Geodesic
On the Abuaf–Ueda Flop via Non-Commutative Crepant Resolutions
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: derived category, non-commutative crepant resolution, flop, tilting bundle. 
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     author = {Wahei Hara},
     title = {On the {Abuaf{\textendash}Ueda} {Flop} via {Non-Commutative} {Crepant} {Resolutions}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a43/}
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Wahei Hara. On the Abuaf–Ueda Flop via Non-Commutative Crepant Resolutions. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a43/

[1] Bondal A., Orlov D., “Derived categories of coherent sheaves”, Proceedings of the International Congress of Mathematicians (Beijing, 2002), v. II, Higher Ed. Press, Beijing, 2002, 47–56, arXiv: math.AG/0206295 | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[4] Hara W., On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists, arXiv: 1706.04417 | MR

[5] Hara W., “Deformation of tilting-type derived equivalences for crepant resolutions”, Int. Math. Res. Not., 2020 (2020), 4062–4102, arXiv: 1709.09948 | DOI | MR | Zbl

[6] Hille L., Van den Bergh M., “Fourier–Mukai transforms”, Handbook of Tilting Theory, London Math. Soc. Lecture Note Ser., 332, Cambridge University Press, Cambridge, 2007, 147–177, arXiv: math.AG/0402043 | DOI | MR

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[12] Kawamata Y., “$D$-equivalence and $K$-equivalence”, J. Differential Geom., 61 (2002), 147–171, arXiv: math.AG/0205287 | DOI | MR | Zbl

[13] Kuznetsov A., “Hyperplane sections and derived categories”, Izv. Math., 70 (2006), 447–547, arXiv: math.AG/0503700 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[15] Li D., “On certain K-equivalent birational maps”, Math. Z., 291 (2019), 959–969, arXiv: 1701.04054 | DOI | MR | Zbl

[16] Namikawa Y., “Mukai flops and derived categories”, J. Reine Angew. Math., 560 (2003), 65–76, arXiv: math.AG/0203287 | DOI | MR | Zbl

[17] Ottaviani G., “Spinor bundles on quadrics”, Trans. Amer. Math. Soc., 307 (1988), 301–316 | DOI | MR | Zbl

[18] Ottaviani G., “On Cayley bundles on the five-dimensional quadric”, Boll. Un. Mat. Ital. A (7), 4 (1990), 87–100 | MR | Zbl

[19] Segal E., “A new 5-fold flop and derived equivalence”, Bull. Lond. Math. Soc., 48 (2016), 533–538, arXiv: 1506.06999 | DOI | MR | Zbl

[20] Toda Y., Uehara H., “Tilting generators via ample line bundles”, Adv. Math., 223 (2010), 1–29, arXiv: 0804.4256 | DOI | MR | Zbl

[21] Ueda K., “$G_2$-Grassmannians and derived equivalences”, Manuscripta Math., 159 (2019), 549–559, arXiv: 1612.08443 | DOI | MR | Zbl

[22] Van den Bergh M., “Non-commutative crepant resolutions”, The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, 749–770, arXiv: math.AG/0211064 | DOI | MR | Zbl

[23] Weyman J., Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, 149, Cambridge University Press, Cambridge, 2003 | DOI | MR | Zbl

[24] Weyman J., Zhao G., Noncommutative desingularization of orbit closures for some representations of ${\rm GL}_n$, arXiv: 1204.0488