Let F be a holomorphic foliation by Riemann surfaces on a compact Kähler surface X. Assume that it is generic in the sense that all the singularities are hyperbolic, and the foliation admits no directed positive closed $(1,1)$-current. Then there exists a unique (up to a multiplicative constant) positive $d{d}^c$-closed $(1,1)$-current directed by F. This is a very strong ergodic property of F showing that all leaves of F have the same asymptotic behavior. Our proof uses an extension of the theory of densities to a class of non-$d{d}^c$-closed currents. This is independent of foliation theory and represents a new tool in pluripotential theory. A complete description of the cone of directed positive $d{d}^c$-closed $(1,1)$-currents is also given when F admits directed positive closed currents.

Conjecturally, the Galois representations that are attached to essentially self-dual regular algebraic cuspidal automorphic representations are Zariski-dense in a polarized Galois deformation ring. We prove new results in this direction in the context of automorphic forms on definite unitary groups over totally real fields. This generalizes the infinite fern argument of Gouvêa–Mazur and Chenevier and relies on the construction of nonclassical p-adic automorphic forms and the computation of the tangent space of the space of trianguline Galois representations. This boils down to a surprising statement about the linear envelope of intersections of Borel subalgebras.

In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also prove that any special test configuration arises from a log canonical place of a bounded complement and establish properties of any minimizer of the stability threshold.

In this article, we study the theta lifting for loop groups and extend the classical tower property established by S. Rallis to the loop setting. As an application, we obtain cusp forms on loop groups, and we give the first example where the cusp forms constructed using this method are nonvanishing.