Coassociative 4-folds with Conical Singularities

0.45 MB·English

by  Jason Lotay

6 Coassociative 4-folds with Conical Singularities
0
0
2
Jason Dean Lotay

n
a University College
J
Oxford
1
3

] 1 Introduction
G
D
Thispaperisdedicatedtothestudyofdeformations ofcoassociative 4-folds ina
.
h
G manifold which have conical singularities. Understanding the deformations
t 2
a ofsuchsingularcoassociative4-foldsshouldbeausefulsteptowardsattempting
m
toprovea7-dimensionalanalogueoftheSYZ conjecture. Theresearchdetailed
[
here is motivated by the work on the deformation theory of special Lagrangian

1
m-folds with conicalsingularitiesby Joycein the seriesof papers[6], [7], [8], [9]
v
2 and [10], and the work of the author in [15] on deformations of asymptotically
6 conical coassociative 4-folds.
7
1 We begin, in Section 2, by discussing the notions of G2 structures, G2 man-
0 ifolds and coassociative 4-folds. In Section 3 we introduce a distinguished class
6
ofsingularmanifolds knownasCS manifolds. CS manifoldshaveconicalsingu-
0
/ larities and their nonsingular part is a noncompact Riemannian manifold. We
h
t also define what we mean by CS coassociative 4-folds.
a
m In order that we may employ various analytic techniques in the course of
: ourstudy,wechoosetouseweighted Banach spaces offormsonthenonsingular
v
part of a CS manifold. These spaces are described in §4. We then focus, in
i
X
Section 5, on a particular linear, elliptic, first-order differential operator acting
r
a between weighted Banach spaces in the case of a 4-dimensional CS manifold.
The Fredholm and index theory of this operator is discussed using the theory
developed in [14].
In Section 6 we stratify the types of deformations allowed into three prob-
lems, each with an associated nonlinear first-order differential operator whose
kernel gives a local description of the moduli space. The main result for each
problem, given in §7, states that the moduli space is locally homeomorphic to
the kernel of a smooth map between smooth manifolds. In each case, the map
in question can be considered as a projection from the infinitesimal deforma-
1
tion space ontothe obstruction space. Thus,whentherearenoobstructionsthe
modulispaceisasmoothmanifold. Furthermore,usingthe materialin§5helps
to provide a lower bound on the expected dimension of the moduli space.
The lastsectionshowsthat, in weakeningthe conditiononthe G structure
2
of the ambient 7-manifold, there is a generic smoothness result for the moduli
spaces of deformations corresponding to our second and third problems.
Notes
(a) Manifolds aretaken to be nonsingular andsubmanifolds to be embedded,
for convenience, unless stated otherwise.
(b) We use the convention that the natural numbers N={0,1,2,...}.
2 Coassociative 4-folds
The key to defining coassociative 4-folds lies with the introduction of a distin-
guished 3-form on R7.
Definition 2.1 Let (x ,...,x ) be coordinates on R7 and write dx for the
1 7 ij...k
form dx ∧dx ∧...∧dx . Define a 3-form ϕ by:
i j k 0
ϕ =dx +dx +dx +dx −dx −dx −dx . (1)
0 123 145 167 246 257 347 356
The 4-form ∗ϕ , where ϕ and ∗ϕ are related by the Hodge star, is given by:
0 0 0
∗ϕ =dx +dx +dx +dx −dx −dx −dx . (2)
0 4567 2367 2345 1357 1346 1256 1247
Our choice of expression(1) for ϕ follows that of [5, Chapter 10]. This form is
0
sometimes known as the G 3-form because the Lie group G is the subgroup
2 2
of GL(7,R) preserving ϕ .
0
Definition 2.2 A 4-dimensional submanifold N of R7 is coassociative if and
only if ϕ | ≡0 and ∗ϕ | >0.
0 N 0 N
This definition is not standard but is equivalent to the usual definition in the
language of calibrated geometry by [3, PropositionIV.4.5 & Theorem IV.4.6].
Remark The condition ϕ | ≡ 0 forces ∗ϕ to be a nonvanishing 4-form on
0 N 0
N. Thus, the positivity of ∗ϕ | is equivalent to a choice of orientation on N.
0 N
So that we may describe coassociative submanifolds of more general 7-
manifolds, we make two definitions following [2, p. 7] and [5, p. 243].
