Almost mathematics, Persistence module, and Tamarkin category
Tatsuki Kuwagaki, Bingyu Zhang
TL;DR
This work establishes a unified higher-categorical framework linking three major theories used in symplectic geometry—Novikov-ring modules, persistence modules, and Tamarkin categories—through almost mathematics. It develops 1D and higher-dimensional APT correspondences, extends to global toric settings via Novikov toric varieties and infinite root stacks, and proves microlocal cut-off and fan-cutoff results that enable gluing across cones and fans. A key contribution is the global toric APT that ties constructible/ Tamarkin-type sheaves to almost quasi-coherent sheaves on Novikov spaces, enabling a sheaf–Fukaya correspondence and a novel form of homological mirror symmetry over the Novikov ring for toric varieties. The framework further yields enhanced persistence theory, Tamarkin torsion, and continuous K-theory, offering a robust higher-algebraic perspective on persistent homology and its symplectic geometry applications, including conjectural Novikov HMS for log Calabi–Yau varieties.
Abstract
We precisely uniform 3 theories that are widely used for symplectic geometers: (Almost) modules over Novikov ring, Persistence module, and Tamarkin category. Along with our method, we also give a neat understanding and language for the related results, in particular, Vaintrob's Novikov/log-perfectoid mirror symmetry for Novikov toric varieties. The results of this paper can also be treated as a study of persistent homology from a higher algebra point of view. Some of our results are shown in the literature, but our method is higher-categorical and sophisticated. As applications, we discuss a Novikov ring coefficient homological mirror symmetry for toric varieties and propose a conjecture for Novikov ring coefficient homological mirror symmetry for log Calabi-Yau varieties.