arXiv subjects

Viktor F. Majewski

Publications and source records attributed to Viktor F. Majewski.

4 recordsLinked to original sources

arXiv2026

Degenerations of exotic Calabi-Yau metrics through Atiyah's flop

We construct new families of complete Calabi-Yau metrics with maximal volume growth on the small resolutions of the conifold $\mathcal{Z} = \{z_1^2 + z_2^2 + z_3^2 + z_4^2 = 0\} \subset \mathbb{C}^4$. These metrics have tangent cone $\mathbb{C} \times (\mathbb{C}^2 / \mathbb{Z}_2)$ at infinity and are parametrised by their K\"ahler class. As the K\"ahler class degenerates, the metrics converge in the pointed Gromov-Hausdorff sense to a Calabi-Yau metric on $\mathcal{Z}$ with an isolated conical singularity modelled on the Stenzel metric at the ordinary double point and tangent cone at infinity $\mathbb{C} \times (\mathbb{C}^2/\mathbb{Z}_2)$, thereby providing a new metric realisation of the Atiyah flop.

math.DG↗

arXiv2026

Non-Existence of Smooth Full-Holonomy Cayley Fibrations

We prove that every Cayley fibration of a compact torsion-free Spin(7)-manifold with full holonomy must have singular fibers. This confirms a long-standing expectation in the study of calibrated fibrations with exceptional holonomy. Our argument first uses the rigidity of the Spin(7)-structure to reduce any hypothetical nonsingular Cayley fibration to two possible topological configurations. We then exclude both by proving a new spinnability theorem for smooth fiber bundles over simply-connected 4-manifolds with fiber homeomorphic to the elliptic surfaces E(2) or E(4). The E(4) case, which constitutes the main new difficulty, is established by combining families Seiberg--Witten theory with parametrized equivariant homotopy theory and equivariant K-theory.

math.DG↗

arXiv2025

Spin(7)-Orbifold Resolutions

We develop an analytic and geometric framework for resolving compact Spin(7)-orbifolds by smooth torsion-free Spin(7)-manifolds. These orbifolds arise naturally as boundary points in the Gromov--Hausdorff compactification of the moduli space of exceptional holonomy metrics, and smooth Gromov--Hausdorff resolutions can be viewed as paths from the boundary back into the smooth part of the moduli space. Our construction replaces the singular strata by adiabatic torsion-free asymptotically conically fibred spaces. The local resolution data are encoded by McKay-type correspondences and Chen--Ruan local systems, while the global deformation problem is controlled by the uniform elliptic theory for Dirac-type operators on orbifold resolutions developed in the author's previous work. In particular, the obstruction map and the associated isentropicity condition from that theory provide the criterion for whether the local harmonic resolution data glue to global harmonic forms on the smooth resolution. In this paper, we link the vanishing of the resulting obstruction map to the string cohomology of the orbifold. When this obstruction vanishes, we deform the preglued Spin(7)-structure to a genuine torsion-free Spin(7)-structure. This extends Joyce's resolution theorem to the nonflat case and yields new families of compact Spin(7)-manifolds. By dimensional reduction, the same framework recovers and extends the Joyce--Karigiannis theory of G2-orbifold resolutions.

math.DG↗

arXiv2025

Dirac Operators on Orbifold Resolutions: Uniform Elliptic Theory

We study Dirac operators on resolutions of Riemannian orbifolds by developing a uniform elliptic theory. The key idea is to view orbifolds as conically fibred singular (CFS) spaces and resolve them by gluing asymptotically conical fibrations (ACF) into the singular strata. This yields smooth Gromov--Hausdorff resolutions that preserve the large--scale structure of the orbifold while replacing its singularities with well--understood local models. Dirac bundles are resolved compatibly with this construction, which allows us to analyse entire families of Dirac operators in a uniform way. Inspired by the linear gluing framework of Hutchings--Taubes, we build uniformly bounded right inverses and describe precisely how kernels and cokernels behave as the geometry degenerates. The analysis relies on weighted spaces adapted to the conically fibred, conically fibred singular and asymptotically conical fibred regions. These allow us to prove Fredholm properties, uniform bounds and to establish exactness of the linear gluing sequence. Consequently, we obtain an index formula decomposing the index into contributions from the orbifold and the ACF models. This framework provides a direct and flexible analytic approach to Dirac operators on orbifold resolutions, avoiding the full edge calculus, and setting the stage for applications to special holonomy metrics and gauge theory.

math.DG↗