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Rational Singularities for Moment Maps of Totally Negative Quivers.

Tanguy Vernet1

  • 1Institute of Science and Technology Austria, Hausel Group, Klosterneuburg, Austria.

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|March 24, 2026
PubMed
Summary

We show that totally negative quiver moment maps have rational singularities, extending this to moduli spaces in 2-Calabi-Yau categories. This research has arithmetic applications, including generalized jet counts and p-adic volumes.

Keywords:
Jet schemesQuiver moment mapsRational singularitiesp-adic integration

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Area of Science:

  • Algebraic Geometry
  • Representation Theory
  • Number Theory

Background:

  • Rational singularities are a key concept in algebraic geometry, influencing the study of moduli spaces.
  • Moment maps and quiver representations are fundamental in understanding algebraic structures.
  • 2-Calabi-Yau categories provide a rich framework for studying homological mirror symmetry and related invariants.

Purpose of the Study:

  • To prove that the zero-fiber of the moment map of a totally negative quiver possesses rational singularities.
  • To extend the property of rational singularities to moduli spaces of objects in 2-Calabi-Yau categories.
  • To explore arithmetic applications related to quiver moment maps and moduli spaces.

Main Methods:

  • Generalizing dimension bounds on jet spaces, building upon Budur's work.
  • Transferring the rational singularities property using recent advancements by Davison.
  • Applying techniques from number theory and representation theory to analyze moduli spaces.

Main Results:

  • The zero-fiber of the moment map for a totally negative quiver is proven to have rational singularities.
  • The rational singularities property is successfully transferred to moduli spaces of objects in 2-Calabi-Yau categories.
  • Generalizations of Wyss's results on the asymptotic behavior of jet counts over finite fields are established.

Conclusions:

  • The study establishes rational singularities for specific quiver-related moduli spaces.
  • Arithmetic applications include new insights into jet counts and the interpretation of moduli spaces via p-adic volumes.
  • This work bridges algebraic geometry, representation theory, and number theory, opening avenues for further research.