Showing papers on "Complex dimension published in 2023"
TL;DR: The minimal total Betti number of a closed almost complex manifold of dimension 2n/ge 8 was shown to be four in this article , thus confirming a conjecture of Sullivan except for dimension 6.
Abstract: We show the minimal total Betti number of a closed almost complex manifold of dimension $2n\ge 8$ is four, thus confirming a conjecture of Sullivan except for dimension $6$. Along the way, we prove the only simply connected closed complex manifold having total Betti number three is the complex projective plane.
TL;DR: For a real analytic family of almost complex structures (M,Jt), t∈T, on an even dimensional compact real analytic manifold M, the authors proved upper semi-continuity with respect to t of the dimension of the space of Δ∂∆t-harmonic (p,q)-forms in Zariski sense.
10 Apr 2023
TL;DR: In this article , a tensor product of two nonzero modules, at least one of which is totally reflexive (or equivalently Gorenstein-projective), has finite projective dimension over commutative Noetherian rings.
Abstract: In this paper we consider a question of Roger Wiegand, which is about tensor products of finitely generated modules that have finite projective dimension over commutative Noetherian rings. We construct modules of infinite projective dimension (and of infinite Gorenstein dimension) whose tensor products have finite projective dimension. Furthermore we determine nontrivial conditions under which such examples cannot occur. For example we prove that, if the tensor product of two nonzero modules, at least one of which is totally reflexive (or equivalently Gorenstein-projective), has finite projective dimension, then both modules in question have finite projective dimension.
09 Feb 2023
TL;DR: In this article , it was shown that 8 is the maximal symmetry dimension of 3-nondegenerate CR structures in dimension 7, which is achieved on the homogeneous model.
Abstract: We investigate 3-nondegenerate CR structures in the lowest possible dimension 7, and one of our goals is to prove Beloshapka's conjecture on the symmetry dimension bound for hypersurfaces in $C^4$. We claim that 8 is the maximal symmetry dimension of 3-nondegenerate CR structures in dimension 7, which is achieved on the homogeneous model. This part (I) is devoted to the homogeneous case: we prove that the model is locally the only homogeneous 3-nondegenerate CR structure in dimension 7.
01 Jan 2023
TL;DR: In this article , the dimension of a set appears as a quantification of the organization of its points and the dimension seems essentially to be a comparison between the entropies of sets.
Abstract: 1 This article is mainly concerned by the concept of dimension. More precisely, our objective is to challenge the conventional zero dimension assigned to the point. This reconsideration allows us to propose two new ways of conceiving the notion of dimension, which are the two sides of the same coin. First, the dimension of a set appears as a quantification of the organization of its points. Secondly, the dimension seems essentially to be a comparison between the entropies of sets. Thus, we started from the point and succeeded in constructing a point-dimension notion allowing us to extend the principle of box dimension in many directions and to finely estimate the dimensions of sets. More precisely, we introduce the notion of point-extended box dimension in the large framework of topological vector spaces, freeing it from the notion of metric. This general setting permits us to treat the case of finite, infinite and invisible dimensions. This first work focuses essentially on general properties and is particularly oriented towards establishing a well founded framework for infinite dimension.
TL;DR: In this article , it was shown that Oguiso's simply connected compact Kähler manifold of dimension four does not have the rational homotopy type of a complex projective manifold.
Abstract: Voisin constructed a series of examples of simply connected compact Kähler manifolds of even dimension, which do not have the rational homotopy type of a complex projective manifold starting from dimension six. In this note, we prove that Voisin's examples of dimension four also do not have the rational homotopy type of a complex projective manifold. Oguiso constructed simply connected compact Kähler manifolds starting from dimension four, which cannot deform to a complex projective manifold under a small deformation. We also prove that Oguiso's examples do not have the rational homotopy type of a complex projective manifold.
29 Jun 2023
TL;DR: In this paper , it was shown that holomorphic geometric structures of affine type on compact complex manifold are locally homogeneous away from an analytic subset of complex codimension at least two, and that they cannot be rigid unless they are an \'etale quotient of a compact complex torus.
Abstract: Let $X$ be a compact complex manifold such that its canonical bundle $K_X$ is numerically trivial. Assume additionally that $X$ is Moishezon or $X$ is Fujiki with dimension at most four. Using the MMP and classical results in foliation theory, we prove a Beauville-Bogomolov type decomposition theorem for $X$. We deduce that holomorphic geometric structures of affine type on $X$ are in fact locally homogeneous away from an analytic subset of complex codimension at least two, and that they cannot be rigid unless $X$ is an \'etale quotient of a compact complex torus. Moreover, we establish a characterization of torus quotients using the vanishing of the first two Chern classes which is valid for any compact complex $n$-folds of algebraic dimension at least $n-1$. Finally, we show that a compact complex manifold with trivial canonical bundle bearing a rigid geometric structure must have infinite fundamental group if either $X$ is Fujiki, $X$ is a threefold, or $X$ is of algebraic dimension at most one.