math.GT daily digest: 7 new submissions for 12 August 2026

Cross submissions

Toric real loci via moment polytopes

• Authors: João Camarneiro; Ana Cannas da Silva
• arXiv: 2608.08918
• Subjects: Symplectic Geometry (math.SG); Algebraic Geometry (math.AG); Geometric Topology (math.GT)
• Comments: 49 pages, 16 figures

Toric real loci are distinguished lagrangian submanifolds of toric symplectic manifolds. We develop a polyhedral model for toric real loci, called a kaleidoscope, based on the restriction of the moment map. This construction provides a direct geometric description of the topology of toric real loci and leads to simple criteria for orientability, together with transparent formulas for the Euler characteristic. We illustrate the theory with numerous examples and count, in every complex dimension up to 9, the smooth toric Fano varieties whose real loci are orientable. The kaleidoscope construction offers a simple, visual, and effective framework for understanding the topology of toric real loci through the combinatorics of their moment polytopes.

Homotopical Robustness of Isometries on the Cayley Plane

• Authors: Thorsten Hertl; Isla Lim
• arXiv: 2608.10559
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)
• Comments: 14 pages, 1 table

We compute the effect of the inclusion on rational homotopy groups by determining the rational homotopy class of the isometry action in terms of a homomorphism between the minimal models of source and target. Although special emphasis is put on the Cayley plane, our results generalise to all simply connected rank $1$-symmetric spaces.

Torsion in asymptotic cones of free groups

• Authors: Jarek Kędra; Assaf Libman; Zipei Nie
• arXiv: 2608.10854
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We prove that the asymptotic cone of any non-abelian free group equipped with the bi-invariant word metric has elements of order 2.

Log Calabi-Yau compactifications of character varieties

• Authors: Hülya Argüz; Pierrick Bousseau
• arXiv: 2608.10918
• Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Geometric Topology (math.GT); Symplectic Geometry (math.SG)
• Comments: 37 pages, 3 figures. Comments welcome!

We prove that the character varieties of compact oriented surfaces and the generic relative character varieties of punctured surfaces admit divisorial log terminal (dlt) log Calabi-Yau compactifications. To do this, we establish a general result giving sufficient conditions for a compactification of an affine variety arising from a filtration of its algebra of regular functions to be log Calabi-Yau. We then apply this result to show that the compactifications constructed by Kutteri-Tehrani-Frohman in the compact case and by Tehrani-Frohman in the punctured case are log canonical and log Calabi-Yau.

Tomter's example revisited

• Authors: Danyu Zhang
• arXiv: 2608.11097
• Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)
• Comments: 16 pages; comments welcome!

We construct examples of fibrewise Anosov flows over the Fenley--Mann--Potrie example of dimension 4, which is a family of non-algebraic Anosov flows with unstable dimension and stable dimension , for every positive integer $n$.

Eigenvalue growth of the discrete Hodge Laplacian across dimensions

• Authors: Philipp Bartmann; Matthias Keller
• arXiv: 2608.11170
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)

We prove several bounds on the largest and smallest eigenvalues of the combinatorial Hodge Laplacian of a finite simplicial complex $\Sigma .$ As a consequence, we obtain new vanishing criteria for cohomology groups and confirm a conjecture of O on the dimensional monotonicity of the largest eigenvalue.

Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups II

• Authors: Simon Lentner; Svea Nora Mierach; Christoph Schweigert; Yorck Sommerhaeuser
• arXiv: 2608.11193
• Subjects: Quantum Algebra (math.QA); Mathematical Physics (math-ph); Geometric Topology (math.GT); Rings and Algebras (math.RA); Representation Theory (math.RT)
• Comments: 105 pages, uses tikz-cd package

In the first part of this work, we have, for a not necessarily semisimple modular category, generalized the action of the mapping class groups of surfaces on the spaces of conformal blocks to the so-called derived block spaces. In the second part presented here, we compute this action explicitly in the case of Drinfel'd doubles of finite groups over fields of positive characteristic. To do that, we connect Lyubashenko's approach to mapping class group representations with the theory of representation varieties. In this way, we are able to show that the mapping class group representations on the derived block spaces are in general different from those on the ordinary block spaces.