Post on 14-Apr-2018
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Canonical Metrics
on Complex
Manifold
University of Michigan
Shing-Tung Yau
Harvard University
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Complex manifolds are topological spaces that
are covered by coordinate charts where the
coordinate changes are given by holomorphic
transformations. Riemann surfaces are one
dimensional complex manifolds.
In order to understand complex manifolds, it
is useful to introduce metrics that are com-
patible with the complex structure. In gen-
eral, we should have a pair (M,ds2M ) where
ds2
M is the metric.
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We say that the metric is canonical if any bi-
holomorphisms of the complex manifolds are
automatically isometries. Such metrics can
then be used to describe invariants of the
complex structures of the manifold.
The first important examples of such metrics
were constructed by Poincare for Riemann
surfaces with genus greater than one.
Note that the flat metric on the torus is not
quite canonical unless we require the biholo-
morphisms to preserve the area. (In higher
dimensions, this is the same as preserving the
Kahler class.)
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Poincare’s metrics are metrics with constant
negative curvature. It requires a proof of the
existence theorem for conformal deformation
of metrics. It is a nonlinear differential equa-
tion and hence difficult higher dimensional
problem. Generalizations of Poincare’s work
took some time.
So far we have discussed Riemannian metrics.
But there are other types of metrics.
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Caratheodory metric:
This is a metric on a Riemann surface X con-
structed by the following procedure:
Given a tangent vector v at a point p ∈ X ,
we define
v = supf ∗(v) : f is a holomorphic map from
X to the unit disk equipped
with the Poincare metric
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In the 1930s, Ahlfors proved a powerful gen-
eralization of the Schwarz lemma in complex
analysis.
Theorem. Holomorphic maps from the Poincare
disk to a Riemann surface with curvature less
than -1 is distance decreasing.
Grauert-Reckziegel generalized Ahlfors-Schwarz
lemma in the 1960s for maps into higher di-
mensional complex manifold.
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Based on these works, Kobayashi introduced
the concept of the Kobayashi metric.
Kobayashi metric:
For a complex tangent vector v at a point p
on a complex manifold, consider all holomor-
phic maps f : D → M such that f (0) = p and
f ∗(a ∂ ∂z ) = v. Define
v = inf f
|a|
It is easy to see that the Kobayashi metric
dominates the Caratheodory metric.
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Both metrics satisfy a stronger property than
the canonical metric condition mentioned above.
If f : M → N is holomorphic, then f decreases
distance in the sense that
f ∗ds2N ≤ ds2
M .
These canonical metrics are not Riemannian
metrics, in general.
In 1970, Royden proved a remarkable theo-
rem that on the Teichmuller space, the Kobayashi
metric is the same as the Teichmuller met-
ric. He also deduced that the group of au-
tomorphisms of the Teichmuller space is the
mapping class group.
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Bergman metric:
Stefan Bergman introduced another intrin-
sic metric. He looked at the space of holo-
morphic n-form f dz1 ∧ · · · ∧ dzn that are L2-
integrable. Note that the inner product
|f |2dz1 ∧ · · · ∧ dzndz1 ∧ · · · ∧ dzn
makes good sense.
With respect to such an inner product, we
can find an orthonormal basis f 1 dz1 ∧ · · · ∧dzn, f 2 dz1 ∧ · · · ∧ dzn, . . . , and Bergman de-
fined a metric (called the Bergman metric)
by
√ −1∂ ∂ log
|f i|2
This metric is canonically defined if there are
enough L2 holomorphic n-forms.
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Kobayashi observed that the Bergman met-
ric is the same as the induced metric from an
embedding of the manifold into the complex
projective space. The embedding is given by
the holomorphic n-forms which are orthonor-
mal.
Many years ago, Lu in China proved that the
Bergman metric dominates the Caratheodory
metric. On the other hand, there is no clear
relationship between the Bergman metric and
the Kobayashi metric.
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Definition. A complex manifold M n is called
holomorphic homogeneous regular if there are
positive constants r < R such that for each
point p ∈ M , there is a one to one holomor-
phic map f : M → Cn such that
1. f ( p) = 0.
2. Br ⊂ f (M ) ⊂ BR where Br and BR are
balls with radius r and R , respectively.
The Bers embedding theorem says that the
Teichmuller space is holomorphic homoge-
neous regular with (r, R) = (2, 6).
