Counting isotropic tangent lines of hypersurfaces

TL;DR

This paper derives an explicit, invariant formula for counting isotropic tangent lines passing through a point and tangent to a hypersurface in symplectic space, accounting for their algebraic signs and homotopy invariance.

Contribution

It introduces a new explicit formula for the algebraic count of isotropic tangent lines, invariant under regular homotopies, and analyzes their behavior during singular degenerations.

Findings

Provides an explicit algebraic formula for the count.

Establishes invariance of the count under regular homotopies.

Identifies how the count jumps during singular degenerations.

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.