1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

58 2. KAHLER-RICCI FLOW

Given an almost complex manifold (M, J), the complexified tangent

bundle is TcM ~ TM ®JR C. For each p E M we may extend the almost

complex structure J to a complex linear map

Jc : TcMp ---+ TcMp·

Since (Jc)^2 = -idTcMp, the eigenvalues of Jc are A and -.;=I. We

define the holomorphic tangent bundle by

T^1 ,^0 M ~{VE TcM: Jc (V) =Av}

and the anti-holomorphic tangent bundle by

T^0 ,^1 M ~{VE TcM: Jc(V) =-RV}.

This gives us a decomposition

TcM = T^1 ,o M EB To,l M.

A vector in T^1 ,o Mis called type (1, 0) and a vector in TO,l M is called type

(0, 1). Alternatively, we can obtain the above decomposition by decomposing

a (real) tangent vector V E TM as V = V^1 ,o + vo,l, where

I.e.,

V^1 '^0 ~ ~ (v - HJV) E T^1 '^0 M,

1

vo,l ~ 2 (v + HJV) E TO,l M.

T^1 ,oM = {V^1 ,o: VE TM},

T^0 ,^1 M = {VO,l: v E TM}.

Since TcM is the complexification of a real vector space, the complex

conjugate (bar operation), X + .j=IY ~ X - .;=IY, where X, YE TM,

is defined on TcM. Given Z E TcM, the real part of Z is defined by

Re (Z) ~ ~ (Z + Z).

Note that if VET M, then

2 Re (V^1 '^0 ) = 2 Re (v^0 ,^1 ) = V.

The almost complex structure satisfies

Jc (V) =Jc (V), VE TcM,

and we have T^1 ,^0 M = T^0 ,^1 M and TO,lM = T^1 ,^0 M.
The complexified cotangent bundle is TC,M = (TcM) = T M ®JR
C, which decomposes into

TC,M = A^1 ,oM EBA^0 ,^1 M,

where A^1 ,^0 M ~ (T^1 ,^0 M) and A^0 ,1M ~ (TO,lM).