Rational cuspidal curves and symplectic fillings

TL;DR

This paper investigates symplectic fillings of contact manifolds arising from rational cuspidal curves with positive self-intersection, using handlebody and blow-down techniques, and classifies some of these fillings.

Contribution

It provides new insights into the symplectic fillings of contact manifolds associated with rational cuspidal curves, including classification results and examples.

Findings

Some contact manifolds are links of surface singularities.

Certain contact manifolds admit no symplectic fillings.

Some fillings can be completely classified.

Abstract

A symplectic rational cuspidal curve with positive self-intersection number admits a concave neighborhood, and thus a corresponding contact manifold on the boundary. In this article, we study symplectic fillings of such contact manifolds, providing a complementary perspective to our earlier article on symplectic isotopy classes of rational cuspidal curves. We explore aspects of these symplectic fillings through Stein handlebodies and rational blow-downs. We give examples of such contact manifolds which are identifiable as links of normal surface singularities, other examples which admit no symplectic fillings, and further examples where the fillings can be fully classified.