I want to know whether ot not Proposition 1.12 (2) in p. 120 of the book [KN] is typo. 


[KN] : Foundations of differential geometry Volume II - Kobayashi and Nomizu
 1969 Interscience Publishing 


Let $\phi$ be a two form associated to Hermitian inner product $h$ such that 
 $\phi(X,Y) = h(X,JY)$ where $J$ is an almost complex structure. 


Then Proposition 1.12 (2) says that 
 $\phi = -2i \sum\_{j,k=1}^n h\_{j\overline{k}} \xi^j \wedge \overline{\xi}^k$ 


I can not understand why 2 is needed in the above equation. 


In the proof,  

 $\phi(Z,W)=-i\sum\_{j,k=1}^n h\_{j\overline{k}}
 (\xi^j(Z)\overline{\xi}^k(W)-\xi^j(W)\overline{\xi}^k (Z))$.
 This equation can be understood. In fact, in (5) of p. 155 the fundamental two form also 
 has 2 (i.e. $\Phi = -2i \sum\_{i, j} g\_{ij} dz^i \wedge d\overline{z}^j$). Summarizingly, do Proposition 1.12 (2) in p. 120 and (5) of p. 155 have typos ?  

 Or, please give me an explanation. 



Algebraic preliminaries of complex manifold 
Some authors define exterior products as quotient spaces of tensor products, letting $\alpha\wedge \beta$ be the coset represented by $\alpha\otimes\beta$. Others define exterior products as subspaces of tensor products, letting $\alpha\wedge \beta$ be $\alpha\otimes\beta-\beta\otimes\alpha$. Or one half of that. You can see that this can lead to confusion. This has been discussed elsewhere at MO.


I think you just have to make sure that you know what the convention is in that book.


