# Sergei Yakovenko's blog: on Math and Teaching

## Calculus on complex manifolds

If $V$ is a complex vector space, then it is naturally also a real vector space (if you allow multiplication by complex numbers, then that by real numbers is automatically allowed). However, forgetting how to multiply by the imaginary unit results in the fact that the dimension $\dim_{\mathbb R}V$ of the space over the real numbers is two times higher. If we regret our decision to forget the complex multiplication, we still can restore it by introducing the $\mathbb R$-linear operator $J\colon V\to V$ such that $J^2=-E$, where $E$ is the identity operator.

An even-dimensional real vector space with such an operator is called an almost complex space, and it obviously can be made into a complex vector space (over $\mathbb C$). However, if we consider an even-dimensional manifold $M$ with the family of operators as above, it is somewhat less than a complex analytic manifold (a topological space equipped with an atlas of charts with biholomophic transition functions). For details, follow the lecture notes that will be available later.