots more ALE scalar-flat Kahler surfaces now known:
Joyce, Calderbank-Singer, Lock-Viaclovsky. . .
But full classification remains an open problem.
122
Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
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....
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
123
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
124
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
125
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
126
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
127
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
128
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
129
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
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130
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
...............................................................................................................................................................................................................................................................................................................................
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131
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
132
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
133
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Seems to depend on choice of coordinates!
134
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
135
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
136
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
137
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
138
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
139
Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
140
Motivation:
141
Motivation:
When n = 3, ADM mass in general relativity.
142
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
143
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
144
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
145
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
146
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g = −(
1− 2m%n−2
)dt2+
(1− 2m
%n−2
)−1
d%2+%2hSn−1
147
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
148
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
149
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
150
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
151
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
152
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
153
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
154
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
But any Ricci-flat ALE manifold has mass zero.
155
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
But any Ricci-flat ALE manifold has mass zero.
Bartnik: Ricci-flat =⇒ faster fall-off of metric!
156
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
But any Ricci-flat ALE manifold has mass zero.
Bartnik: Ricci-flat =⇒ faster fall-off of metric!
=⇒ “gravitational instantons” have mass zero.
157
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2⊂ C2 × CP1
158
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1
159
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
160
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1
161
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1
162
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:
ω =i
2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2
163
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:
ω =i
2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2
164
Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:
ω =i
2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2
also has mass m .
165
Positive Mass Conjecture:
166
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
167
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
168
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
169
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Physical intuition:
170
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Physical intuition:
Local matter density ≥ 0 =⇒ total mass ≥ 0.
171
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Physical intuition:
Local matter density ≥ 0 =⇒ total mass ≥ 0.
172
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
173
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
174
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
175
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
176
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n.)
177
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
178
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
179
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
180
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1987:
181
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1987:
ALE counter-examples.
182
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1987:
ALE counter-examples.
Scalar-flat Kahler metrics
183
Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1987:
ALE counter-examples.
Scalar-flat Kahler metrics
on line bundles L→ CP1 of Chern-class ≤ −3.
184
Mass of ALE Kahler manifolds?
185
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
186
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
187
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
188
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
189
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
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..................................
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190
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
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n = 2m ≥ 4
191
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
192
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Main Point:
193
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Main Point:
Mass of an ALE Kahler manifold is unambiguous.
194
Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Main Point:
Mass of an ALE Kahler manifold is unambiguous.
Does not depend on the choice of an end!
195
We begin with the scalar-flat Kahler case.
196
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
197
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
198
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
199
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
200
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
201
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
202
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
203
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
204
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and
205
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and
• the Kahler class [ω] ∈ H2(M) of the metric.
206
We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and
• the Kahler class [ω] ∈ H2(M) of the metric.
In fact, we will see that there is an explicit formulafor the mass in terms of these data!
207
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
208
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
209
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
210
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M, g) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
211
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M, g) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
212
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
213
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
214
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
215
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
Note that minimality is essential here.
216
The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
Note that minimality is essential here.
Non-minimal resolutions typically admit families ofsuch metrics for which the mass can be continuouslydeformed from negative to positive.
217
Explicit formula depends on a topological fact:
218
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. x Then the natural map
219
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. x Then the natural map
220
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
221
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
222
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
223
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
224
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Here
Hpc (M) :=
ker d : Epc (M)→ Ep+1c (M)
dEp−1c (M)
225
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Here
Hpc (M) :=
ker d : Epc (M)→ Ep+1c (M)
dEp−1c (M)
where
Epc (M) := Smooth, compactly supported p-forms on M.
226
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
227
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
228
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
229
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
230
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
231
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
to denote the inverse of the natural map
232
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
to denote the inverse of the natural map
H2c (M)→ H2
dR(M)
induced by the inclusion of compactly supportedsmooth forms into all forms.
233
Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
to denote the inverse of the natural map
H2c (M)→ H2
dR(M)
induced by the inclusion of compactly supportedsmooth forms into all forms.
234
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
235
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)ofcomplex dimension m has mass given by
236
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
237
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
238
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
239
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
240
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
241
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
242
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
243
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
244
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
245
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
246
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
• c1 = c1(M,J) ∈ H2(M) is first Chern class;
247
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
• c1 = c1(M,J) ∈ H2(M) is first Chern class;
• [ω] ∈ H2(M) is Kahler class of (g, J); and
248
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
• c1 = c1(M,J) ∈ H2(M) is first Chern class;
• [ω] ∈ H2(M) is Kahler class of (g, J); and
• 〈 , 〉 is pairing between H2c (M) and H2m−2(M).
249
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
250
4πm(2m−1)(m−1)!
m(M, g) = − 4π(m−1)!
〈♣(c1), [ω]m−1〉+∫Msgdµg
251
For a compact Kahler manifold (M2m, g, J),
∫Msgdµg =
4π
(m− 1)!〈c1, [ω]m−1〉
252
For a compact Kahler manifold (M2m, g, J),
0 = − 4π
(m− 1)!〈c1, [ω]m−1〉 +
∫Msgdµg
253
For an ALE Kahler manifold (M2m, g, J),
4πm(2m−1)(m−1)!
m(M, g) = − 4π(m−1)!
〈♣(c1), [ω]m−1〉+∫Msgdµg
254
For an ALE Kahler manifold (M2m, g, J),
4πm(2m−1)(m−1)!
m(M, g) = − 4π(m−1)!
〈♣(c1), [ω]m−1〉+∫Msgdµg
So the mass is a “boundary correction” to the topo-logical formula for the total scalar curvature.
255
We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
256
We can now state our mass formula:
Corollary. Any ALE scalar-flat Kahler mani-fold (M, g, J) of complex dimension m has massgiven by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
.
