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822 MARCO ABATE, FILIPPO BRACCI, AND FRANCESCA TOVENA
Finally, in Section 8 we shall present a couple of applications of our results
to the discrete dynamics of holomorphic self-maps of complex surfaces, thus
closing the circle and coming back to the arguments that originally inspired
our work.
1. The order of contact
Let M be an n-dimensional complex manifold, and S ⊂ M an irreducible
subvariety of codimension m. We shall denote by O
M
the sheaf of holomorphic
functions on M, and by I
S
the subsheaf of functions vanishing on S. With a
slight abuse of notations, we shall use the same symbol to denote both a germ
at p and any representative defined in a neighborhood of p. We shall denote
by TM the holomorphic tangent bundle of M, and by T
M
the sheaf of germs
of local holomorphic sections of TM. Finally, we shall denote by End(M,S)
the set of (germs about S of) holomorphic self-maps of M fixing S pointwise.
Let f ∈ End(M,S) be given, f ≡ id
M
, and take p ∈ S. For every h ∈O
M,p
the germ h ◦ f is well-defined, and we have h ◦ f − h ∈I
S,p
.
Definition 1.1. The f-order of vanishing at p of h ∈O
M,p
is given by
ν
f
(h; p) = max{µ ∈ N | h ◦ f − h ∈I
µ
S,p
},
and the order of contact ν
f
(p) of f at p with S by
ν
f
(p) = min{ν
f
(h; p) | h ∈O
M,p
}.
We shall momentarily prove that ν
f
(p) does not depend on p.
Let (z
1
, ,z
n
) be local coordinates in a neighborhood of p.Ifh is any
holomorphic function defined in a neighborhood of p, the definition of order of
contact yields the important relation
(1.1) h ◦ f − h =
n

j=1
(f
j
− z
j
)
∂h
∂z
j
(mod I
2ν
f
(p)
S,p
),
where f
j
= z
j
◦ f.
As a consequence we have
Lemma 1.1. (i) Let (z
1
, ,z
n
) be any set of local coordinates at p ∈ S.
Then
ν
f
(p) = min
j=1, ,n
{ν
f
(z
j
; p)}.
(ii) For any h ∈O
M,p
the function p → ν
f
(h; p) is constant in a neighborhood
of p.
(iii) The function p → ν
f
(p) is constant.
INDEX THEOREMS FOR HOLOMORPHIC SELF-MAPS
823
Proof. (i) Clearly, ν
f
(p) ≤ min
j=1, ,n
{ν
f
(z
j
; p)}. The opposite inequality
follows from (1.1).
(ii) Let h ∈O
M,p
, and choose a set {
1
, ,
k
} of generators of I
S,p
. Then
we can write
(1.2) h ◦ f − h =

|I|=ν
f
(h;p)

I
g
I
,
where I =(i
1
, ,i
k
) ∈ N
k
is a k-multi-index, |I| = i
1
+ ··· + i
k
, 
I
=
(
1
)
i
1
···(
k
)
i
k
and g
I
∈O
M,p
. Furthermore, there is a multi-index I
0
such
that g
I
0
/∈I
S,p
. By the coherence of the sheaf of ideals of S, the relation (1.2)
holds for the corresponding germs at all points q ∈ S in a neighborhood of p.
Furthermore, g
I
0
/∈I
S,p
means that g
I
0
|
S
≡ 0 in a neighborhood of p, and
thus g
I
0
/∈I
S,q
for all q ∈ S close enough to p. Putting these two observations
together we get the assertion.
(iii) By (i) and (ii) we see that the function p → ν
f
(p) is locally constant
and since S is connected, it is constant everywhere.
We shall then denote by ν
f
the order of contact of f with S, computed at
any point p ∈ S.
As we shall see, it is important to compare the order of contact of f with
the f-order of vanishing of germs in I
S,p
.
Definition 1.2. We say that f is tangential at p if
min

ν
f
(h; p) | h ∈I
S,p

>ν
f
.
Lemma 1.2. Let {
1
, ,
k
} be a set of generators of I
S,p
. Then
ν
f
(h; p) ≥ min{ν
f
(
1
; p), ,ν
f
(
k
; p),ν
f
+1}
for all h ∈I
S,p
. In particular, f is tangential at p if and only if
min{ν
f
(
1
; p), ,ν
f
(
k
; p)} >ν
f
.
Proof. Let us write h = g
1

