Let $(X,T^{1,0}X)$ be a compact orientable embeddable three dimensional strongly pseudoconvex CR manifold and let ${\rm P\,}$ be the associated CR Paneitz operator. In this paper, we show that (I) ${\rm P\,}$ is self-adjoint and ${\rm P\,}$ has $L^2$ closed range. Let $N$ and $\Pi$ be the associated partial inverse and the orthogonal projection onto ${\rm Ker\,}{\rm P\,}$ respectively, then $N$ and $\Pi$ enjoy some regularity properties. (II) Let $\hat{\mathcal{P}}$ and $\hat{\mathcal{P}_0}$ be the space of $L^2$ CR pluriharmonic functions and the space of real part of $L^2$ global CR functions respectively. Let $S$ be the associated Szeg\"o projection and let $\tau$, $\tau_0$ be the orthogonal projections onto $\hat{\mathcal{P}}$ and $\hat{\mathcal{P}_0}$ respectively. Then, $\Pi=S+\ol S+F_0$, $\tau=S+\ol S+F_1$, $\tau_0=S+\ol S+F_2$, where $F_0, F_1, F_2$ are smoothing operators on $X$. In particular, $\Pi$, $\tau$ and $\tau_0$ are Fourier integral operators with complex phases and $\hat{\mathcal{P}}^\perp\bigcap{\rm Ker\,}{\rm P\,}$, $\hat{\mathcal{P}_0}^\perp\bigcap\hat{\mathcal{P}}$, $\hat{\mathcal{P}_0}^\perp\bigcap{\rm Ker\,}{\rm P\,}$ are all finite dimensional subspaces of $C^\infty(X)$ (it is well-known that $\hat{\mathcal{P}_0}\subset\hat{\mathcal{P}}\subset{\rm Ker\,}{\rm P\,}$). (III) ${\rm Spec\,}{\rm P\,}$ is a discrete subset of $\Real$ and for every $\lambda\in{\rm Spec\,}{\rm P\,}$, $\lambda\neq0$, $\lambda$ is an eigenvalue of ${\rm P\,}$ and the associated eigenspace $H_\lambda({\rm P\,})$ is a finite dimensional subspace of $C^\infty(X)$.

In this paper, we prove that in any projective manifold, the complements of general hypersurfaces of sufficiently large degree are Kobayashi hyperbolic. We also provide an effective lower bound on the degree. This confirms a conjecture by S. Kobayashi in 1970. Our proof, based on the theory of jet differentials, is obtained by reducing the problem to the construction of a particular example with strong hyperbolicity properties. This approach relies the construction of higher order logarithmic connections allowing us to construct logarithmic Wronskians. These logarithmic Wronskians are the building blocks of the more general logarithmic jet differentials we are able to construct. As a byproduct of our proof, we prove a more general result on the orbifold hyperbolicity for generic geometric orbifolds in the sense of Campana, with only one component and large multiplicities. We also establish a Second Main Theorem type result for holomorphic entire curves intersecting general hypersurfaces, and we prove the Kobayashi hyperbolicity of the cyclic cover of a general hypersurface, again with an explicit lower bound on the degree of all these hypersurfaces.

We study Nakai-Moishezon type question and Donaldson’s “tamed to compatible” question for almost complex structures on rational four manifolds. By extending Taubes’ subvarieties-currentform technique to J-nef genus 0 classes, we give affirmative answers of these two questions for all tamed almost complex structures on S^2 bundles over S^2 as well as for many geometrically interesting tamed almost complex structures on other rational four manifolds, including the del Pezzo ones.

Given a hypersurface coamoeba of a Laurent polynomial$f$, it is an open problem to describe the structure of the set of connected components of its complement. In this paper we approach this problem by introducing the lopsided coamoeba. We show that the closed lopsided coamoeba comes naturally equipped with an order map, i.e. a map from the set of connected components of its complement to a translated lattice inside the zonotope of a Gale dual of the point configuration $\operatorname{supp}(f)$ . Under a natural assumption, this map is a bijection. Finally we use this map to obtain new results concerning coamoebas of polynomials of small codimension.

We construct Coleff–Herrera products and Bochner–Martinelli type residue currents associated with a tuple$f$of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case. This include the transformation law, the Poincaré–Lelong formula and the equivalence of the Coleff–Herrera product and the Bochner–Martinelli type residue current associated with$f$when$f$defines a complete intersection.