Enumeration of Rational Contact Curves via Torus Actions

We prove that some Gromov-Witten numbers associated to rational contact (Legendrian) curves in any contact complex projective space with arbitrary incidence conditions are enumerative. Also, we use the Bott formula on the Kontsevich space to find the exact value of those numbers. As an example, the numbers of rational contact curves of low degree in P3 and P5 are computed. The results are consistent with existing results.

Automated summary

This article proves that some Gromov-Witten numbers associated with rational contact (Legendrian) curves in any contact complex projective space with arbitrary incidence conditions are enumerative, and uses the Bott formula on the Kontsevich space to find the exact value of these numbers. The numbers of rational contact curves of low degree in P3 and P5 are computed as examples, and the results are consistent with existing results.