1
+ ···+ g
k

k
for suitable g
1
, ,g
k
∈O
M,p
.
Then
h ◦ f − h =
k

j=1

(g
j
◦ f)(
j
◦ f − 
j
)+(g
j
◦ f − g
j
)
j

,
and the assertion follows.
Corollary 1.3. If f is tangential at one point p ∈ S, then it is tangential
at all points of S.
Proof. The coherence of the sheaf of ideals of S implies that if {
1
, ,
k
}
are generators of I
S,p
then the corresponding germs are generators of I
S,q
for
824 MARCO ABATE, FILIPPO BRACCI, AND FRANCESCA TOVENA
all q ∈ S close enough to p. Then Lemmas 1.1.(ii) and 1.2 imply that both
the set of points where f is tangential and the set of points where f is not
tangential are open; hence the assertion follows because S is connected.
Of course, we shall then say that f is tangential along S if it is tangential
at any point of S.
Example 1.1. Let p be a smooth point of S, and choose local coordinates
z =(z
1
, ,z
n
) defined in a neighborhood U of p, centered at p and such that
S ∩ U = {z
1
= ··· = z
m
=0}. We shall write z

=(z
1
, ,z
m
) and z

=
(z
m+1
, ,z
n
), so that z

yields local coordinates on S. Take f ∈ End(M, S),
f ≡ id
M
; then in local coordinates the map f can be written as (f
1
, ,f
n
)
with
f
j
(z)=z
j
+

h≥1
P
j
h
(z

,z

),
where each P
j
h
is a homogeneous polynomial of degree h in the variables z

,
with coefficients depending holomorphically on z

. Then Lemma 1.1 yields
ν
f
= min{h ≥ 1 |∃1 ≤ j ≤ n : P
j
h
≡ 0}.
Furthermore, {z
1
, ,z
m
} is a set of generators of I
S,p
; therefore by Lemma 1.2
the map f is tangential if and only if
min{h ≥ 1 |∃1 ≤ j ≤ m : P
j
h
≡ 0} > min{h ≥ 1 |∃m +1≤ j ≤ n : P
j
h
≡ 0}.
Remark 1.1. When S is smooth, the differential of f acts linearly on the
normal bundle N
S
of S in M.IfS is a hypersurface, N
S
is a line bundle, and
the action is multiplication by a holomorphic function b;ifS is compact, this
function is a constant. It is easy to check that in local coordinates chosen as in
the previous example the expression of the function b is exactly 1 + P
1
1
(z)/z
1
;
therefore we must have P
1
1
(z)=(b
f
− 1)z
1
for a suitable constant b
f
∈ C.In
particular, if b
f
= 1 then necessarily ν
f
= 1 and f is not tangential along S.
Remark 1.2. The number µ introduced in [BT, (2)] is, by Lemma 1.1, our
order of contact; therefore our notion of tangential is equivalent to the notion
of nondegeneracy defined in [BT] when n = 2 and m = 1. On the other hand,
as already remarked in [BT], a nondegenerate map in the sense defined in [A2]
when n =2,m = 1 and S is smooth is tangential if and only if b
f
= 1 (which
was the case mainly considered in that paper).
Example 1.2. A particularly interesting example (actually, the one inspir-
ing this paper) of map f ∈ End(M,S) is obtained by blowing up a map tangent
to the identity. Let f
o
be a (germ of) holomorphic self-map of C
n
(or of any
complex n-manifold) fixing the origin (or any other point) and tangent to the
INDEX THEOREMS FOR HOLOMORPHIC SELF-MAPS
825
identity, that is, such that d(f
o
)
O
= id. If π : M → C
n
denotes the blow-
up of the origin, let S = π
−1
(O)
∼
=
P
n−1
(C) be the exceptional divisor, and
f ∈ End(M,S) the lifting of f
o
, that is, the unique holomorphic self-map of M
such that f
o
◦ π = π ◦ f (see, e.g., [A1] for details). If
f
j
o
(w)=w
j
+

h≥2
Q
j
h
(w)
is the expansion of f
j
o
in a series of homogeneous polynomials (for j =1, ,n),
then in the canonical coordinates centered in p =[1:0:··· : 0] the map f is
given by
f
j
(z)=











z
1
+

h≥2
Q
1
h
(1,z

)(z
1
)
h
for j =1,
z
j
+

h≥2

Q
j
h
(1,z

) − z
j
Q
1
h
(1,z

)