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Theorem. (Liu, Sun, Yau. S.K.Yeung) For
holomorphic homogeneous regular manifolds,
the Bergman metric, the Kobayashi metric,
and the Caratheodory metric are equivalent.
The main estimate behind this theorem and
the definition of Kobayashi metric and Caratheodor
metric is the Schwarz lemma, which in turn
depends on the negativity of the holomorphic
sectional curvature.
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Perhaps a more convenient definition is the
following:
Let P (T ∗(M )) be the projectified cotangent
bundle of M , and let O(1) be the canonical
holomorphic line bundle over P (T ∗(M )). If
O(1) admits a metric with positive curvature,
then M would admit a Finsler metric with
negative holomorphic sectional curvature.
In general, we do not expect O(1) > 0, but
there can exist a line bundle L > 0 over
P (T ∗(M )) such that for some m > 0 , (m O(1))L−1
admits a non-trivial holomorphic section. This
condition gives rise to a Finsler metric which
may degenerate along some divisor on P (T ∗(M )).
But the holomorphic sectional curvature is
strongly negative.
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Hence, I proposed the following definition to
replace the Caratheodory metric.
Definition. For any tangent vector v ∈ T p(M ),
define
v = supvD : where D is a Finsler metric
which may degenerate along a divisor
in the tangent bundle of M and
the holomorphic sectional curvature
of
D is less than
−1
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Based on a theorem of Bogomolov, Lu and
I observed that for an algebraic surface M
such that c21 > c2, the metric exists and
that either
(1) there is a divisor D ⊆ M such that for
every holomorphic map f : C → M , the
image f (C) ⊆ D ; or
(2) there is an algebraic foliation with singu-
larity on M such that f maps C to a leaf
of such foliation.
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If one assumes the stronger condition, thatof c2
1 > 2 c2, then an argument of Miyaoka
allows one to conclude that the above con-
dition (2) is unnecessary.
This says that Lang’s conjecture holds for
algebraic surface with c21 > 2 c2.
The metric defined above can be made to
be more algebraic geometric if one uses the
metric obtained by the projective embedding
for L where (m O(1))L−1 admits a section.
The Kobayashi metric always dominates the
metric I just defined. However, it is a much
more transcendental object.
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However we can propose metric which is more
algebraic than the Kobayashi metric. Namely
in the definition of Kobayashi metric, we can
replace the Poincare disc by an algebraic curve
equipped with a metric whose curvature is
equal to
−1.
In this way, we obtain a more algebraic ge-
ometric definition of the Kobayashi metric
while part of the properties of the Kobayashi
metrics are kept.
On the other hand, the definition is good
only when the manifold is algebraic so that
we have plenty of algebraic curves.
Note that in all of the above definitions, there
is corresponding definitions for canonical vol-
ume forms. It is interesting to find relation-
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ship between canonical volumes with canon-
ical metrics.
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So far, the canonical metrics we reviewed
have no clear properties for the curvature.
Without such information, it is difficult to ap-
ply tools from Riemannian geometry to study
complex structures of the manifold.
The only resonable curvature constrain is the
Ricci tensor of the metric. The natural equa-
tion is the Einstein equation.
Namely, the Ricci tensor should be a multi-
ple of the metric. These metrics are called
Kahler-Einstein metrics.
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The existence of such metrics was posed by
Calabi. For the case where the first Chern
class is negative, this was proved indepen-
dently by Aubin and Yau. For the zero first
Chern class case, the existence was proved
by me and such metrics has been especially
important for string theory.
These metrics are canonical. But we need to
generalize the concept of canonical metric inthe following way.
If f : M → N is biholomorphic and if ωM
and ωN are the Kahler forms of the Ricci flat
metric of M and N respectively such that
f ∗ωN is cohomologous to ωM , then f is an
isometry.
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I have applied the existence of such Kahler-
Einstein metrics to questions in algebraic man-
ifolds. For examples:
(1) global rigidity of complex sturcture on
CPn;
(2) algebraic geometric criterions for a man-
ifold to be a Shimura variety.
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For c1 > 0, the problem of the existence of a
Kahler-Einstein metric has not been solved.
The conjecture that I made twenty-five years
ago gives the guiding principle: for such man-
ifolds, the existence of a Kahler-Einstein met-
ric is equivalent to the stability of the mani-
fold.
