46
J. Zhang and G. Khan
X
+
= X + α
−1 L X ∈ T
+
p M, X
−
= X − α
−1 L X ∈ T
−
p M.
The (1,0) or “plus”-eigenspace T
+
p M and the (0,1) or “minus”-eigenspace T
−
p M
are formed, respectively, by such X
+ and X
− having been supplemented with the
imaginary parts α
−1 L X or −α
−1 L X:
T
+
p M := {X
+
= X + α
−1 L X : X ∈ T p M},
T
−
p M := {X
−
= X − α
−1 L X : X ∈ T p M}.
Note that for any (para-)complexified vector field Z ∈ T
L
M, the vector field
Z
+
= Z + α
−1 L Z is always in T
+
p M and the vector field Z
−
= Z − α
−1 L Z is
always in T
−
p M. This can be easily shown:
L Z
+
= α Z
+
, L Z
−
= −α Z
−
because
L(Z + α
−1 L Z) ≡ α(Z + α
−1 L Z),
L(Z − α
−1 L Z) ≡ −α(Z − α
−1 L Z).
Thus, any vector Z ∈ T
L
p M can be readily decomposed into T
+
p M and T
−
p M. Without loss of generality, T
+
p M is formed by vectors of the form X
+
= X + α
−1 L X,
and T
−
p M of vectors of the form X
−
= X − α
−1 L X, when X exhausts the real
vector field as sections of T p M.
3.3.2 (Para-)holomorphicity of ∇
The splitting of T M ⊗ L by L into direct sum of T
+
p M and T
−
p M subbundles gives
rise to questions of whether/how ∇ respects such splitting.
3.3.2.1 (Para)-Dolbeault Operator ¯
∂
The (para-)Dolbeault operator ¯
∂ on M for a given L is defined as [8]
¯
∂ X Y =
1
4
[X, Y ] − L
−1 [L X, Y ] + L
−1 [X, LY ] − L
2 [L X, LY ]
(3.14)
for any vector fields X and Y in (T M). It can be checked that ¯
∂ X Y is tensorial in
X , such that ¯
∂ f X Y = f
¯
∂ X Y
, and is a derivation in Y , such that the Leibniz rule is
satisfied. It can be easily verified that
¯
∂ L X Y = −L( ¯
∂ X Y ),
¯
∂ X (LY ) = L( ¯
∂ X Y ).
Using definition Eq. (3.7) for N L , we can rewrite the definition (3.14) of ¯
∂ as