(z
1
)
h−1
1+

h≥2
Q
1
h
(1,z

)(z
1
)
h−1
for j =2, ,n,
where z

=(z
2
, ,z
n
). Therefore b
f
=1,
ν
f
(z
1
; p) = min{h ≥ 2 | Q
1
h
(1,z

) ≡ 0},
and
ν
f
= min

ν
f
(z
1
; p),
min{h ≥ 1 |∃2 ≤ j ≤ n : Q
j
h+1
(1,z

) − z
j
Q
1
h+1
(1,z

) ≡ 0}

.
Let ν(f
o
) ≥ 2 be the order of f
o
, that is, the minimum h such that Q
j
h
≡ 0
for some 1 ≤ j ≤ n. Clearly, ν
f
(z
1
; p) ≥ ν(f
o
) and ν
f
≥ ν(f
o
) − 1. More
precisely, if there is 2 ≤ j ≤ n such that Q
j
ν(f
o
)
(1,z

) ≡ z
j
Q
1
ν(f
o
)
(1,z

), then
ν
f
= ν(f
o
)−1 and f is tangential. If on the other hand we have Q
j
ν(f
o
)
(1,z

) ≡
z
j
Q
1
ν(f
o
)
(1,z

) for all 2 ≤ j ≤ n, then necessarily Q
1
ν(f
o
)
(1,z

) ≡ 0, ν
f
(z
1
; p)=
ν(f
o
)=ν
f
, and f is not tangential.
Borrowing a term from continuous dynamics, we say that a map f
o
tangent
to the identity at the origin is dicritical if w
h
Q
k
ν(f
o
)
(w) ≡ w
k
Q
h
ν(f
o
)
(w) for all
1 ≤ h, k ≤ n. Then we have proved that:
Proposition 1.4. Let f
o
∈ End(C
n
,O) be a (germ of ) holomorphic self -
map of C
n
tangent to the identity at the origin, and let f ∈ End(M,S) be its
blow-up. Then f is not tangential if and only if f
o
is dicritical. Furthermore,
ν
f
= ν(f
o
) − 1 if f
o
is not dicritical, and ν
f
= ν(f
o
) if f
o
is dicritical.
In particular, most maps obtained with this procedure are tangential.
826 MARCO ABATE, FILIPPO BRACCI, AND FRANCESCA TOVENA
2. Comfortably embedded submanifolds
Up to now S was any complex subvariety of the manifold M. However,
some of the proofs in the following sections do not work in this generality; so
this section is devoted to describe the kind of properties we shall (sometimes)
need on S.
Let E

and E

be two vector bundles on the same manifold S. We recall
(see, e.g., [Ati, §1]) that an extension of E

by E

is an exact sequence of vector
bundles
O−→ E

ι
−→ E
π
−→ E

−→ O.
Two extensions are equivalent if there is an isomorphism of exact sequences
which is the identity on E

and E

.
A splitting of an extension O−→ E

ι
−→ E
π
−→ E

−→ O is a morphism
σ : E

→ E such that π ◦ σ =id
E

. In particular, E = ι(E

) ⊕ σ(E

), and
we shall say that the extension splits. We explicitly remark that an exten-
sion splits if and only if it is equivalent to the trivial extension O → E

