arXiv subjects

Michael Albanese

Publications and source records attributed to Michael Albanese.

11 recordsLinked to original sources

arXiv2026

Hodge theoretic results for nearly Kähler manifolds in all dimensions

We generalize to nearly Kähler manifolds of arbitrary dimensions most of the Hodge-theoretic results for nearly Kähler $6$-manifolds that were established by Verbitsky. In particular, for a compact nearly Kähler manifold of any dimension, the (appropriately defined) Hodge numbers are related to the Betti numbers in the same way as on a compact Kähler manifold. In the $6$-dimensional case, Verbitsky was able to say slightly more using the induced $\mathrm{SU}(3)$ structure. We discuss potential extensions of this to twistor spaces over positive scalar curvature quaternionic-Kähler manifolds, which are a particular class of $(4n+2)$-dimensional nearly Kähler manifolds equipped with a special $\mathrm{SU}(n) \! \cdot \! \mathrm{U}(1)$ structure.

math.DG↗

arXiv2025

Obstructions to almost complex structures following Massey

We provide proofs of two theorems stated by Massey in 1961, concerning the obstructions to finding complex structures on real vector bundles. In addition, we determine the second obstruction to a complex structure on a rank six orientable real vector bundle. The obstructions are fractional parts of integral Stiefel-Whitney classes, and a fourth of an appropriate combination of Pontryagin, Chern, and Euler classes.

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arXiv2025

Aspherical Complex Surfaces, the Singer Conjecture, and Gromov-Lück Inequality $χ\geq |σ|$

We discuss the Singer conjecture and Gromov-Lück inequality $χ\geq |σ|$ for aspherical complex surfaces. We give a proof of the Singer conjecture for aspherical complex surface with residually finite fundamental group that does not rely on Gromov's Kähler groups theory. Without the residually finiteness assumption, we observe that this conjecture can be proven for all aspherical complex surfaces except possibly those in Class $\mathrm{VII}_0^+$ (a positive answer to the global spherical shell conjecture would rule out the existence of aspherical surfaces in this class). We also sharpen Gromov's inequality for aspherical complex surfaces that are not in Class $\mathrm{VII}_0^+$. This is achieved by connecting the circle of ideas of the Singer conjecture with the study of Reid's conjecture.

math.DG↗

arXiv2025

Examples of biquotients whose tangent bundle is not a biquotient vector bundle

A biquotient vector bundle is any vector bundle over a biquotient $G/\!\!/ H$ of the form $G\times_{H} V$ for an $H$-representation $V$. Over most biquotients, biquotient vector bundles are the only vector bundles known to admit metrics of non-negative sectional curvature, and hence they play a crucial role in the context of the converse to the Soul Theorem of Cheeger and Gromoll. In this article, we study the question of when the tangent bundle of $G/\!\!/ H$ is a biquotient vector bundle. We find infinite families of examples of biquotients $M\cong G/\!\!/ H$ for which the tangent bundle is not a biquotient vector bundle for any presentation of $M$ as a biquotient. In addition, we find infinite families of manifolds which arise as biquotients in two ways: one for which the tangent bundle is a biquotient bundle, and one for which it is not. Some of these results depend on an observation of Hirzebruch which relates the signature and Euler characteristic of an almost complex manifold. We include a proof of this fact as it seems to be missing from the literature.

math.DG↗

arXiv2024

Aspherical 4-Manifolds, Complex Structures, and Einstein Metrics

We show that extended graph 4-manifolds with positive Euler characteristic cannot support a complex structure. This result stems from a new proof of the fact that a closed real-hyperbolic 4-manifold cannot support a complex structure. Finally, we construct infinitely many extended graph 4-manifolds with positive Euler characteristic which support almost complex structures.

math.DG↗

arXiv2024

Generalized almost-Kähler-Ricci solitons

We generalize Kähler-Ricci solitons to the almost-Kähler setting as the zeros of Inoue's moment map \cite{MR4017922}, and show that their existence is an obstruction to the existence of first-Chern-Einstein almost-Kähler metrics on compact symplectic Fano manifolds. We prove deformation results of such metrics in the $4$-dimensional case. Moreover, we study the Lie algebra of holomorphic vector fields on $2n$-dimensional compact symplectic Fano manifolds admitting generalized almost-Kähler-Ricci solitons. In particular, we partially extend Matsushima's theorem \cite{MR0094478} to compact first-Chern-Einstein almost-Kähler manifolds.

math.DG↗

arXiv2023

Spin^h and further generalisations of spin

The question of which manifolds are spin or spin^c has a simple and complete answer. In this paper we address the same question for spin^h manifolds, which are less studied but have appeared in geometry and physics in recent decades. We determine that the first obstruction to being spin^h is the fifth integral Stiefel-Whitney class W_5. Moreover, we show that every compact orientable manifold of dimension 7 or lower is spin^h, and that there are orientable manifolds which are not spin^h in all higher dimensions. We are then led to consider an infinite sequence of generalised spin structures. In doing so, we show that there is no integer k such that every manifold embeds in a spin manifold with codimension k.

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arXiv2021

Kodaira Dimension and the Yamabe Problem, II

For compact complex surfaces (M^4, J) of Kaehler type, it was previously shown that the sign of the Yamabe invariant Y(M) only depends on the Kodaira dimension Kod (M, J). In this paper, we prove that this pattern in fact extends to all compact complex surfaces except those of class VII. In the process, we give a simplified proof of a result that explains why the exclusion of class VII is essential here.

math.DG↗

arXiv2020

The Yamabe invariants of Inoue surfaces, Kodaira surfaces, and their blowups

Shortly after the introduction of Seiberg-Witten theory, LeBrun showed that the sign of the Yamabe invariant of a compact Kähler surface is determined by its Kodaira dimension. In this paper, we show that LeBrun's Theorem is no longer true for non-Kähler surfaces. In particular, we show that the Yamabe invariants of Inoue surfaces and their blowups are all zero. We also take this opportunity to record a proof that the Yamabe invariants of Kodaira surfaces and their blowups are all zero, as previously indicated by LeBrun.

math.DG↗

arXiv2019

Connected sums of almost complex manifolds, products of rational homology spheres, and the twisted spin^c Dirac operator

We record an answer to the question "In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?". In the process of doing so, we are naturally led to ask "For which values of l is the connected sum of l closed almost complex manifolds necessarily an almost complex manifold?". We answer this question, along with its non-compact analogue, using obstruction theory and Yang's results on the existence of almost complex structures on (n-1)-connected 2n-manifolds. Finally, we partially extend Datta and Subramanian's result on the nonexistence of almost complex structures on products of two even spheres to rational homology spheres by using the index of the twisted spin^c Dirac operator.

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arXiv2018

On the minimal sum of Betti numbers of an almost complex manifold

We show that the only rational homology spheres which can admit almost complex structures occur in dimensions two and six. Moreover, we provide infinitely many examples of six-dimensional rational homology spheres which admit almost complex structures, and infinitely many which do not. We then show that if a closed almost complex manifold has sum of Betti numbers three, then its dimension must be a power of two.

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