## Geometry of Real Hypersurfaces

The literature on the extension of CR functions to one or both sides of a hypersurface is extensive. Therefore the Levi form is the complex Hessian of the defining function, restricted to the space of vectors tangent to the hypersurface. The Levi form is fundamental in studying the geometry of hypersurfaces. By higher differentiation, or by passing to a nonsingular point, one obtains the result that there can be no complex analytic varieties in a strongly pseudoconvex hypersurface. Detecting whether there is a complex variety in a real hypersurface involves more than a glance at the Levi form. The chapter considers real hypersurfaces and more generally real subvarieties that are defined by polynomial equations. If the loci have a common component, then there are infinitely many solutions. If they have no common component, then the number of solutions equals the product of the degrees of the defining polynomials.