arXiv subjects

Maximilian Schimpf

Publications and source records attributed to Maximilian Schimpf.

5 recordsLinked to original sources

arXiv2026

Coherent sheaves in logarithmic geometry

This paper introduces an abelian category of logarithmic coherent sheaves that arranges coherent sheaves across all expansions and root stacks of a simple normal crossing degeneration. Formally, logarithmic coherent sheaves are coherent sheaves in the full logarithmic étale topology. We develop a suite of tools that reduces the evaluation of the basic functors of homological algebra to the conventional calculation on a computable logarithmic alteration. A second paper will establish good properties of the associated logarithmic derived category. We thus offer a unified perspective on logarithmic moduli spaces of coherent sheaves: The logarithmic Quot spaces motivated by Maulik and Ranganathan's logarithmic Donaldson--Thomas theory, the logarithmic Picard group constructed by Molcho and Wise, and moduli spaces of logarithmic parabolic sheaves as developed by Borne, Talpo, and Vistoli. In establishing the connection with logarithmic Picard groups, we offer a new interpretation of chip firing as the combinatorial shadow to a logarithmic version of S-equivalence.

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arXiv2025

Grothendieck topologies with logarithmic modifications

Many concepts in log geometry are invariant under log blow-ups. To formalize this invariance, we introduce the m-open, m-\'etale, m-smooth, m-fppf, and m-fpqc topologies for fs log schemes. These refine the standard topologies from scheme theory by treating abstract log modifications as covers. For example, the m-\'etale topology is a subtopology of full log \'etale topology, characterized by a stronger lifting property than for log \'etale maps. Along the way, we identify and correct an error in the definition of the full log \'etale topology. We also prove a global integralization theorem by logarithmic blow-ups and use it to describe the m-open topos as a limit over log blow-ups. Finally, we characterize the sheaves on the m-type sites and connect the m-open site to Kato's valuative space.

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arXiv2025

Admissible covers and stable maps

Moduli spaces of admissible covers and stable maps of target curves give rise to cycles on $\overline{M}_{g,n}$. We prove a formula relating these cycles. It recovers both the Ekedahl-Lando-Shapiro-Vainshtein formula and the Gromov-Witten/Hurwitz correspondence, providing a cycle-theoretic refinement thereof. The formula is verified computationally in low-genus cases with the help of Johannes Schmitt.

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arXiv2024

Stable pairs on local curves and Bethe roots

We give an explicit formula for the descendent stable pair invariants of all (absolute) local curves in terms of certain power series called Bethe roots, which also appear in the physics/representation theory literature. We derive new explicit descriptions for the Bethe roots which are of independent interest. From this we derive rationality, functional equation and a characterization of poles for the full descendent stable pair theory of local curves as conjectured by Pandharipande and Pixton. We also sketch how our methods give a new approach to the spectrum of quantum multiplication on $\mathsf{Hilb}^n(\mathbf{C}^2)$.

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arXiv2023

Pandharipande-Thomas theory of elliptic threefolds, quasi-Jacobi forms and holomorphic anomaly equations

Let $π: X \to B$ be an elliptically fibered threefold satisfying $c_3(T_X \otimes ω_X)=0$. We conjecture that the $π$-relative generating series of Pandharipande-Thomas invariants of $X$ are quasi-Jacobi forms and satisfy two holomorphic anomaly equations. For elliptic Calabi-Yau threefolds our conjectures specialize to the Huang-Katz-Klemm conjecture. The proposed formulas constitute the first case of holomorphic anomaly equations in Pandharipande-Thomas theory. We prove our conjectures for the equivariant Pandharipande-Thomas theory of $\mathbb{C}^2 \times E$ when specialized to the anti-diagonal action. For $K3 \times \mathbb{C}$ we state reduced versions of our conjectures. As a corollary we find an explicit conjectural formula for the stationary theory generalizing the Katz-Klemm-Vafa formula for K3 surfaces. Further evidence is available for $\mathbb{P}^2 \times E$ based on earlier work of the second author. To deal with elliptic threefolds with $c_3(T_X \otimes ω_X) \neq 0$ we show that the moduli space of $π$-stable pairs is represented by a proper algebraic space. We conjecture that the associated $π$-stable pair invariants form quasi-Jacobi forms.

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