Counting isotropic tangent lines of hypersurfaces

Abstract

Consider the standard symplectic $(\RR^{2n}, \omega_0)$, a point $p\in\RR^{2n}$ and an immersed closed orientable hypersurface $\Sigma\subset\RR^{2n}\minus\{p\}$, all in general position. We study the following passage/tangency question: how many lines in $\RR^{2n}$ pass through $p$ and tangent to $\Sigma$ parallel to the 1-dimensional characteristic distribution $\ker\left(\omega_0\big|_{T\Sigma}\right)\subset T\Sigma$ of $\omega_0$. 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 $(p, \Sigma)$, 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.