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The aim of this paper is to prove a uniform Fourier restriction estimate forcertain $2-$dimensional surfaces in $\mathbb R^{2n}$. These surfaces are theimage of complex polynomial curves $\gamma(z) = (p_1(z), \dots, p_n(z))$,equipped with the complex equivalent to the affine arclength measure. Thisresult 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 analternative proof of a geometric inequality by Dendrinos and Wright [DW10] thatextends the result to complex polynomials.

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