2
Definition 2.3 Let M be an oriented 7-manifold. For each x∈M there exists
an orientation preserving isomorphism ι : T M → R7. Since dim G = 14,
x x 2
dim GL (T M)=49 and dimΛ3T∗M =35, the GL (T M) orbit of ι∗(ϕ ) in
+ x x + x x 0
Λ3T∗M, denoted Λ3T∗M, is open. A 3-form ϕ on M is definite, or positive, if
x + x
ϕ| ∈Λ3T∗M forallx∈M. Denotethebundleofdefinite3-formsΛ3T∗M.
TxM + x +
It is a bundle with fibre GL (7,R)/G which is not a vector subbundle of
+ 2
Λ3T∗M.
Essentially, a definite 3-form is identified with the G 3-form on R7 at each
2
point in M. Therefore, to each definite 3-form ϕ we can uniquely associate a
4-form ∗ϕ and a metric g on M such that the triple (ϕ,∗ϕ,g) corresponds to
(ϕ ,∗ϕ ,g ) at each point. This leads us to our next definition.
0 0 0
Definition 2.4 Let M be an oriented 7-manifold, let ϕ be a definite 3-formon
M andlet g be the metric associatedto ϕ. We call (ϕ,g) a G structure onM.
2
If ϕ is closed (or coclosed) then (ϕ,g) is a closed (or coclosed) G structure. A
2
closed and coclosed G structure is called torsion-free.
2
Our choice of notation here agrees with [2].
Remark There is a 1-1 correspondence between pairs (ϕ,g) and principal G
2
subbundles of the frame bundle.
Our definition of torsion-free G structure is not standard, but agrees with
2
other definitions by the following result [19, Lemma 11.5].
Proposition 2.5 Let (ϕ,g) be a G structure and let ∇ be the Levi–Civita
2
connection of g. The following are equivalent:
dϕ=d∗ϕ=0; ∇ϕ=0; and Hol(g)⊆G with ϕ as the associated 3-form.
2
Definition 2.6 Let M be an oriented 7-manifoldendowedwith a G structure
2
(ϕ,g), denoted (M,ϕ,g). We say that (M,ϕ,g) is a ϕ-closed, or ϕ-coclosed,
7-manifold if (ϕ,g) is a closed, respectively coclosed, G structure. If (ϕ,g) is
2
torsion-free, we call (M,ϕ,g) a G manifold.
2
We are now able to complete our definitions.
Definition 2.7 A 4-dimensional submanifold N of (M,ϕ,g) is coassociative if
and only if ϕ| ≡0 and ∗ϕ| >0.
N N
We end this section with a result, which follows from [16, Proposition 4.2],
that is invaluable in describing the deformation theory of coassociative 4-folds.
3
Proposition 2.8 Let N be a coassociative 4-fold in (M,ϕ,g). There is an
isomorphism between the normal bundle ν(N) of N in M and Λ2T∗N given by
+
v 7→(v·ϕ)| .
TN
3 Conical singularities
3.1 CS manifolds
Definition 3.1 Let M be a connected Hausdorff topological space and let
z ,...,z ∈ M. Suppose that Mˆ = M \ {z ,...,z } has the structure of a
1 s 1 s
(nonsingular) n-dimensional Riemannian manifold, with Riemannian metric g,
compatible with its topology. Then M is a manifold with conical singularities
(at z ,...,z with rate λ) if there exist constants ǫ > 0 and λ > 1, a compact
1 s
(n−1)-dimensionalRiemannianmanifold(Σ ,h ),anopensetU ∋z inM with
i i i i
U ∩U = ∅ for j 6= i and a diffeomorphism Ψ : (0,ǫ)×Σ → U \{z } ⊆ Mˆ,
i j i i i i
for i=1,...,s, such that
|∇j(Ψ∗(g)−g )|=O(rλ−1−j) for j ∈N as r →0, (3)
i i i i i
wherer isthecoordinateon(0,∞)ontheconeC =(0,∞)×Σ ,g =dr2+r2h
i i i i i i i
is the conical metric on C , ∇ is the Levi–Civita connection derived from g
i i i
and |.| is calculated using g . We call C the cone at the singularity z and let
i i i
the ends Mˆ of Mˆ be the disjoint union
∞
s
Mˆ = U \{z }.
