We study the exponentially small singular values of the semiclassical d-bar operator on an exponentially weighted L^2 space on a two-dimensional torus. We will assume that the Laplacian of the exponential weight changes sign along a curve. We will introduce the notion of upper and lower bound weights which give together with the orthogonal Bergman projection precise upper and lower bounds on the number of small singular values. Solving a free boundary value problem we obtain optimal weights which yield Weyl asymptotics for the counting function of exponentially small singular values. We also provide a precise description of the leading term of the Weyl asymptotics in the regime of small exponential decay rates of the singular values. This talk is based on joint work with J. Söstrand and M. Hitrik.