Global well-posedness for a NLS-KdV system on $\mathbb{T}$

We prove that the Cauchy problem of the Schr\"odinger - Korteweg - deVries (NLS-KdV) system on $\mathbb{T}$ is globally well-posed for initial data $(u_0,v_0)$ below the energy space $H^1\times H^1$. More precisely, we show that the non-resonant NLS-KdV is globally well-posed for initial data $(u_0,v_0)\in H^s(\mathbb{T})\times H^s(\mathbb{T})$ with $s>11/13$ and the resonant NLS-KdV is globally well-posed for initial data $(u_0,v_0)\in H^s(\mathbb{T})\times H^s(\mathbb{T})$ with $s>8/9$. The idea of the proof of this theorem is to apply the I-method of Colliander, Keel, Staffilani, Takaoka and Tao in order to improve the results of Arbieto, Corcho and Matheus concerning the global well-posedness of the NLS-KdV on $\mathbb{T}$ in the energy space $H^1\times H^1$.