arXane › Mathematics › Mathematics
Wednesday, 4 November 2026: 5 new papers, 3 cross-lists and 2 replacements in Complex Variables (math.CV), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We study the numerical constructions of almost complex and complex structures on spheres and products of spheres up to real dimension six. Our general set-up parametrises almost complex structures via a global matrix field in an ambient Euclidean space, compressed by the tangent projector. The projector's algebraic residual measures the almost complex condition, while its Nijenhuis tensor is evaluated in the intrinsic norm of the round product metric. Polynomial features and neural networks are trained without prescribed targets to perform general searches for these structures. The almost complex search passes numerical admissibility checks on all nine examples that admit such structures and rejects $S^4$, where a topological obstruction is known. A matrix correction then enforces the almost complex identity, subject to a novel uniform admissibility condition. For integrability, a free polynomial representation with Levenberg-Marquardt optimisation learns the complex structures on $S^2\times S^2$, $S^3\times S^1$, $S^3\times S^3$ and $S^5\times S^1$, with the largest Nijenhuis norms found by adversarial search all below $10^{-9}$. These general searches were unsuccessful on $S^6$, $S^4\times S^2$ and $S^4\times T^2$. Therefore, for these three cases direct numerical reconstructions were built in torus-fibred coordinates. They are integrable by construction, satisfy their compatibility conditions to better than $10^{-11}$, and corroborate the recent constructions on $S^6$ and $S^4\times S^2$.
Let $D\subset\mathbb{C}^n$, $n\ge2$, be a bounded domain that is biholomorphic to a finite-dimensional Teichm\"uller space. We prove that $\partial D$ has no locally convex boundary patch containing a complex $(n-1)$-dimensional manifold through each of its points. Note that in complex dimension two this excludes every locally convex boundary point, without any boundary regularity assumption. In every dimension, we also exclude locally convex $C^2$ boundary patches.
In this note, we study the regularity of solutions to the geodesic equation in the space of plurisubharmonic functions in a general Reinhardt domain. Our findings resolve a problem left open from our previous note [AD26].
We study the distortion of the lower dimension and lower spectrum of sets under quasiconformal mappings. As an application, we verify weighted Hardy--Sobolev inequalities in Euclidean domains arising as quasiconformal images of domains with sufficiently regular boundaries. The regularity in question is phrased in terms of estimates for either the Assouad or the lower dimension of the complement. We also provide dilatation-dependent classification results for certain planar open sets.
Using a sharp bound for convolutions of non-negative trigonometric polynomials with the Fej\'er kernel resembling a pointwise variant of Bernstein's inequality for polynomials, we prove new properties of extremal functions for the Krzy\.z conjecture. In particular, we resolve a 1993 conjecture of Peretz by showing that if $f$ is extremal for some coefficient, then $|f(0)|\leq 1/e$. Furthermore, we show that if equality holds in this estimate for every extremal function, then the Krzy\.z conjecture is true.
We construct a canonical projectivization of every smooth proper algebraic variety $X$ in characteristic zero. Once a principalization procedure is fixed, the construction gives a finite sequence of blowups in smooth centers with smooth projective endpoint, natural under semilinear isomorphisms and preserving the quasi-projective core, the intersection of all maximal quasi-projective open subsets. At each stage we select finitely many ideals near the generic points of the image in $X$ of the boundary of a maximal quasi-projective open. Principalizing their product strictly lowers the dimension of that image. The selection is defined intrinsically, without a polarization or a chosen projective modification; a Hilbert scheme argument proves its finiteness. Naturality embeds the automorphism group in that of a smooth projective variety of the same dimension. We then extend the construction to smooth proper integral algebraic spaces, obtaining canonical projectivizations of all compact connected Moishezon manifolds.
In dimension $n\ge2$, a homeomorphism of $R^n$ preserving BMO space therein was proved to be quasiconformal, up to an orientation reversing reflection.
We study Jack Lee's conjecture that, for a compact strictly pseudoconvex CR manifold $M$, the condition $c_1(T^{1,0}M)=0$ in $H^2(M,\mathbb{R})$ implies the existence of a pseudo-Einstein structure. For links of isolated singularities, we construct counterexamples in every odd dimension at least three, with $c_1(T^{1,0}M)=0$ in $H^2(M,\mathbb{Z})$ and $b_1(M)=0$. For isolated algebraic singularities of log canonical type and complex dimension at least three, we prove that the real Chern class condition is equivalent to $\mathbb{Q}$-Cartierness of the canonical divisor, log canonicity, and the existence of a pseudo-Einstein structure. Finally, we construct a Du Bois threefold singularity whose link satisfies the real Chern class condition but admits no pseudo-Einstein structure. The initial versions of the proofs were generated with AI assistance and were subsequently revised and verified by the authors.