The symplectic mapping class group of $\CC P^2 n{\bar{\CC P^2}}$ with $n\leq4$

TL;DR

This paper determines the symplectic mapping class group of up to four-point blow-ups of the projective plane, showing the Torelli part is trivial and that the group is generated by specific reflections.

Contribution

It provides a complete description of the symplectic mapping class group for these blow-ups, identifying generators and triviality of the Torelli subgroup.

Findings

Torelli part of the symplectomorphism group is trivial for n ≤ 4.

The symplectic mapping class group is generated by reflections on certain spherical classes.

Explicit structure of the symplectic mapping class group is established.

Abstract

In this paper we prove that the Torelli part of the symplectomorphism groups of the $n$-point ($n\leq 4$) blow-ups of the projective plane is trivial. Consequently, we determine the symplectic mapping class group. It is generated by reflections on $K_{\omega}- $spherical class with zero $\omega$ area.