∞ i i
i=1
G
We saythatM is CS ora CS manifold (with rateλ) ifit is amanifold with
conicalsingularities which haverate λ and it is compact as a topologicalspace.
In these circumstances it may be written as the disjoint union
s
M =K⊔ U ,
i
i=1
G
where K is compact as it is closed in M.
The condition λ > 1 guarantees that the metric on Mˆ genuinely converges
to the conical metric on C , as is evident from (3). Since M is supposed to
i
be Hausdorff, the set U \{z } is open in Mˆ for all i. Moreover, the condition
i i
that the U are disjoint may be easily satisfied since, if i6=j, z and z may be
i i j
separated by two disjoint open sets and, by hypothesis, there are only a finite
number of singularities.
4
Remark If M is a CS manifold, Mˆ is a noncompact manifold.
Definition 3.2 LetM be aCSmanifold. Usingthe notationofDefinition 3.1,
a radius function on Mˆ is a smooth function ρ:Mˆ →(0,1], bounded below by
a positive constant on Mˆ \Mˆ , such that there exist positive constants c <1
∞ 1
and c >1 with
2
c r <Ψ∗(ρ)<c r
1 i i 2 i
on (0,ǫ)×Σ for i=1,...,s.
i
IfM isCSwemayconstructaradiusfunctiononMˆ asfollows. Letρ(x)=1
forallx∈Mˆ \Mˆ . Define ρ :Ψ ((0,ǫ/2)×Σ )→(0,1)tobe equalto r /ǫfor
∞ i i i i
i = 1,...,s and then define ρ by interpolating smoothly between its definition
on Mˆ \Mˆ and ρ on each of the disjoint sets Ψ ((ǫ/2,ǫ)×Σ ).
∞ i i i
3.2 CS coassociative 4-folds
LetB(0;η)denotethe openballabout0inR7 withradiusη >0,i.e. B(0;η)=
{v ∈ R7 : |v| < η}. We define a preferred choice of local coordinates on a G
2
manifold near a finite set of points.
Definition 3.3 Let (M,ϕ,g) be a G manifold as in Definition 2.6 and let
2
z ,...,z be points in M. There exist a constant η > 0, an open set V ∋ z
1 s i i
in M with V ∩V = ∅ for j 6= i and a diffeomorphism χ : B(0;η) ⊆ R7 → V
i j i i
with χ (0) = z , for i = 1,...,s, such that ζ = dχ | : R7 → T M is an
i i i i 0 zi
isomorphismidentifying the standardG structure (ϕ ,g ) onR7 with the pair
2 0 0
(ϕ| ,g| ). We call the set {χ : B(0;η) → V : i = 1,...,s} a G
TziM TziM i i 2
coordinate system near z ,...,z .
1 s
We say that two G coordinate systems near z ,...,z , with maps χ and
2 1 s i
χ˜ for i=1,...,s respectively, are equivalent if dχ˜ | =dχ | =ζ for all i.
i i 0 i 0 i
The definition above is an analogue of the local coordinate system for almost
Calabi–Yau manifolds used by Joyce [6, Definition 3.6]. Although the family
of G coordinate systems near z ,...,z is clearly infinite-dimensional, there
2 1 s
are only finitely many equivalence classes, given by the number of possible sets
{ζ ,...,ζ }. Moreover,the family of choices for each ζ is isomorphic to G .
1 s i 2
Note Definition 3.3 does notrequire the G structure (ϕ,g) to be torsion-free.
