Page 170 - Curvature and Homology

152 V. COMPLEX MANIFOLDS

Applying J to both sides of this relation gives

Hence, the eigenvalues are I/ - 1 and - <-7, and so since J is a
real operator, that is JY= Jc, the eigenvectors of - -1 are the
conjugates of those of 1/?. The vector space V must therefore be
even dimensional, that is m = 2n. The eigenvectors of 1/7 form a
vector space of complex dimension n which we denote by F0 and those
corresponding to - I/? form the vector space VOgl = V1sO; moreover,

that is Vc = Po @ VOJ (direct sum). Thus, the tensor J defines a
complex structure on V.
An element of the eigenspace Vpo will be called a vector of bidegree
(or type) (1,O) and an element of VOJ a vector of bidegree (or type) (0,l).
A complex structure may be defined on the dual space of V in the
obvious manner. The tensor product

may then be decomposed into a direct sum of tensor products of vector
spaces each identical with one of the spaces V1sO, VJ, V*lpO and V*Osl.
A term in this decomposition is said to be a pure tensor space and an
element of such a space is called a tensor of type ($2) if Po occurs
q, times, 'VOtl - r, times, V*lsO - q, times and V*OJ - r, times.
A skew-symmetric tensor or, equivalently, an element of the Grassman
algebra over Vc (or (VC)*) is a sum of pure forms each of which is said
to be of bidegree (q,, r,) (or (q,, r,)). For example,

that is, an element of the tensor space VC @ Vc is a sum of tensors
of types (i g), (: i) and (g 9. We denote by //gpr the space of forms of
bidegree (q, r).
In the sequel, we shall employ the following systems of indices unless
otherwise indicated: upper case Latin letters A, B, ..- run from 1, ..., 2n
and lower case Latin letters i, j, run from 1, .-., n; moreover, i* = i + n
and (i + n)* = i.
Let {e,, ..a, e,) be a basis of V1pO. Denote the conjugate vectors Zd
by ed,, i = 1, ..., n. Apparently, they form a basis of VOJ, and since
VC = PC @ Pa1, the 2n vectors {ed, e,,} form a basis of VC. Such a