Donaldson has made the most important con-
tribution. They are related to a proposalthat I made on how to approximate Kahler-
Einstein metrics by suitably defined Bergman
metrics. The stability of the manifold is re-
lated to the embedding of the manifold intoa complex projective space by high multiples
of a line bundle. In the present case, we use
the canonical line bundle.
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The embedding gives rise to the Bergman
metric. The proof of the approximation was
done in Tian’s thesis, using ideas from Siu-
Yau.
When we embed the manifold, it is important
to be able to make the embedding balanced
and this was observed by my student Lo in his
thesis. The balanced condition here can be
interpreted to be related to the moment map
and was used by Donaldson to understand
the related concept of stability.
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When we deform the complex structure of a
manifold, one can form a metric on the mod-
uli space using the Kahler-Einstein metric as
a background metric. This metric is called
the Weil-Petersson metric.
For a Riemann surface, the Weil-Petersson
metric is not complete and is very differentfrom the other metrics. It turns out that mi-
nus of its Ricci tensor defines a Kahler met-
ric which is equivalent to the Kahler-Einstein
metric and the Bergman metric (work of Liu,
Sun, and Yau).
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Liu, Sun, and Yau have also proved that theKahler-Einstein metric is good in the sense of
Mumford on the Deligne-Mumford compact-
ification of the moduli space. In particular,
the moduli space is log stable.
Kahler-Einstein metrics can also be general-
ized to manifolds with singularities. However,
the manifold has to be Kahler. In dimension
greater than two, there are many complex
manifolds which are not Kahler. I conjecture:
Every almost complex manifold with dimen-
sion greater than two admit an integrable
complex structure.
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When a manifold is not Kahler, it is difficult
to find canonical metric that are useful. Here,
physics can provide some help.
In 1986, Strominger proposed the following
system of equations for a bundle V
defined
on a complex manifold M which admits a
holomorphic three-form Ω.
(1) F h ∧ ω2
= 0
(2) F 2,0h = F
0,2h = 0
(3) ∂ ∂ω = √ −1tr(F h ∧ F h) − √ −1 trRg ∧ Rg
(4) d∗ω =√ −1(∂ − ∂ )log Ωω
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Above, h is a Hermitian metric on V and ωis a Hermitian metric on M .
Equation (1) and (2) simply means V satis-
fies the Hermitian-Yang-Mills condition. Equa-
tion (4) means that the metric is conformally
balanced. And (3) is the anomaly equation
required for quantum consistency.
For a general complex manifold, the balanced
metric is a Hermitian metric given by the Her-
mitian form ω =√ −1
gi jdzi ∧ dz j such that
d(ωn
−1) = 0 .
This class of metrics was first studied by
Michelsohn in early 1980s.
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It is difficult to solve Strominger’s four equa-
tions. But we have made progress in the pasttwo years.
Below I present two types of solutions to
Strominger’s system:
1. Li and Yau: On a Calabi-Yau manifold,
the equations can be solved by perturb-
ing the Calabi-Yau metric and vector bun-
dles.
2. Fu and Yau: On the non-Kahler mani-
fold, T
2
bundle over K 3, a metric ansatzcan be used and the delicate estimates
required for a smooth solution have been
obtained.
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I. Li-Yau: Perturbation method
Let E s be a smooth family of holomorphic
vector bundles over a Calabi-Yau space X .
Let h0 be a Hermitian-Yang-Mills connection
on E 0.
We would like to extend h0 to a smooth fam-
ily of Hermitian-Yang-Mills connections.
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The interesting case is when h0 is reducible.
Let (X, ω0) be Kahler.
Let (E 1, D1) and (E 2, D
2) be degree zero and
slope-stable vector bundles.
Let h1 and h2 be the Hermitian metrics on
E 1 and E 2 respectively.
Then h1⊕ et h2 is still a Hermitian metric that
corresponds to the connection D0 = D1⊕D2.
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Suppose we are given a deformation of holo-
morphic structure Ds of D
0. Then Kodaira-
Spencer identifies the first order deformationof D
s at 0 to an element
k ∈ H 1(X, ε∗ ⊗ ε)
where ε is the sheaf of holomorphic section s
of (E, D0).
Therefore
k
∈ ⊕2i,j=1H 1(ε∗
i
⊗ε j).
