Global $H^2$-solutions for the generalized derivative NLS on $\mathbb{T}$

We prove global existence of $H^2$ solutions to the Cauchy problem for the generalized derivative nonlinear Schr\"{o}dinger equation on the 1-d torus. This answers an open problem posed by Ambrose and Simpson (2015). The key is the extraction of the terms that cause the problem in energy estimates and the construction of suitable energies so as to cancel the problematic terms out by effectively using integration by parts and the equation.