  We consider the cubic non-linear Schr\"odinger equation on general closed
(compact without boundary) Riemannian surfaces. The problem is known to be
locally well-posed in $H^s(M)$ for $s>1/2$. Global well-posedness for $s\geq 1$
follows easily from conservation of energy and standard arguments. In this
work, we extend the range of global well-posedness to $s>2/3$. This
generalizes, without any loss in regularity, a similar result on $\T^2$. The
proof relies on the I-method of Colliander, Keel, Staffilani, Takaoka, and Tao,
a semi-classical bilinear Strichartz estimate proved by the author, and
spectral localization estimates for products of eigenfunctions, which is
essential to develop multilinear spectral analysis on general compact
manifolds.
