Counting isotropic tangent lines of hypersurfaces

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• 作者：Sergei Lanzat ^{lanzat.sergei@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：53A07 ; 53C15 ; 53D05 ; 53D15 ; 57N16 ; 57N35 ; 57R42
• 刊名：Differential Geometry and its Applications
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：47
• 期：Complete
• 页码：246-255
• 全文大小：330 K

文摘

Consider the standard symplectic (R²ⁿ,ω₀)$({\mathbb{R}}^{2n},{\omega }_0)$, a point p∈R²ⁿ$p\in {\mathbb{R}}^{2n}$ and an immersed closed orientable hypersurface Σ⊂R²ⁿ∖{p}$\mathrm{\Sigma }\subset {\mathbb{R}}^{2n}\setminus \{p\}$, all in general position. We study the following passage/tangency question: how many lines in R²ⁿ${\mathbb{R}}^{2n}$ pass through p   and tangent to Σ parallel to the 1-dimensional characteristic distribution d05cc37e4fbda0ec16793625" title="Click to view the MathML source">ker⁡(ω₀|_TΣ)⊂TΣ$\ker \left({\omega }_0{|}_{T\mathrm{\Sigma }}\right)\subset T\mathrm{\Sigma }$ of ω₀${\omega }_0$. We count each such line with a certain sign, and present an explicit formula for their algebraic number. This number is invariant under regular homotopies in the class of a general position of the pair (p,Σ)$(p,\mathrm{\Sigma })$, but jumps (in a well-controlled way) when during a homotopy we pass a certain singular discriminant. It provides a low bound to the actual number of these isotropic lines.