A geometric lemma for complex polynomial curves with applications in Fourier restriction theory

The aim of this paper is to prove a uniform Fourier restriction estimate for certain $2-$dimensional surfaces in ${\mathbb{R}}^{2n}$. These surfaces are the image of complex polynomial curves $\gamma (z)=(p_1(z),\dots ,p_n(z))$, equipped with the complex equivalent to the affine arclength measure. This result is a complex-polynomial counterpart to a previous result by Stovall [Sto16] in the real setting. As a means to prove this theorem we provide an alternative proof of a geometric inequality by Dendrinos and Wright [DW10] that extends the result to complex polynomials.