2
Definition 3.4 Let (M,ϕ,g) be a G manifold, let N ⊆ M be compact and
2
connected and let z ,...,z ∈N. We say that N is a 4-fold in M with conical
1 s
singularitiesatz ,...,z withrateλ,denotedaCS4-fold,ifNˆ =N\{z ,...,z }
1 s 1 s
5
is a (nonsingular) 4-dimensional submanifold of M and there exist constants
0<ǫ<η andλ>1,acompact3-dimensionalRiemanniansubmanifold(Σ ,h )
i i
ofS6 ⊆R7,whereh is therestrictionofthe roundmetriconS6 toΣ ,anopen
i i
setU ∋z inN withU ⊆V andasmoothmapΦ :(0,ǫ)×Σ →B(0;η)⊆R7,
i i i i i i
fori=1,...,s,suchthatΨ =χ ◦Φ :(0,ǫ)×Σ →U \{z }isadiffeomorphism
i i i i i i
and Φ satisfies
i
|∇j(Φ (r ,σ )−ι (r ,σ ))|=O(rλ−j) for j ∈N as r →0, (4)
i i i i i i i i i
where ι (r ,σ ) = r σ ∈ B(0;η), ∇ is the Levi–Civita connection of the cone
i i i i i i
metric g =dr2+r2h on C =(0,∞)×Σ coupled with partial differentiation
i i i i i i
on R7, |.| is calculated with respect to g and {χ : B(0;η)→ V : i = 1,...,s}
i i i
is a G coordinate system near z ,...,z .
2 1 s
We callC the cone atthe singularityz andΣ the link of the cone C . We
i i i i
may write N as the disjoint union
s
N =K⊔ U ,
i
i=1
G
where K is compact.
If Nˆ is coassociative in M, we say that N is a CS coassociative 4-fold.
Suppose N is a CS 4-fold at z ,...,z with rate λ in (M,ϕ,g) and use the
1 s
notation of Definition 3.4. The induced metric on Nˆ, g| , makes Nˆ into a
Nˆ
Riemannian manifold. Moreover, it is clear from (4) that the maps Ψ satisfy
i
(3) in Definition 3.1 with the same constant λ. Thus, N may be considered as
a CS manifold with rate λ.
It is important to note that, if λ ∈ (1,2), Definition 3.4 is independent of
the choice of G coordinate system near the singularities, up to equivalence.
2
Suppose we have two equivalent coordinate systems defined using maps χ and
i
χ˜ . These maps must agree up to second order since the zero and first order
i
behaviour of each is prescribed, as stated in Definition 3.3. Therefore, the
transformedmapsΦ˜ correspondingtoχ˜ suchthatΨ˜ =χ˜ ◦Φ˜ =χ ◦Φ =Ψ
i i i i i i i i
are defined by:
Φ˜ =(χ˜−1◦χ )◦Φ .
i i i i
Hence
|∇j(Φ˜ (r ,σ )−Φ (r ,σ ))|=O(r2−j) for j ∈N as r →0,
i i i i i i i i i
where∇ and|.|arecalculatedasinDefinition3.4. Thus,inorderthattheterms
i
generatedby the transformationof the G coordinate system neither dominate
2
nor be of equal magnitude to the O(rλ−j) terms given in (4), we need λ<2.
i
6
We now make a definition whichalso depends only on equivalence classesof
G coordinate systems near the singularities.
2
Definition 3.5 Let N be a CS 4-fold at z ,...,z in a G manifold (M,ϕ,g).
1 s 2
Use the notation of Definitions 3.3 and 3.4. For i=1,...,s define a cone Cˆ in
i
T M by Cˆ =(ζ ◦ι )(C ). We call Cˆ the tangent cone at z .
zi i i i i i i
One can show that Cˆ is a tangent cone to N at z in the sense of geometric
i i
measure theory (see, for example, [4, p. 233]). We also have a straightforward
resultrelatingtothetangentconesatsingularpointsofCScoassociative4-folds.
Proposition 3.6 Let N be a CS coassociative 4-fold at z ,...,z in a G man-
1 s 2
ifold (M,ϕ,g). The tangent cones at z ,...,z are coassociative.
1 s
Proof: Use the notation of Definitions 3.3 and 3.4.
