Compute the pushforward $F_*(\omega)$ of a 1-holomorphic differential form $\omega$ through $F:U\subset \mathbb C^3\to\mathbb C^3$

Question

Let $\omega = (x-y)dx + ({z}^{2} - x)dy + xydz$ be a 1-holomorphic form in a open subset $U \subset \mathbb{C}^{3}$ and $F : U \subset \mathbb{C}^{3} \longrightarrow \mathbb{C}^{3}$ defined by : $F(x,y,z) = (x^{2}, y-z, z^{2} + x)$.

What is the pushforward $F_{*}(\omega)$?

Can someone help me? Thank you.

Answer 1

Given two differentiable manifolds $M$ and $N$ and a diffeomorphism $F:M\to N$, you can define, for a given $\omega \in \Omega^1(M)$, $$F_{\star}(\omega):=(F^{-1})^{\star}(\omega)\in \Omega^1(N).$$ So, if you assume $u,v,w$ as coordinates system for $F(U)$ you can compute the pull back of $\omega$ through $G:=F^{-1}$ doing $$F_{\star}(\omega)=G^{\star}(\omega)=(\omega_1\circ G)d(x\circ G)+(\omega_2\circ G)d(y\circ G)+(\omega_3\circ G)d(z\circ G),$$ where $\omega_j(x,y,z)\in \mathcal C^{\infty}(U)$ are the coefficients of the $1$-form and the composition of the coordinate functions of $U$ with $G$ gives the components of $G$.

Answer 2

Question: "Let $ω=(x−y)dx+(z^2−x)dy+xydz$ be a 1-holomorphic form in a open subset $U⊂C^3$ and $F:U⊂C^3⟶C^3$ defined by : $F(x,y,z)=(x^2,y−z,z^2+x)$. What is the pushforward F∗(ω) ?"

Answer: If you view the map as a map of algebraic varieties (it is a polynomial map) consider the map $\phi: A:=\mathbb{C}[x,y,z] \rightarrow \mathbb{C}[x,y,z]$ defined by $\phi(x)=x^2, \phi(y):=y-z, \phi(z):=z^2+x$. You get an induced map at modules of differentials

$$d\phi: \Omega^1_{A} \rightarrow \Omega^1_A$$

defined by $d\phi(d(a)):=d(\phi(a))$. The form $\omega$ is defined for all $(x,y,z)$.