→
E

⊕ E

→ E

→ O.
Let S now be a complex submanifold of a complex manifold M. We shall
denote by TM|
S
the restriction to S of the tangent bundle of M, and by
N
S
= TM|
S
/T S the normal bundle of S into M. Furthermore, T
M,S
will be
the sheaf of germs of holomorphic sections of TM|
S
(which is different from
the restriction T
M
|
S
to S of the sheaf of holomorphic sections of TM), and N
S
the sheaf of germs of holomorphic sections of N
S
.
Definition 2.1. Let S be a complex submanifold of codimension m in an
n-dimensional complex manifold M. A chart (U
α
,z
α
)ofM is adapted to S if
either S ∩U
α
= ∅ or S∩U
α
= {z
1
α
= ···= z
m
α
=0}, where z
α
=(z
1
α
, ,z
n
α
). In
particular, {z
1
α
, ,z
m
α
} is a set of generators of I
S,p
for all p ∈ S ∩U
α
. An atlas
U = {(U
α
,z
α
)} of M is adapted to S if all charts in U are. If U = {(U
α
,z
α
)}
is adapted to S we shall denote by
U
S
= {(U

α
,z

α
)} the atlas of S given by
U

α
= U
α
∩ S and z

α
=(z
m+1
α
, ,z
n
α
), where we are clearly considering only
the indices such that U
α
∩ S = ∅.If(U
α
,z
α
) is a chart adapted to S, we shall
denote by ∂
α,r
the projection of ∂/∂z
r
α
|
S∩U
α
in N
S
, and by ω
r
α
the local section
of N
∗
S
induced by dz
r
α
|
S∩U
α
;thus{∂
α,1
, ,∂
α,m
} and {ω
1
α
, ,ω
m
α
} are local
frames for N
S
and N
∗
S
respectively over U
α
∩ S, dual to each other.
From now on, every chart and atlas we consider on M will be adapted
to S.
Remark 2.1. We shall use the Einstein convention on the sum over re-
peated indices. Furthermore, indices like j, h, k will run from 1 to n; indices
like r, s, t, u, v will run from 1 to m; and indices like p, q will run from m +1
to n.
INDEX THEOREMS FOR HOLOMORPHIC SELF-MAPS
827
Definition 2.2. We shall say that S splits into M if the extension O →
TS → TM|
S
→ N
S
→ O splits.
Example 2.1. It is well-known that if S is a rational smooth curve with
negative self-intersection in a surface M, then S splits into M.
Proposition 2.1. Let S be a complex submanifold of codimension m in
an n-dimensional complex manifold M. Then S splits into M if and only if
there is an atlas
ˆ
U = {(
ˆ
U
α
, ˆz
α
)} adapted to S such that
(2.1)
∂ˆz
p
β
∂ˆz
r
α





S
≡ 0,
for all r =1, ,m, p = m +1, ,n and indices α and β.
Proof. It is well known (see, e.g., [Ati, Prop. 2]) that there is a one-to-one
correspondence between equivalence classes of extensions of N
S
by TS and the
cohomology group H
1

S, Hom(N
S
, T
S
)

, and an extension splits if and only if
it corresponds to the zero cohomology class.
The class corresponding to the extension O → TS → TM|
S
→ N
S
→ O
is the class δ(id
N
S
), where δ: H
0

S, Hom(N
S
, N
S
)

→ H
1

S, Hom(N
S
, T
S
)

is
the connecting homomorphism in the long exact sequence of cohomology asso-
ciated to the short exact sequence obtained by applying the functor Hom(N
S
, ·)
to the extension sequence. More precisely, if
U is an atlas adapted to S, we get
a local splitting morphism σ
α
: N
U

α
→ TM|
U

α
by setting σ
α
(∂
r,α
)=∂/∂z
r
α
,
and then the element of H
1

U
S
, Hom(N
S
, T
S
)

associated to the extension is
{σ
β
− σ
α
}.Now,
(σ
β
− σ
α
)(∂
r,α
)=
∂z
s
β
∂z
r
α




S
∂
∂z
s
β
−
∂
∂z
r
α
=
∂z
s
β
∂z
r
α
∂z
p
α
∂z
s
β





S
∂
∂z
p
α
.
So, if (2.1) holds, then S splits into M. Conversely, assume that S splits
into M ; then we can find an atlas
U adapted to S and a 0-cochain {c
α
}∈
H
0
(U
S
, T
S
⊗N
∗
S
) such that
(2.2)
∂z
s
β
∂z
r
α
∂z
p
α
∂z
s
β





S
=(c
β
)
q
s
∂z
s
β
∂z
r
α
∂z
p
α
∂z
q
β





S
− (c
α
)
p
r
on U
α
∩ U
β
∩ S. We claim that the coordinates
(2.3)