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Theorem. Suppose k12 and k21 are nonzero.
Then there is a unique t so that for s suffi-
ciently small h0(t) = h1 ⊕ eth2 extends to a
smooth family of Hermitian-Yang-Mills met-rics on (E, D
s ).
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The fourth equation of Strominger system is
equivalent to
dΩωω2
= 0 .
Let H(X ) be the cone of positive definite
Hermitian form on X .
Let H(E )0 be the space of determinant one
Hermitian metric on the bundle E (i.e. the
induced metric on ∧rE CX is the constant
one metric).
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We define
L = L1
⊕L2
⊕L3 :
H(E )0 × H(X ) −→Ω3,3(End0E ) ⊕ Im
√ −1∂ ∂ ⊕ Im d∗0
where
L1(H, ω) = √ −1F H ∧ ω2
L2(H, ω) =√ −1∂ ∂ω + trE (F H ∧ F H )
− trT (Rg ∧ Rg)
L3(H, ω) = ∗0d Ωωω2
.
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We shall apply the implicit function theorem
to L.
Fix a determinant one Hermitian metric
,
on E . We can write other determinant one
Hermitian metric on E by a unique positive
definite , Hermitian symmetric endomor-
phism H of Z satisfying det H = 1.
Such spaces H will be denoted by Γ(End+h E ),
identity I ∈ Γ(End+h E ).
The tangent space at I is Γ(End0hE ) traceless
symmetric endomorphisms of E .
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δL1(I, ω0)(δh,δω)
= DDH δh + 2F H ∧ ω0 ∧ δω
δL2(I, ω0)(δh,δω)
=√
−1∂ ∂ (δω) + 2(trE δF I (δh)
∧ F I )
− trT δRg0(δg) ∧ Rg0
δL3(I, ω0)(δh,δω)
= 2d∗0
(δω)−
d∗0
((δω,ω0
)ω0
).
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We can construct irreducible solutions to Stro-
minger’s system perturbatively.
Start with a Calabi–Yau manifold,
(E, D0) = C
⊕(r−3)X
⊕T X ,
the metric is identified with I : E −→ E .
For all c > 0, (I , c ω0) is a solution to L = 0.
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Let
W 1 = Ω3,3R
(End0hE )L
pk−2
W 2 = (Im√ −1 ∂ ∂ )L p
k−2⊕ (Im d∗
0)L pk−1
V 0 =
A ⊕ aI T X | A ∈ EndC
⊕(r−3)X
are constant matrices such that
A = A−t, tr A + 3a = 0
V 1 = ω3
0 ⊗ V 0.
Then ∃ C > 0 such that for all c > C ,
δL1(I,cω0) ⊕ δL2(I,cω0) ⊕ δL3(I,cω0)
: Γ(End0
hE )
L pk ⊕
Ω1,1(X
)−→ W 1/V 1 ⊕ W 2
is surjective.
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Theorem. Let X be a Calabi-Yau three-fold
with ω a Ricci-flat Kahler form. Let Ds be
a smooth deformation of holomorphic struc-
ture D0 on E = CX ⊕T X . Suppose the associ-
ated cohomology classes [C 12] and [C 21] are
non-zero. Then for sufficiently large c , there
is a family of pairs of Hermitian metrics and
Hermitian forms (H s, ωs) for 0 ≤ s < ε such
that
1. ω0 = c ω and the harmonic part of ωs is
equal to c ω.
2. The pair (H s, ωs) is a solution to Stro-
minger’s system for the holomorphic vec-
tor bundle (E, Ds ).
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Let
Ds =D
0 + A3, As ∈ Ω0,1(End E )
A0 =
C 11 C 12C 21 C 22
∈ Ω0,1(End E ).
We can assume C ij are D0 harmonic. Since
H 1(X, OX ) = 0, C 11 = 0.
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In general, we consider the r +3 holomorphic
vector bundle C⊕rX ⊕ T X . We also have
D0
= 0 C 12
C 21 C 22
where
C 12 =(α1, . . . , αr)t ∈ Ω0,1(T X )⊕ j
C 21 =(β 1, . . . , β r)
∈Ω0,1(T rX )
⊕ j
C 22 ∈ Ω0,1(End T X ).
