1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

56 2. KAHLER-RICCI FLOW

are holomorphic (complex analytic), and hence biholomorphic, whenever

ui n Uj i= 0. Two systems of holomorphic coordinates {Ui, Zi} and {Vj, Wj}

on M are equivalent if whenever Ui n Vj i= 0,

wj o z;-^1 : zi (Ui n Vj)--+ wj (Ui n Vj)

is a biholomorphism. A complex structure on a differentiable manifold

M is an equivalence class of systems of holomorphic coordinates. A com-

plex manifold is simply a differentiable manifold with a complex structure.

A complex manifold has a natural real analytic structure and is orientable

(using holomorphic coordinates { za ~ xa + y'=Iya} :=l , an orientation is

determined by requiring that the frame { 8 ~ 1 , 8 ~1 , •.. , 8 ~n, 8 ~n } be posi-

tively oriented). In this chapter and only in this chapter we shall use Mn

to denote a complex manifold M of complex dimension n (half of the real

dimension), i.e., n ~dime M =! dimIR M.

A real submanifold N of a complex manifold Mn is a complex sub-

manifold if for every p E N there exist holomorphic coordinates { za} ~=l

in a neighborhood U of p such that

N nu= { q Eu: zk+l (q) = ... =Zn (q) = 0}.

Clearly a complex submanifold is itself a complex manifold. In the above

definition, k is the complex dimension of N.

Some elementary examples of complex manifolds are

(1) complex Euclidean space en,

(2) complex projective space epn'

(3) complex submanifold of epm =algebraic manifold= zero set of a

finite number of homogeneous polynomials,

( 4) complex torus en ;r' where r is a lattice.

A complex structure defines at each point p E M a map J : T Mp --+

T Mp by J = d (z-^1 o y'=I oz), where z are holomorphic coordinates de-

fined in a neighborhood of p (this definition is independent of the choice

of z). Since y'=I o v'=I = - idrcn, it is easy to see J^2 = - idT M. In gen-

eral, given an even-dimensional differentiable manifold M, an automorphism

J : TM --+ TM is called an almost complex structure if J^2 = - idT M.

(A manifold with an almost complex structure is called an almost com-

plex manifold.) Hence a complex structure induces an almost complex

structure. We say that an almost complex structure is integrable if there

exists a complex structure which induces the almost complex structure.

Given an almost complex manifold (M, J), the Nijenhuis tensor is

defined by

N1 (X, Y) ~ [JX, JY] - J [JX, Y] - J [X, JY] - [X, Y]

for X, Y E TM. By the Newlander-Nirenberg Theorem, a necessary

and sufficient condition that an almost complex structure be integrable is

that the Nijenhuis tensor vanish.