ˆz
r
α
= z
r
α
,
ˆz
p
α
= z
p
α
+(c
α
)
p
s
(z

α
)z
s
α
828 MARCO ABATE, FILIPPO BRACCI, AND FRANCESCA TOVENA
satisfy (2.1) when restricted to suitable open sets
ˆ
U
α
⊆ U
α
. Indeed, (2.2) yields
∂ˆz
p
β
∂ˆz
r
α
=
∂ˆz
p
β
∂z
s
α
∂z
s
α
∂ˆz
r
α
+
∂ˆz
p
β
∂z
q
α
∂z
q
α
∂ˆz
r
α
=
∂ˆz
p
β
∂z
r
α
− (c
α
)
q
r
∂ˆz
p
β
∂z
q
α
+ R
1
=
∂z
p
β
∂z
r
α
+(c
β
)
p
s
∂z
s
β
∂z
r
α
− (c
α
)
q
r
∂z
p
β
∂z
q
α
+ R
1
= R
1
,
where R
1
denotes terms vanishing on S, and we are done.
Definition 2.3. Assume that S splits into M . An atlas
U = {(U
α
,z
α
)}
adapted to S and satisfying (2.1) will be called a splitting atlas for S.Itis
easy to see that for any splitting morphism σ : N
S
→ TM|
S
there exists a
splitting atlas
U
such that σ(∂
r,α
)=∂/∂z
r
α
for all r =1, m and indices α;
we shall say that
U is adapted to σ.
Example 2.2. A local holomorphic retraction of M onto S is a holomorphic
retraction ρ: W → S, where W is a neighborhood of S in M. It is clear that the
existence of such a local holomorphic retraction implies that S splits into M.
Example 2.3. Let π : M → S be a rank m holomorphic vector bundle
on S. If we identify S with the zero section of the vector bundle, π becomes
a (global) holomorphic retraction of M on S. The charts given by the trivi-
alization of the bundle clearly give a splitting atlas. Furthermore, if (U
α
,z
α
)
and (U
β
,z
β
) are two such charts, we have z

β
= ϕ
βα
(z

α
) and z

β
= a
βα
(z

α
)z

α
,
where a
βα
is an invertible matrix depending only on z

α
. In particular, we have
∂z
p
β
∂z
r
α
≡ 0 and
∂
2
z
r
β
∂z
s
α
∂z
t
α
≡ 0
for all r, s, t =1, ,m, p = m +1, ,n and indices α and β.
The previous example, compared with (2.1), suggests the following
Definition 2.4. Let S be a codimension m complex submanifold of an
n-dimensional complex manifold M. We say that S is comfortably embedded
in M if S splits into M and there exists a splitting atlas
U = {(U
α
,z
α
)} such
that
(2.4)
∂
2
z
r
β
∂z
s
α
∂z
t
α





S
≡ 0
for all r, s, t =1, ,m and indices α and β.
An atlas satisfying the previous condition is said to be comfortable for S.
Roughly speaking, then, a comfortably embedded submanifold is like a first-
order approximation of the zero section of a vector bundle.
INDEX THEOREMS FOR HOLOMORPHIC SELF-MAPS
829
Let us express condition (2.4) in a different way. If (U
α
,z
α
) and (U
β
,z
β
)
are two charts about p ∈ S adapted to S, we can write
(2.5) z
r
β
=(a
βα
)
r
s
z
s
α
for suitable (a
βα
)
r
s
∈O
M,p
. The germs (a
βα
)
r
s
(unless m = 1) are not uniquely
determined by (2.5); indeed, all the other solutions of (2.5) are of the form
(a
βα
)
r
s
+ e
r
s
, where the e
r
s
’s are holomorphic and satisfy
(2.6) e
r
s
z
s
α
≡ 0.
Differentiating with respect to z
t
α
we get
(2.7) e
r
t
+
∂e
r
s
∂z
t
α
z
s
α
≡ 0;
in particular, e
r
t
|
S
≡ 0, and so the restriction of (a
βα
)
r
s
to S is uniquely de-
termined — and it indeed gives the 1-cocycle of the normal bundle N
S
with
respect to the atlas
U
S
.
Differentiating (2.7) we obtain
(2.8)
∂e
r
t
∂z
s
α
+
∂e
r
s
∂z
t
α
+
∂
2
e
r
u
∂z
s
α
∂z
t
α
z
u
α
≡ 0;
in particular,

