Restriction estimates to complex hypersurfaces

The restriction problem is better understood for hypersurfaces and recent progresses have been made by bilinear and multilinear approaches and most recently polynomial partitioning method which is combined with those estimates. However, for surfaces with codimension bigger than 1, bilinear and multilinear generalization of restriction estimates are more involved and effectiveness of these multilinear estimates is not so well understood yet. Regarding the restriction problem for the surfaces with codimensions bigger than 1, the current state of the art is still at the level of $TT^*$ method which is known to be useful for obtaining $L^q$--$L^2$ restriction estimates. In this paper, we consider a special type of codimension 2 surfaces which are given by graphs of complex analytic functions and attempt to make progress beyond the $L^2$ restriction estimates.