It is enough to show that ι (C ) is coassociative in R7 for all i, since ζ :
i i i
R7 →T M is an isomorphism identifying (ϕ ,g ) with (ϕ| ,g| ). This
zi 0 0 TziM TziM
is equivalent to the condition ι∗(ϕ )≡0 for i=1,...,s.
i 0
Note that ϕ| ≡ 0 implies that, for all i, ϕ| ≡ 0. Hence, Ψ∗(ϕ) =
Nˆ Ui\{zi} i
Φ∗(χ∗(ϕ)) vanishes on C for all i. Using (4),
i i i
|Φ∗(χ∗(ϕ))−ι∗(χ∗(ϕ))|=O(rλ−1) as r →0
i i i i i i
for all i. Moreover,
|ι∗(χ∗(ϕ))−ι∗(ϕ )|=O(r ) as r →0
i i i 0 i i
since
χ∗(ϕ)=ϕ +O(r ) and |∇ι |=O(1) as r →0.
i 0 i i i
Therefore, because λ>1,
|ι∗(ϕ )|→0 as r →0
i 0 i
for all i. As T ι (C )=T ι (C ) for all (r ,σ )∈C , |ι∗(ϕ )| is independent
riσi i i σi i i i i i i 0
of r and thus vanishes for all i as required. (cid:3)
i
4 Weighted Banach spaces
For this section let M be an n-dimensional CS manifold and let Mˆ be its non-
singular part as in Definition 3.1. We define weighted Banach spaces of forms
as in [1, §1], as well as the usual ‘unweighted’ spaces.
7
Definition 4.1 Let p ≥ 1 and let k,m ∈ N with m ≤ n. The Sobolev space
Lp(ΛmT∗Mˆ) is the set of m-forms ξ on Mˆ which are k times weakly differen-
k
tiable and such that the norm
1
k p
kξkLpk =j=0ZMˆ |∇jξ|pdVg (5)
X
 
is finite. The normed vector space Lp(ΛmT∗Mˆ) is a Banachspace for all p≥1
k
and L2(ΛmT∗Mˆ) is a Hilbert space.
k
We introduce the space of m-forms
Lp (ΛmT∗Mˆ)={ξ : fξ ∈Lp(ΛmT∗Mˆ)for allf ∈C∞(Mˆ)}
k,loc k cs
where C∞(Mˆ) is the space of smooth functions on Mˆ with compact support.
cs
Let µ∈R and let ρ be a radius function on Mˆ. The weighted Sobolev space
Lp (ΛmT∗Mˆ)ofm-formsξ onMˆ isthesubspaceofLp (ΛmT∗Mˆ)suchthat
k,µ k,loc
the norm
1
k p
kξkLpk,µ =j=0ZMˆ |ρj−µ∇jξ|pρ−ndVg (6)
X
 
isfinite. ThenLp (ΛmT∗Mˆ)isaBanachspaceandL2 (ΛmT∗Mˆ)isaHilbert
k,µ k,µ
space.
We may note here, trivially, that Lp(ΛmT∗Mˆ) is equal to the standard Lp-
0
space of m-forms on Mˆ. Further, by comparing equations (5) and (6) for the
respective norms, Lp(ΛmT∗Mˆ)=Lp (ΛmT∗Mˆ). In particular,
0,−n
p
L2(ΛmT∗Mˆ)=L2 (ΛmT∗Mˆ). (7)
0,−n
2
For the following two definitions we take Ck (ΛmT∗Mˆ) to be the vector
loc
space of k times continuously differentiable m-forms.
Definition 4.2 Let ρ be a radius function on Mˆ, let µ ∈ R and let k,m ∈ N
with m ≤ n. The weighted Ck-space Ck(ΛmT∗Mˆ) of m-forms ξ on Mˆ is the
µ
subspace of Ck (ΛmT∗Mˆ) such that the norm
loc
k
kξk = sup|ρj−µ∇jξ|
Ck
µ
j=0 Mˆ
X
is finite. We also define
C∞(ΛmT∗Mˆ)= Ck(ΛmT∗Mˆ).
µ µ
k≥0
\
8
Then Ck(ΛmT∗Mˆ) is a Banach space but in general C∞(ΛmT∗Mˆ) is not.
µ µ
InthenextdefinitionwerefertotheusualnormedvectorspaceCk(ΛmT∗Mˆ)
of k times continuously differentiable m-forms such that the following norm is
finite:
k
kξk = sup|∇jξ|.