257
We can now state our mass formula:
Corollary. Any ALE scalar-flat Kahler mani-fold (M, g, J) of complex dimension m has massgiven by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
.
So Theorem A is an immediate consequence!
258
Rough Idea of Proof:
259
Rough Idea of Proof:
Special Case: Suppose
260
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
261
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
262
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
263
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
264
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
265
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
(z1, z2) = (x1 + ix2, x3 + ix4)
are harmonic. So xj are harmonic, too, and
266
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
(z1, z2) = (x1 + ix2, x3 + ix4)
are harmonic. So xj are harmonic, too, and
267
Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
(z1, z2) = (x1 + ix2, x3 + ix4)
are harmonic. So xj are harmonic, too, and
gjk(gj`,k − gjk,`
)ν`αE = −?d log
(√det g
)+O(%−3−ε).
268
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
269
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
270
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
271
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
−?d log(√
det g)
= 2 θ ∧ ω.
272
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
−?d log(√
det g)
= 2 θ ∧ ω.Thus
m(M, g) = − lim%→∞
1
6π2
∫S%/Γ
θ ∧ ω
273
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
−?d log(√
det g)
= 2 θ ∧ ω.Thus
m(M, g) = − lim%→∞
1
6π2
∫S%/Γ
θ ∧ ω
However, since s = 0,
d(θ ∧ ω) = ρ ∧ ω =s
4ω2 = 0.
274
m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
−?d log(√
det g)
= 2 θ ∧ ω.Thus
m(M, g) = − 1
6π2
∫S%/Γ
θ ∧ ω
275
Let f : M → R be smooth cut-off function:
276
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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277
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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278
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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279
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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“end”
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280
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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radius
“end”
1
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281
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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282
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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radius
M%
%
1
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283
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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M%
%
1
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284
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
285
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
286
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
Compactly supported, because dθ = ρ near infinity.
287
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
288
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 =
∫Mψ ∧ ω
289
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 =
∫M%
ψ ∧ ω
where M% defined by radius ≤ %.
290
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 =
∫M%
ψ ∧ ω
291
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 =
∫M%
[ρ− d(fθ)] ∧ ω
292
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 =
∫M%
[ρ− d(fθ)] ∧ ω
because scalar-flat =⇒ ρ ∧ ω = 0.
293
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫M%
d(fθ) ∧ ω
because scalar-flat =⇒ ρ ∧ ω = 0.
294
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫M%
d(fθ ∧ ω)
because scalar-flat =⇒ ρ ∧ ω = 0.
295
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫M%
d(fθ ∧ ω)
296
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫∂M%
fθ ∧ ω
by Stokes’ theorem.
297
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫∂M%
θ ∧ ω
by Stokes’ theorem.
298
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
299
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
300
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
So
m(M, g) = − 1
6π2
∫S%/Γ
θ ∧ ω
301
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
So
m(M, g) = − 1
3π〈♣(c1), [ω]〉
302
Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
So
m(M, g) = − 1
3π〈♣(c1), [ω]〉
as claimed.
303
We assumed:
304
We assumed:
•m = 2;
• s ≡ 0; and
• Complex structure J standard at infinity.
305
General case:
306
General case:
•General m ≥ 2: straightforward. . .
307
General case:
•General m ≥ 2: straightforward. . .
308
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
309
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
310
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
311
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
312
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
The last point is serious.
313
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
Seen in “gravitational instantons”
and other explicit examples.
314
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
One argument proceeds by osculation:
315
General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
One argument proceeds by osculation:
J = J0 + O(%−3), OJ = O(%−4)
in suitable asymptotic coordinates adapted to g.
316
To understand J at infinity:
317
To understand J at infinity:
Let M∞ be universal over of end M∞.
318
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
319
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
320
To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
Complete analytic family encodes info about J .
321
To understand J at infinity:
322
To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
323
To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
Linear system of CPm−1 gives holomorphic map
(M ∪ CPm−1)→ CPmwhich is biholomorphism near CPm−1.
324
To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
Linear system of CPm−1 gives holomorphic map
(M ∪ CPm−1)→ CPmwhich is biholomorphism near CPm−1.
Thus obtain holomorphic map
Φ : M → Cm
which is biholomorphism near infinity.
325
To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
Linear system of CPm−1 gives holomorphic map
(M ∪ CPm−1)→ CPmwhich is biholomorphism near CPm−1.
Thus obtain holomorphic map
Φ : M → Cm
which is biholomorphism near infinity.
This has some interesting consequences. . .
326
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
327
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
328
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
329
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
330
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
331
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
Moreover, m = 0 ⇐⇒
332
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
Moreover, m = 0⇐⇒ (M, g) is Euclidean space.
333
Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
Moreover, m = 0⇐⇒ (M, g) is Euclidean space.
Proof actually shows something stronger!
334
Theorem E (Penrose Inequality). Let (M, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
335
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
336
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
337
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
338
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
339
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
340
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
341
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
342
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
343
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
344
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
345
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
346
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
with = ⇐⇒
347
Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
with = ⇐⇒ (M, g, J) is scalar-flat Kahler.
348
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
349
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
350
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
351
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
352
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
353
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
D =∑
njDj
354
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
D =∑
njDj
and
−〈♣(c1),ωm−1
(m− 1)!〉 =
∑njVol (Dj)
355
This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
D =∑
njDj
and
−〈♣(c1),ωm−1
(m− 1)!〉 =
∑njVol (Dj)
so the mass formula implies the claim.
356
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
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357
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
358
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
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359
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
360
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
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361
362
Happy Birthday, Roger!
363
Happy Birthday, Roger!
Happy Birthday, Twistors!
364
Happy Birthday, Roger!
Happy Birthday, Twistors!
Happy Non-Retirement, Nick!
365
