# Elementary Question on Sections of $\Lambda^2(X \times Y)$

Let $$X$$ and $$Y$$ be complex manifolds with local coordinates $$(x_1, ..., x_n)$$ and $$(y_1, ..., y_m)$$, respectively. The exterior algebra $$\Lambda^2(X \times Y)$$ decomposes as $$\Lambda^2(X \times Y) = \Lambda^2(X) \oplus (\Lambda^1X \otimes \Lambda^1Y) \oplus \Lambda^2(Y).$$ Therfore, a smooth $$2$$-form on $$X \times Y$$ may written as $$\begin{eqnarray*} \omega &=& \sum_{i,j=1}^n \varphi_{ij} dx_i \wedge dx_j + \eta + \sum_{k,\ell =1}^m \psi_{k\ell} dy_k \wedge dy_{\ell}, \end{eqnarray*}$$

where $$\eta \in \Omega^1(X) \otimes \Omega^1(Y)$$. I'm not sure how to write $$\eta$$ in local coordinates. Any help is appreciated.

## 1 Answer

This is easy enough: $$\eta = \sum\limits_{i,k} f_{ik}(x,y) dx_i\wedge dy_k$$. (Be careful, by the way. What you're writing is valid for smooth, real manifolds. If you're doing complex manifolds and $$x_i$$ and $$y_k$$ are local holomorphic coordinates, then you need not only $$dx_i$$ but also $$d\bar x_i$$ and a general $$2$$-form on a complex manifold is a linear combination of forms of type $$(2,0)$$, $$(1,1)$$, and $$(0,2)$$.)

• Yes, of course! Thank you very much for that! – AmorFati Jun 24 at 4:17