1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

66 2. KAHLER-RICCI FLOW

to be

K (Z W)::::: Rm (X, JX, JU, U) = Rmic (Z, Z, W, W)

ic ' · 1x1^2 1u1^2 1z1^2 1w1^2

Clearly we have Kie (Z, Z) =Kc (Z). We say that the bisectional curvature

is positive (respectively, nonnegative) if Kie (Z, W) > 0 (respectively,

Kc (Z, W) 2:: 0) for all Z, WE T^1 ,o M ~ { o}. Clearly positive (respectively,

nonnegative) bisectional sectional curvature implies positive (respectively,

nonnegative) holomorphic sectional curvature. We also have the following

(see Proposition 1 on p. 32 of [270] for example).

LEMMA 2.16. Po.sitive (respectively, nonnegative) Riemannian sectional

curvature implies positive (respectively, nonnegative) bisectional sectional

curvature.

Similarly to the notation for the components of the complex Riemann

curvature tensor, we define

\1 a.Rf3'?o'f/ ~ ( ('\7 ic) a~o: Rm IC) ( 8~f3' 8~'Y' 3~0' 8~'f/).

The second Bianchi identity says that

(2.9)

since

(\7 ~ a Rmic)(~ 8#'~'~'~ ~~~)+('Va ~ Rmic)(~ 8~'8#'~'~ ~ ~ ~)

+ ( \7 ~ Rm ic) ( 8~-r' 3~a.' 8~5' 8~'f/) =^0

(i.e., \7 a.Rf3'?Mi + \7f3R;ya.or;+\7 1 Ra.f3or; = 0, where \7 1 Ra.f3or; = 0). Taking the
trace, we have

(2.10)

Note that \7 a.Rf3'? = o~"' Rf3'? - r~f3R 01.

The volume form is dμg ~ ~! wn, where wn ~ w A · · · A w ( n times) and

n =dime J\11, which in local coordinates is

wn = n! (A) n det(9a.,B)dz^1 A dz^1 A · · · A dzn A dzn.

The volume is

(2.11) Vol (M) = Volg (M) = ~ f wn.

n.JM

The total scalar curvature may be written as

(2.12) { Rdμ--.:. ( 1 )1 { p/\wn-1.

JM n-l .JM