Let f be a holomorphic automorphism of a compact Kähler manifold. Assume that f admits a unique maximal dynamic degree d_p with only one eigenvalue of maximal modulus. Let μ be its equilibrium measure. In this paper, we prove that μ is exponentially mixing for all d.s.h. test functions.

We study singular real analytic Levi-flat subsets invariant by singular holomorphic foliations in complex projective spaces. We give sufficient conditions for a real analytic Levi-flat subset to be the pull-back of a semianalytic Levi-flat hypersurface in a complex projective surface under a rational map or to be the pull-back of a real algebraic curve under a meromorphic function. In particular, we give an application to the case of a singular real analytic Levi-flat hypersurface. Our results improve previous ones due to Lebl and Bretas–Fernández-Pérez–Mol.

The Separatrix Theorem of C. Camacho and P. Sad says that there exists at least one invariant curve (separatrix) passing through the singularity of a germ of holomorphic foliation on complex surface, when the surface underlying the foliation is smooth or when it is singular and the dual graph of resolution surface singularity is a tree. Under some assumptions, we obtain existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. It will be necessary to require an extra condition of the foliation, namely, absence of saddle-node in its reduction of singularities.

Let <i>L</i> be a holomorphic line bundle over a compact Khler manifold <i>X</i>. Motivated by mirror symmetry, we study the deformed HermitianYangMills equation on <i>L</i>, which is the line bundle analogue of the special Lagrangian equation in the case that <i>X</i> is CalabiYau. We show that this equation is the Euler-Lagrange equation for a positive functional, and that solutions are unique global minimizers. We provide a necessary and sufficient criterion for existence in the case that <i>X</i> is a Khler surface. For the higher dimensional cases, we introduce a line bundle version of the Lagrangian mean curvature flow, and prove convergence when <i>L</i> is ample and <i>X</i> has non-negative orthogonal bisectional curvature.

We address the question “when the local image of a map is well defined” and answer it in case of holomorphic map germs with target (C^2,0). We prove a criterion for holomorphic map germs (X,x)→(Y,y) to be locally open, solving a conjecture by Huckleberry in all dimensions.