- INTRODUCTION TO KAHLER MANIFOLDS 59
A covariant tensor of type (p, q) is a section of @p,q M ~ ( tg)P A l,O M) Q9
( @q A O,l M).^3 The complex conjugate extends to the tensor bundles and
Q9p,qM = Q9q,pM.
We say that a differential (p + q)-form 'r/ is of type (p, q) if it is a section
of the vector bundle AP,qM ~ (APTl,O M) A (AqTO,l M) c AP+qTcM. We
denote the space of (p, q)-forms by n,p,q (M). We say that a (p, p )-tensor (or
form) rJ is real if fj = rJ.
When we are considering complex manifolds, it is most natural to carry
out calculations in holomorphic coordinates. Let {za} be local holomorphic
coordinates. We may write za ~ xa + .J=Iya, where xa and ya are real-
valued functions. Defineandso that
dza ~ dxa + Hdya,
dza ~ dxa - Hdya
{)~a ~ t ( 0 ~a - H 0 ~a) '
{)~a ~ t ({)~a + H 0 ~a) '
dz~ ( o~f3) = 5$, d;P ( o~f3) = 5$,
dza (a~f3) = dza (a~f3) = o.
Note that 8 ~0!. = 8 ~0!. and a~f3 = 8 ~f3 • The holomorphic tangent bundle
T^1 ,o M is locally the span of the vectors { 8 ~0!.} :=l. Given two overlapping
local holomorphic coordinates { za} and { wf3} , the transition matrix relating
aza. a an d Bwf3 a lS. { Bwf3 aza. }n a,{3=l , w h" lC h sa t" is fi es
8 n oza 8
8wf3 = L 8wf3. {)za·
a=lThe anti-holomorphic tangent bundle TO,l Mis locally the span of { a~f3} ;=l.
NOTATION 2.2. Henceforth we shall use the Einstein summation con-
vention where each pair of repeated indices consisting of an upper index and
a lower index is summed from 1 to n (sometimes, as a reminder, we include
the summation symbol, however we always sum over repeated indices unless
otherwise indicated).(^3) Not to be confused with a (p, q)-tensor in Riemannian geometry which is a section
of (@PT*M) ® (@qTM).