∂e
r
t
∂z
s
α
+
∂e
r
s
∂z
t
α





S
≡ 0,
and so the restriction of
∂(a
βα
)
r
t
∂z
s
α
+
∂(a
βα
)
r
s
∂z
t
α
to S is uniquely determined for all r, s, t =1, ,m.
With this notation, we have
∂
2
z
r
β
∂z
s
α
∂z
t
α
=
∂(a
βα
)
r
s
∂z
t
α
+
∂(a
βα
)
r
t
∂z
s
α
+
∂
2
(a
βα
)
r
u
∂z
s
α
∂z
t
α
z
u
α
;
therefore (2.4) is equivalent to requiring
(2.9)

∂(a
βα
)
r
t
∂z
s
α
+
∂(a
βα
)
r
s
∂z
t
α





S
≡ 0
for all r, s, t =1, ,m, and indices α and β.
Example 2.4. It is easy to check that the exceptional divisor S in Exam-
ple 1.2 is comfortably embedded into the blow-up M.
Then the main result of this section is
830 MARCO ABATE, FILIPPO BRACCI, AND FRANCESCA TOVENA
Theorem 2.2. Let S be a codimension m complex submanifold of an
n-dimensional complex manifold M . Assume that S splits into M, and let
U = {(U
α
,z
α
)} be a splitting atlas. Define a 1-cochain {h
βα
} of N
S
⊗N
∗
S
⊗N
∗
S
by setting
h
βα
=
1
2
∂z
r
α
∂z
u
β
∂
2
z
u
β
∂z
s
α
∂z
t
α





S
∂
α,r
⊗ ω
s
α
⊗ ω
t
α
(2.10)
=
1
2
(a
αβ
)
r
u

∂(a
βα
)
u
s
∂z
t
α
+
∂(a
βα
)
u
t
∂z
s
α





S
∂
α,r
⊗ ω
s
α
⊗ ω
t
α
.
Then:
(i) {h
βα
} defines an element [h] ∈ H
1
(S, N
S
⊗N
∗
S
⊗N
∗
S
) independent of
U;
(ii) S is comfortably embedded in M if and only if [h]=0.
Proof. (i) Let us first prove that {h
βα
} is a 1-cocycle with values in
N
S
⊗N
∗
S
⊗N
∗
S
. We know that
(a
αβ
)
r
u
(a
βα
)
u
s
= δ
r
s
+ e
r
s
,
where δ
r
s
is Kronecker’s delta, and the e
r
s
’s satisfy (2.6) . Differentiating we get
∂(a
αβ
)
r
u
∂z
t
α
(a
βα
)
u
s
+(a
αβ
)
r
u
∂(a
βα
)
u
s
∂z
t
α
=
∂e
r
s
∂z
t
α
;
therefore (2.8) yields
(a
βα
)
u
s
∂(a
αβ
)
r
u
∂z
t
α




S
+(a
βα
)
u
t
∂(a
αβ
)
r
u
∂z
s
α




S
= −(a
αβ
)
r
u

∂(a
βα
)
u
s
∂z
t
α
+
∂(a
βα
)
u
t
∂z
s
α





S
.
Hence
h
αβ
=
1
2
(a
βα
)
r
u

∂(a
αβ
)
u
s
∂z
t
β
+
∂(a
αβ
)
u
t
∂z
s
β






S
∂
β,r
⊗ ω
s
β
⊗ ω
t
β
=
1
2
(a
βα
)
r
u
(a
αβ
)
r
1
r
(a
βα
)
s
s
1
(a
βα
)
t
t
1
×

