Research on the geometry of the projective hypersutfaces and plane curves

对于复杂解析超曲面奇点，已有Milnor纤维及其单调、Zeta函数等基本工具，以及Milnor纤维上奇点、接触结构和辛结构的分解。我们已经为混合奇点建立了这些基本工具。这将为该领域的进一步研究提供指导。事实上，基本的工具已经在两篇论文中提供了：Oka，Mutsuo：Polal Weight齐次超曲面的拓扑，Kodai Math。J.No.2,163-182,2008和Oka，Mutsuo，非退化混合函数，Kodai Math.J.33(2010年)，1-62 in Oka，Mutsuo，混合曲线的交理论，arxiv：1202.2166我们给出了混合曲线的交理论。这涉及到负交点，而且比全纯曲线更复杂。Inoka，Mutsuo，关于混合投影曲线，arxiv：0910.2523和Oka，Mutsuo，关于极次混合平面曲线，日本-澳大利亚实奇点和复奇点研讨会--JARCS III，67-74，Proc.数学应用中心。南方人。纳特。大学，43岁，澳大利亚。纳特。在堪培拉大学，2010年，我们证明了对于任何亏格，我们可以将该曲面嵌入为1次的混合射影曲线。这些结果已经被法国和墨西哥的研究人员用于这一方向的进一步研究。

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.