Suppose [α1], . . . , [αr] ∈ H 1(X, T ∗X ) are lin-
early independent and [β 1], . . . , [β r] ∈ H 1(X, T ∗X )
are linearly independent. Then the above
theorem holds.
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Example
Consider
X = z50 + · · · + z5
x = 0 ∈ P4
0 0
0 −→ T X −→ T X P4 −→ OX (5) −→ 0
0 −→ F −→ OX (1)⊕5 −→ OX (5) −→ 0 OX OX
0 0
Here F is the cokernel of OX
(1)⊕
5
−→ OX (5)
and fill in
0 −→ OX −→ F −→ T X −→ 0.
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The above sequence is a non-split extension.
Making use of this element in Ext1(T X , OX )
we can perform a deformation of the holo-
morphic structure Dt with C 12 = 0 and C 21 = 0.
Hence we have proved:
Let X be a smooth quintic three-fold and ω
be any Kahler form on X . Then for large c >
0, there is a smooth deformation of CX ⊕ T X
such that for small s, there are pairs (H s, ωs)
of Hermitian metrics on E and Hermitian forms
ωs on X that solves Strominger’s system.
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For the Calabi–Yau manifold with three gen-
erations that I constructed:
X ⊂ P3 × P3
given by
x3
i = 0
y
3
i = 0
xiyi = 0
quotient by Z3. One can also construct ir-
reducible solution to Strominger’s system on
T X ⊕ C⊕2X .
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II. Fu-Yau: Non-Kahler manifolds
Let (S, ωS , ΩS ) be the K3 surface. Let ω12π ,ω2
2π
∈H 2(S,Z) and let ω1 and ω2 be anti-self-dual
(1,1)-forms. Then there is a non-Kahler man-
ifold X such that π : X → S is a holomorphic
T 2 bundle over S .
If we write locally ω1 = dα1 and ω2 = dα2,
then there are coordinates of the T 2 fiber, x
and y, such that dx + π∗α1 and dy + π∗α2 are
globally defined 1-forms on X .
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Let
θ = dx + π∗α1 +√ −1(dy + π∗α2) .
Then the Hermitian form on X is
ω0 = π∗ωS + √ −12
θ ∧ θ
and the holomorphic 3-form is
Ω = π∗ΩS ∧ θ .
ω0 satisfies the fourth equation d( Ω ω0 ω20) = 0.
Let u be any smooth function on S and let
ωu = π∗(euωS ) +
√ −1
2 θ ∧ θ .
Then ωu is a Hermitian metric on X and
(ωu, Ω) also satisfies d( Ω ωu ω2u) = 0.
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If we let Ru be the curvature of the Hermitian
connection of the metric ωu on the holomor-
phic tangent bundle, then
tr Ru ∧ Ru =tr RS ∧ RS + 2 ∂ ∂u ∧ ∂ ∂u
+2√ −1 ∂ ∂ (e−uf ωS ),
where for ω2 = n ω1, n ∈ Z, f = 1+n2
4 ω12ωS
.
So the third equation in Strominger’s system
can be reduced to√ −1∂ ∂eu ∧ ωS − 2 ∂ ∂u ∧ ∂ ∂u − 2
√ −1 ∂ ∂ (e−uf ωS )
=tr RS ∧
RS −
tr F H ∧
F H −
(
ω1
2 +
ω2
2)ω2
S
2!.
(1)
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Let (E, H ) be the Hermitian-Yang-Mills vec-
tor bundle over S with the gauge group SU (r).
Then (V = π∗E, H ) is also the Hermitian-
Yang-Mills vector bundle over X . We can
consider equation (1) as the equation on the
K 3 surface S . Integrating equation (1) over
S , S trRS ∧ RS − trF H ∧ F H
= S
(
ω1
2ωS
+
ω2
2ωS
)ω2
S
2!
.
We use Q( ωi2π) to denote the intersection num-
ber of anti-self-dual (1,1)-form ωi2π. As
18π2 S trRS ∧ RS = 24, the above condition
can be written as
2(24 − c2(E )) = −
Q
ω1
2π
+ Q
ω2
2π
.
(2)
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Certainly we can choose ω1 and ω2 and SU (r)
vector bundle E such that they satisfy the
condition (2). Then there is a smooth func-
tion µ such that
trRS ∧ RS − trF H ∧ F H
−ω12 + ω22
ω2S
2! = −µ
ω2S
2!.