(a
αβ
)
t
2
t
∂(a
αβ
)
u
s
∂z
t
2
α
+(a
αβ
)
s
2
s
∂(a
αβ
)
u
t
∂z
s
2
α





S
∂
α,r
1
⊗ ω
s
1
α
⊗ ω
t
1
α
=
1
2

(a
βα
)
s
s
1
∂(a
αβ
)
r
1
s
∂z
t
1
α
+(a
βα
)
t
t
1
∂(a
αβ
)
r
1
t
∂z
s
1
α





S
∂
α,r
1
⊗ ω
s
1
α
⊗ ω
t
1
α
= −h
βα
,
where in the second equality we used (2.1). Analogously one proves that h
αβ
+
h
βγ
+ h
γα
= 0, and thus {h
βα
} is a 1-cocycle as claimed.
Now we have to prove that the cohomology class [h] is independent of the
atlas
U. Let
ˆ
U = {(
ˆ
U
α
, ˆz
α
)} be another splitting atlas; up to taking a common
INDEX THEOREMS FOR HOLOMORPHIC SELF-MAPS
831
refinement we can assume that U
α
=
ˆ
U
α
for all α. Choose (A
α
)
r
s
∈O(U
α
)so
that ˆz
r
α
=(A
α
)
r
s
z
s
α
; as usual, the restrictions to S of (A
α
)
r
s
and of
∂(A
α
)
r
s
∂z
t
α
+
∂(A
α
)
r
t
∂z
s
α
are uniquely defined. Set, now,
C
α
=
1
2
(A
−1
α
)
r
u

∂(A
α
)
u
s
∂z
t
α
+
∂(A
α
)
u
t
∂z
s
α





S
∂
α,r
⊗ ω
s
α
⊗ ω
t
α
;
then it is not difficult to check that
h
βα
−
ˆ
h
βα
= C
β
− C
α
,
where {
ˆ
h
βα
} is the 1-cocycle built using
ˆ
U, and this means exactly that both
{h
βα
} and {
ˆ
h
βα
} determine the same cohomology class.
(ii) If S is comfortably embedded, using a comfortable atlas we immedi-
ately see that [h] = 0. Conversely, assume that [h] = 0; therefore we can find a
splitting atlas
U and a 0-cochain {c
α
} of N
S
⊗N
∗
S
⊗N
∗
S
such that h
βα
= c
α
−c
β
.
Writing
c
α
=(c
α
)
r
st
∂
α,r
⊗ ω
s
α
⊗ ω
t
α
,
with (c
α
)
r
ts
symmetric in the lower indices, we define ˆz
α
by setting

ˆz
r
α
= z
r
α
+(c
α
)
r
st
(z

α
) z
s
α
z
t
α
for r =1, ,m,
ˆz
p
α
= z
p
α
for p = m +1, ,n,
on a suitable
ˆ
U
α
⊆ U
α
. Then
ˆ
U = {(
ˆ
U
α
, ˆz
α
)} clearly is a splitting atlas; we
claim that it is comfortable too. Indeed, by definition the functions
(ˆa
βα
)
r
s
=[δ
r
u
+(c
β
)
r
uv
(a
βα
)
v
t
z
t
α
](a
βα
)
u
u
1
d
u
1
s
satisfy (2.5) for
ˆ
U
, where the d
u
1
s
’s are such that z
u
1
α
= d
u
1
s
ˆz
s
α
. Hence

∂(ˆa
βα
)
r
s
∂ˆz
t
α
+
∂(ˆa
βα
)
r
t
∂ˆz
s
α





S
=2(c
β
)
r
uv
(a
βα
)
u
s
(a
βα
)
v
t
|
S
+

∂(a
βα
)
r
s
∂z
t
α
+
∂(a
βα
)
r
t
∂z
s
α





S
+(a
βα
)
r
u

∂d
u
s
∂z
t
α
+
∂d
u
t
∂z
s
α





S
.
Now, differentiating
z
u
α
= d
u
v

z
v
α
+(c
α
)
v
rs
z
r
α
z
s
α

we get
δ
u
t
=
∂d
u
v
∂z
t
α

z
v
α
+(c
α
)
v
rs
z
r
α
z
s
α

+ d
u
v

δ
v
t
+2(c
α
)
v
rt
z
r
α

and
0=

∂d
u
s
∂z
t
α
+
∂d
u
t
∂z
s
α





S
+2(c
α
)
u
st
.
Recalling that h
βα
= c
α
− c
β
we then see that
ˆ
U satisfies (2.9), and we are
done.