Research on the geometry of the projective hypersutfaces and plane curves

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For complex analytic hypersurface singularities, there are established fundamental tools such as Milnor fibration and its monodromy, zeta functions and the resolution of the singularities, contact structures and syplectic structures on the Milnor fibration. We have established these fundamental tools for mixed singularities. This will give a guide line for the further research in this domain. In fact, the basic tools are provided in two papers:Oka, Mutsuo : Topology of polar weighted homogeneous hypersurfaces, Kodai Math. J. No.2, 163-182, 2008 andOka, Mutsuo, Non-degenerate mixed functions, Kodai Math. J. 33(2010), 1-62In Oka, Mutsuo, Intersection theory on mixed curves, arXiv : 1202.2166 we presented an intersection theory for mixed curves. This involves negative intersections and more complicated than the holomorphic curves. InOka, Mutsuo, On mixed projective curves, arXiv : 0910.2523 andOka, Mutsuo, On mixed plane curves of polar degree 1.The Japanese-Australian Workshop on Real and Complex Singularities-JARCS III, 67-74, Proc. Centre MathAppl. Austral. Nat. Univ., 43, Austral. Nat. Univ., Canberra, 2010, we proved that for any genus, we can embed this surface as a mixed projective curve with degree 1These results are already used among the researchers in France and Mexico for further research in this direction.