Ck
j=0 Mˆ
X
Definition 4.3 Let d(x,y) be the geodesic distance between points x,y ∈ Mˆ
and let ρ be a radius function on Mˆ. Let a ∈ (0,1) and let k,m ∈ N with
m≤n. Let
H ={(x,y)∈Mˆ ×Mˆ : x6=y, c ρ(x)≤ρ(y)≤c ρ(x) and
1 2
there exists a geodesic in Mˆ of length d(x,y) from x to y},
where 0 < c < 1 < c are constant. A section s of a vector bundle V on Mˆ,
1 2
endowed with a connection, is Ho¨lder continuous (with exponent a) if
|s(x)−s(y)|
[s]a = sup V <∞.
d(x,y)a
(x,y)∈H
We understand the quantity |s(x)−s(y)| as follows. Given (x,y) ∈ H, there
V
exists a geodesic γ of length d(x,y) connecting x and y. Parallel translation
along γ using the connection on V identifies the fibres over x and y and the
metrics on them. Thus, with this identification, |s(x)−s(y)| is well-defined.
V
The Ho¨lder space Ck,a(ΛmT∗Mˆ) is the set of ξ ∈ Ck(ΛmT∗Mˆ) such that
∇kξ is Ho¨lder continuous (with exponent a) and the norm
kξk =kξk +[∇kξ]a
Ck,a Ck
is finite. The normed vector space Ck,a(ΛmT∗Mˆ) is a Banach space.
We also introduce the notation
Ck,a(ΛmT∗Mˆ)
loc
={ξ ∈Ck (ΛmT∗Mˆ):fξ ∈Ck,a(ΛmT∗Mˆ)for allf ∈C∞(Mˆ)}.
loc cs
Let µ ∈ R. The weighted Ho¨lder space Ck,a(ΛmT∗Mˆ) of m-forms ξ on Mˆ
µ
is the subspace of Ck,a(ΛmT∗Mˆ) such that the norm
loc
kξk =kξk +[ξ]k,a
Cµk,a Cµk µ
is finite, where
[ξ]k,a =[ρk+a−µ∇kξ]a.
µ
9
Then Ck,a(ΛmT∗Mˆ) is a Banach space. It is clear that we have an embedding
µ
Ck,a(ΛmT∗Mˆ)֒→Cl(ΛmT∗Mˆ) whenever l≤k.
µ µ
WeshallneedtheanalogueoftheSobolevEmbeddingTheoremforweighted
spaces, which is adapted from [14, Lemma 7.2] and [1, Theorem 1.2].
Theorem 4.4 (Weighted Sobolev Embedding Theorem) Let p, q ≥1,
a∈(0,1), µ,ν ∈R and k,l,m∈N with m≤n.
(a) If k ≥l, k− n ≥l− n and either
p q
(i) p≤q and µ≥ν or
(ii) p>q and µ>ν,
there is a continuous embedding Lp (ΛmT∗Mˆ)֒→Lq (ΛmT∗Mˆ).
k,µ l,ν
(b) If k − n ≥ l + a, there is a continuous embedding Lp (ΛmT∗Mˆ) ֒→
p k,µ
Cl,a(ΛmT∗Mˆ).
µ
WeshallalsorequireanImplicitFunctionTheoremforBanachspaces,which
follows immediately from [12, Chapter 6, Theorem 2.1].
Theorem 4.5 (Implicit Function Theorem)LetX andY beBanachspaces
and let W ⊆ X be an open neighbourhood of 0. Let G : W → Y be a Ck map
(k ≥ 1) such that G(0) = 0. Suppose further that dG| : X → Y is surjective
0
with kernel K such that X = K ⊕A for some closed subspace A of X. There
exist open sets V ⊆ K and V′ ⊆ A, both containing 0, with V ×V′ ⊆ W, and
a unique Ck map V :V →V′ such that
KerG∩(V ×V′)={(x,V(x)) : x∈V}
in X =K⊕A.
5 The operator d + d∗
In this section we let M be a 4-dimensional CS manifold and let Mˆ be as in
Definition 3.1. An essential part of our study is the use of the Fredholm and
index theory for the elliptic operator d+d∗ acting from Λ2T∗Mˆ ⊕Λ4T∗Mˆ to
+
Λ3T∗Mˆ. We therefore consider
d+d∗ :Lp (Λ2T∗Mˆ ⊕Λ4T∗Mˆ)→Lp (Λ3T∗Mˆ), (8)
k+1,µ + k,µ−1
where p≥2, k∈N and µ∈R.
10