Green currents of holomorphic correspondences on compact K\"ahler manifolds

Consider a holomorphic correspondence $f$ on a compact K\"ahler manifold $X$ of dimension $k$. Let $1\le q\le k$ be any integer such that the dynamical degrees of $f$ satisfy $d_{q-1}<d_q$. We construct the Green currents $T_c$ of $f$ associated with the classes $c$ belonging to the dominant eigenspace for the action of $f^*$ on $H^{q,q}(X,\mathbb{R})$. We also show that the super-potential of $T_c$ is $\log$-H\"older-continuous. When $f$ has simple action on cohomology and its graph satisfies an assumption on the local multiplicity, we prove the exponential equidistribution of all positive closed currents towards the main Green current, i.e., the only one associated to the unique maximal degree $d_q$.