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

SIGMA 17 (2021), 044, 22 pages      arXiv:1812.10688
Contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji Saito for his 77th birthday

On the Abuaf-Ueda Flop via Non-Commutative Crepant Resolutions

Wahei Hara
The Mathematics and Statistics Building, University of Glasgow, University Place, Glasgow, G12 8QQ, UK

Received September 30, 2020, in final form April 18, 2021; Published online April 30, 2021

The Abuaf-Ueda flop is a 7-dimensional flop related to $G_2$ homogeneous spaces. The derived equivalence for this flop was first proved by Ueda using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence using tilting bundles. Our proof also shows the existence of a non-commutative crepant resolution of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over this non-commutative crepant resolution.

Key words: derived category; non-commutative crepant resolution; flop; tilting bundle.

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  1. Bondal A., Orlov D., Derived categories of coherent sheaves, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 47-56, arXiv:math.AG/0206295.
  2. Buchweitz R.O., Leuschke G.J., Van den Bergh M., Non-commutative desingularization of determinantal varieties I, Invent. Math. 182 (2010), 47-115, arXiv:0911.2659.
  3. Hara W., Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops, Adv. Math. 318 (2017), 355-410, arXiv:1704.07192.
  4. Hara W., On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists, arXiv:1706.04417.
  5. Hara W., Deformation of tilting-type derived equivalences for crepant resolutions, Int. Math. Res. Not. 2020 (2020), 4062-4102, arXiv:1709.09948.
  6. Hille L., Van den Bergh M., Fourier-Mukai transforms, in Handbook of Tilting Theory, London Math. Soc. Lecture Note Ser., Vol. 332, Cambridge University Press, Cambridge, 2007, 147-177, arXiv:math.AG/0402043.
  7. Ito A., Miura M., Okawa S., Ueda K., Calabi-Yau complete intersections in exceptional Grassmannians, arXiv:1606.04076.
  8. Ito A., Miura M., Okawa S., Ueda K., The class of the affine line is a zero divisor in the Grothendieck ring: via $G_2$-Grassmannians, J. Algebraic Geom. 28 (2019), 245-250, arXiv:1606.04210.
  9. Iyama O., Wemyss M., Singular derived categories of $\mathbb{Q}$-factorial terminalizations and maximal modification algebras, Adv. Math. 261 (2014), 85-121, arXiv:1108.4518.
  10. Kanemitsu A., Mukai pairs and simple K-equivalence, arXiv:1812.05392.
  11. Karmazyn J., Quiver GIT for varieties with tilting bundles, Manuscripta Math. 154 (2017), 91-128, arXiv:1407.5005.
  12. Kawamata Y., $D$-equivalence and $K$-equivalence, J. Differential Geom. 61 (2002), 147-171, arXiv:math.AG/0205287.
  13. Kuznetsov A., Hyperplane sections and derived categories, Izv. Math. 70 (2006), 447-547, arXiv:math.AG/0503700.
  14. Kuznetsov A., Derived equivalence of Ito-Miura-Okawa-Ueda Calabi-Yau 3-folds, J. Math. Soc. Japan 70 (2018), 1007-1013, arXiv:1611.08386.
  15. Li D., On certain K-equivalent birational maps, Math. Z. 291 (2019), 959-969, arXiv:1701.04054.
  16. Namikawa Y., Mukai flops and derived categories, J. Reine Angew. Math. 560 (2003), 65-76, arXiv:math.AG/0203287.
  17. Ottaviani G., Spinor bundles on quadrics, Trans. Amer. Math. Soc. 307 (1988), 301-316.
  18. Ottaviani G., On Cayley bundles on the five-dimensional quadric, Boll. Un. Mat. Ital. A (7) 4 (1990), 87-100.
  19. Segal E., A new 5-fold flop and derived equivalence, Bull. Lond. Math. Soc. 48 (2016), 533-538, arXiv:1506.06999.
  20. Toda Y., Uehara H., Tilting generators via ample line bundles, Adv. Math. 223 (2010), 1-29, arXiv:0804.4256.
  21. Ueda K., $G_2$-Grassmannians and derived equivalences, Manuscripta Math. 159 (2019), 549-559, arXiv:1612.08443.
  22. Van den Bergh M., Non-commutative crepant resolutions, in The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, 749-770, arXiv:math.AG/0211064.
  23. Weyman J., Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, Vol. 149, Cambridge University Press, Cambridge, 2003.
  24. Weyman J., Zhao G., Noncommutative desingularization of orbit closures for some representations of ${\rm GL}_n$, arXiv:1204.0488.

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