In this note, I recall various important contributions made by D’Angelo to the $\bar{\partial}$-Neumann problem. I briefly describe some of the issues that D’Angelo addressed and comment on the significance of his results.

This paper discusses developments in complex analysis and CR geometry in the last forty years related to the Cauchy–Riemann equations, proper holomorphic mappings between balls, and positivity conditions in complex analysis. The paper includes anecdotes about some of the contributors to these developments.

Generalizing and strengthening a recent result of Koziarz, we prove a version of the Ohsawa–Takegoshi–Manivel theorem for $\bar{\partial}$-closed forms.

We consider holomorphic mappings $H$ between a smooth real hypersurface $M\subset\mathbb{C}^{n+1}$ and another $M'\subset\mathbb{C}^{N+1}$ with $N\geq n$. We provide conditions guaranteeing that $H$ is transversal to $M'$ along all of $M$. In the strictly pseudoconvex case, this is well known and follows from the classical Hopf boundary lemma. In the equidimensional case ($N=n$), transversality holds for maps of full generic rank provided that the source is of finite type in view of recent results by the authors (see also a previous paper by the first author and L. Rothschild). In the positive codimensional case ($N>n$), the situation is more delicate as examples readily show. In recent work by S. Baouendi, the first author, and L. Rothschild, conditions were given guaranteeing that the map $H$ is transversal outside a proper subvariety of $M$, and examples were given showing that transversality may fail at certain points.

One of the results in this paper implies that if $N\le 2n-2$, $M'$ is Levi-nondegenerate, and $H$ has maximal rank outside a complex subvariety of codimension $2$, then $H$ is transversal to $M'$ at all points of $M$. We show by examples that this conclusion fails in general if $N\geq 2n$, or if the set $W_{H}$ of points where $H$ is not of maximal rank has codimension one. We also show that $H$ is transversal at all points if $H$ is assumed to be a finite map (which allows $W_{H}$ to have codimension one) and the stronger inequality $N\leq 2n-3$ holds, provided that $M$ is of finite type.

The notion of the projective hull of a compact set in a complex projective space $\mathbb{P}^{n}$ was introduced by Harvey and Lawson in 2006. In this paper, we describe the projective hull by Poletsky sequences of analytic discs, in analogy to the known descriptions of the holomorphic and the plurisubharmonic hull.

We discuss and expand upon our previous results on comparison of the Bergman and the Szegö kernels. We also study the relations between the Bergman growth exponent and the finite type conditions.

By the work of Siegel it is well known that as a rule the Birkhoff normal form of a real analytic Hamiltonian system whose eigenvalues satisfies suitable non-resonance condition cannot be realized by convergent symplectic transformations. We show the existence of divergent Birkhoff normal forms for suitable Hamiltonian systems. Our calculation shows how the small divisors appear in the normal forms, from which the divergence is derived by using Siegel’s methods of small divisors.

We give a simple and more or less elementary proof of a classical result of E. Calabi on the global extension of a local holomorphic isometry into a complex space form.

We define a category of formal CR manifolds in a purely algebraic way, study their basic properties, and prove the Baouendi–Ebenfelt–Rothschild minimality criterion in this setting. The objects of our category include deformations of classical CR manifolds (real-analytic or formal).

Let $D\subset\mathbb{C}^{n}$ be a bounded, strongly Levi-pseudoconvex domain with minimally smooth boundary. We prove $L^{p}(D)$-regularity for the Bergman projection $B$, and for the operator $|B|$ whose kernel is the absolute value of the Bergman kernel with $p$ in the range $(1,+\infty)$. As an application, we show that the space of holomorphic functions in a neighborhood of $\overline{D}$ is dense in $\vartheta L^{p}(D)$.

We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of nonnegative distinct monomials. This bound was conjectured by John P. D’Angelo, proved in two dimensions by D’Angelo, Kos and Riehl and in three dimensions by the authors. The current work builds upon these results to settle the conjecture in all dimensions. We also give a complete description of all polynomials in dimensions 4 and higher for which the sharp bound is obtained. The results prove the sharp degree bounds for monomial CR mappings of spheres in all dimensions.

Buchberger and, independently, Kandri-Rody and Kapur, defined a strong Gröbner basis for a polynomial ideal over a Euclidean domain in a way that gives rise to canonical reductions. This retains what is perhaps the most important property of Gröbner bases over fields. A difficulty is that these can be substantially harder to compute than their field counterparts. We extend their results for computing these bases to give an algorithm that is effective in practice. In particular, we show how to use S-polynomials (rather than “critical pairs”) so that the algorithm becomes quite similar to that for fields, and thus known strategies for the latter may be employed. We also show how Buchberger’s important criteria for detection of unneeded S-polynomials can be extended to work over Euclidean domains. We illustrate with some examples.

A new proof of Oka’s lemma is given for smoothly bounded, pseudoconvex domains $\Omega\subset\subset\mathbb{C}^{n}$. The method of proof is then also applied to other convexity-like hypotheses on the boundary of $\Omega$.

The subtle distinction between hermitian sums of squares and sums of squares, regarded as positivity certificates of a polynomial restricted to a real algebraic variety, is analyzed on the simplest, yet very relevant, example: an ellipse.

We solve a problem on filling by Levi-flat hypersurfaces for a class of totally real 2-tori in a real 4-manifold with an almost complex structure tamed by an exact symplectic form. As an application, we obtain a simple proof of Gromov’s non-squeezing theorem in dimension 4 and new results on rigidity of symplectic structures.

We present results on $L^{2}$ estimates for solutions of $\bar{\partial}$-equations on a Stein manifold with a divisor. The structure of the divisor allows us to introduce weights with certain types of singularities, and the geometry of the manifold near the divisor allows us, by exploiting twisted techniques, to weaken the usual curvature hypotheses that guarantee a solution. We investigate two situations; one in which the weights are not locally integrable, and another in which they can be.

We consider a fundamental solution for the $\overline{\partial}$-operator on a complex $n$-manifold, which is given by an $(n,n-1)$-form of the Cauchy–Leray type $\Theta=\theta\wedge(\overline{\partial}\theta)^{n-1}$, where $\theta $ is a suitable $(1,0)$-form. On the open submanifold $M^{n}$ where $\theta$ is smooth and nonzero, its multiples generate a complex line sub-bundle $E\subset T^{*}_{(1,0)}M$, which we assume to satisfy a certain integrability condition. To such an $E$ we attach a global holomorphic invariant, in the form of a complex Godbillon–Vey $\partial $-cohomology class, provided a certain primary obstruction class vanishes. If $\theta $ is also Levi nondegenerate, in that $\Theta\neq0$, then it determines an invariant connection on the hyperplane bundle given by $\theta=0$. This provides $\theta $ formally with a complete system of local holomorphic invariants.

We prove an analog of Whitney’s embedding theorem for pseudo-holomorphic discs.

We propose a unified normal form for all Levi-nondegenerate hypersurface type almost CR structures.