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.

We continue the study in [2] in the setting of weighted pluripotential theory arising from polynomials associated to a convex body P in (R+)d. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of P-pluripotential-theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge–Ampère equation in an appropriate finite energy class. This is achieved using a variational approach.

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.

It is shown that an HKT space with closed parallel potential 1-form has D (2, 1; 1) symmetry. Every locally conformally hyper-Khler manifold generates this type of geometry. The HKT spaces with closed parallel potential 1-form arising in this way are characterized by their symmetries and an inhomogeneous cubic condition on their torsion.

Let X be a compact quotient of the product of the real Heisenberg group H _4m+1 of dimension 4m + 1 and the three-dimensional real Euclidean space R ³ . A left-invariant hypercomplex structure on H _4m+1 R ³ descends onto the compact quotient X. The space X is a hyperholomorphic fibration of 4-tori over a 4m-torus. We calculate the parameter space and obstructions to deformations of this hypercomplex structure on X. Using our calculations, we show that all small deformations generate invariant hypercomplex structures on X but not all of them arise from deformations of the lattice. This is in contrast to the deformations on the 4m-torus.