Zero sets of real polynomials containing complex varieties

Abstract

We give necessary conditions for real algebraic hypersurfaces (possibly with singularities) to contain nontrivial germs of complex hypersurfaces. Moreover, if a real hypersurface $S$ in $\mathbf{C}^2$ is defined by a real polynomial of a sufficiently general form and if $S$ contains a nontrivial analytic disk, then, using the above result, we show that $S$ must contain certain complex lines.