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Perverse schobers and Orlov equivalences.

Naoki Koseki1, Genki Ouchi2

  • 1The University of Liverpool, Mathematical Sciences Building, Liverpool, L69 7ZL UK.

European Journal of Mathematics
|May 3, 2023
PubMed
Summary

This study introduces perverse schobers, a new mathematical concept categorifying perverse sheaves. Examples are constructed on the Riemann sphere, linking mirror symmetry for Calabi-Yau hypersurfaces to intersection complexes.

Keywords:
Calabi–Yau hypersurfacesDerived factorization categoriesMirror symmetryPerverse schobers

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Area of Science:

  • Algebraic Geometry
  • Representation Theory
  • Mathematical Physics

Background:

  • Perverse sheaves are fundamental objects in algebraic geometry, used to study singularities and cohomology.
  • Perverse schobers, proposed by Kapranov-Schechtman, offer a higher categorical framework for understanding perverse sheaves.
  • Mirror symmetry connects different Calabi-Yau manifolds and has deep implications in string theory and algebraic geometry.

Purpose of the Study:

  • To construct concrete examples of perverse schobers.
  • To demonstrate the categorification of intersection complexes using perverse schobers.
  • To explore the connection between mirror symmetry for Calabi-Yau hypersurfaces and these novel mathematical structures.

Main Methods:

  • Construction of perverse schobers on the Riemann sphere.
  • Utilizing natural local systems derived from mirror symmetry constructions.
  • Leveraging the Orlov equivalence as a key theoretical tool.

Main Results:

  • Successful construction of specific examples of perverse schobers on the Riemann sphere.
  • Demonstration that these perverse schobers categorify intersection complexes of local systems.
  • Highlighting the crucial role of the Orlov equivalence in this categorification process.

Conclusions:

  • The study provides explicit examples of perverse schobers, validating their construction.
  • Perverse schobers offer a powerful new lens for studying mirror symmetry and related phenomena in algebraic geometry.
  • The Orlov equivalence is confirmed as a vital component in bridging these mathematical concepts.