So we obtain the following equation:
√ −1∂ ∂eu ∧ ωS − ∂ ∂u ∧ ∂ ∂u
− √ −1∂ ∂ (e−uf ωS ) + µω2
S
2! = 0 .
(3)
which can be rewritten as the standard com-
plex Monge-Ampere equation:
∆(e−u − f e−u) + 8det ui j
det gi j
+ µ = 0 .
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We solve equation (3) by the continuity method
and get
Theorem. Equation (3) has a smooth solu-
tion u such that
ω = (eu + f e−u)ωS + 2√
−1∂ ∂u
is a Hermitian metric on S .
The requirement that ω is hermitian is an
elliptic condition that we need to impose.
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Theorem. Let S be a K 3 surface with Calabi-
Yau metric ωS . Let ω1 and ω2 be anti-self-
dual (1, 1)-forms on S such that ω12π ∈ H 2(S,Z)
and ω22π ∈ H 2(S,Z). Let X be a T 2-bundle
over S constructed by ω1 and ω2. Let E bea stable bundle over S with the gauge group
SU (r). Suppose ω1, ω2 and c2(E ) satisfy the
topological constraint (2). Then there exists
a smooth function u and a Hermitian-Yang-Mills metric H on E such that (V = π∗E, π∗F H , X , ωu
is a solution of Strominger’s system.
We will use the normalization condition
S
e−4u ω2S
2!
14
= A ,
S 1
ω2S
2! = 1 .
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Zeroth order estimate
Let P = 2gi j ∂ 2
∂zi∂ z j. We have two methods of
calculating
S P (eku)
det gi j
det gi j
ω2S
2! .
Then using the Sobolev inequality, Moser it-
eration and Poincare inequality, we obtain
Proposition. If A < 1, then there is a con-
stant C 1 which depends on f , µ and the
Sobolev constant of ωS such that
inf S
u ≥ − ln(C 1A).
Moreover, if A is small enough such thatA < (C 1)−1, then there is an upper bound
of supS u which depends on f , µ, Sobolev
constant of ωS and A.
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An estimate of the determinant
We need to estimate the lower bound of the
determinant
F =det g
i j
det gi j
.
We apply the maximum principle to the func-tion
G = 1 − e−u | u |2 +2e−u − 2e− inf u
and obtain
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Proposition. Given any constant κ ∈ (0, 1),
we fix some positive constant satisfying
< min1, 2−1κ.
Suppose that A satisfies
A < min1, C −11 , 2(1 + sup f )−1
2C −11 ,
1
−κ
2C 3
1
C −1
1 ,
3
−6
C 4 C −1
1 , C 5,
where C 3 and C 4 depend on f and µ, C 4 also
depends on the curvature bound of ωS , and
C 5 depends on κ, and C 3. Then F > κe2u ≥
κ(C 1A)−2.
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Second order estimate
Since
eu + f e−u + u ≥ F 12 > κ
12(C 1A)−1 > 0,
it is sufficient to have an upper estimate of
eu + f e−u + u. Applying the maximum prin-
ciple to the function
e−λ1u+λ2|u|2 · (eu + f e−u + u),
where λ1 and λ2 are positive constants that
can be determined, we can obtain the second
order estimate.
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Third order estimate
Let
Γ = gi jgklu,iku, jl
Θ = girgs jgktu,i jku,rst
Ξ = gi jgklg pqu,ikpu, jlq
Φ = gi j
gkl
g pq
grs
u,ilpru, jkqs
Ψ = gi jgklg pqgrsu,ilpsu, jkqr,
where indices preceded by a comma indicate
covariant differentiation with respective to
the metric ωS .
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Again, we apply the maximum principle to
the function
(κ1+u)Θ+κ2(m+u)Γ+κ3 | u |2 Γ+κ4Γ,
where all κi are positive constants that can
be determined and m is a fixed constant such
that m + u > 0. We can then obtain the
third order estimate.
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Topology of the total space:
(1) h0,1(X ) = h0,1(S ) + 1
(2) h1,0(X ) = h1,0(S )
(3) b1(X ) = b1(S ) + 1
(4a) b2(X ) = b2(S ) − 1
if ω1 is a multiple of ω2
(4b) b2(X ) = b2(S ) − 2
if ω1 is not a multiple of ω2 .