Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 16 (2020), 096, 22 pages      arXiv:1911.03288      https://doi.org/10.3842/SIGMA.2020.096

Torus-Equivariant Chow Rings of Quiver Moduli

Hans Franzen
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany

Received March 14, 2020, in final form September 16, 2020; Published online September 30, 2020

Abstract
We compute rational equivariant Chow rings with respect to a torus of quiver moduli spaces. We derive a presentation in terms of generators and relations, use torus localization to identify it as a subring of the Chow ring of the fixed point locus, and we compare the two descriptions.

Key words: torus actions; equivariant Chow rings; torus localization; quiver moduli.

pdf (501 kb)   tex (28 kb)  

References

  1. Altmann K., Hille L., Strong exceptional sequences provided by quivers, Algebr. Represent. Theory 2 (1999), 1-17.
  2. Assem I., Skowroński A., Simson D., Elements of the representation theory of associative algebras, Vol. 1, Techniques of representation theory, London Mathematical Society Student Texts, Vol. 65, Cambridge University Press, Cambridge, 2006.
  3. Brion M., Equivariant Chow groups for torus actions, Transform. Groups 2 (1997), 225-267.
  4. Chang T., Skjelbred T., The topological Schur lemma and related results, Ann. of Math. 100 (1974), 307-321.
  5. Edidin D., Graham W., Equivariant intersection theory, Invent. Math. 131 (1998), 595-634, arXiv:alg-geom/9609018.
  6. Franzen H., Reineke M., Semistable Chow-Hall algebras of quivers and quantized Donaldson-Thomas invariants, Algebra Number Theory 12 (2018), 1001-1025, arXiv:1512.03748.
  7. Franzen H., Reineke M., Sabatini S., Fano quiver moduli, arXiv:2001.10556.
  8. Goresky M., Kottwitz R., MacPherson R., Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25-83.
  9. King A.D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 515-530.
  10. Kresch A., Cycle groups for Artin stacks, Invent. Math. 138 (1999), 495-536, arXiv:math.AG/9810166.
  11. Le Bruyn L., Procesi C., Semisimple representations of quivers, Trans. Amer. Math. Soc. 317 (1990), 585-598.
  12. Molina Rojas L.A., Vistoli A., On the Chow rings of classifying spaces for classical groups, Rend. Sem. Mat. Univ. Padova 116 (2006), 271-298, arXiv:math.AG/0505560.
  13. Mumford D., Fogarty J., Kirwan F., Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2), Vol. 34, Springer-Verlag, Berlin, 1994.
  14. Pabiniak M., Sabatini S., Canonical bases for the equivariant cohomology and K-theory rings of symplectic toric manifolds, J. Symplectic Geom. 16 (2018), 1117-1165, arXiv:1503.04730.
  15. Reineke M., Moduli of representations of quivers, in Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, 589-637, arXiv:0802.2147.
  16. Reineke M., Stoppa J., Weist T., MPS degeneration formula for quiver moduli and refined GW/Kronecker correspondence, Geom. Topol. 16 (2012), 2097-2134, arXiv:1110.4847.
  17. Reineke M., Weist T., Refined GW/Kronecker correspondence, Math. Ann. 355 (2013), 17-56, arXiv:1103.5283.
  18. Rupel D., Weist T., Cell decompositions for rank two quiver Grassmannians, Math. Z. 295 (2020), 993-1038, arXiv:1803.06590.
  19. Weist T., Localization in quiver moduli spaces, Represent. Theory 17 (2013), 382-425, arXiv:0903.5442.

Previous article  Next article  Contents of Volume 16 (2020)