Non-commutative deformations of perverse coherent sheaves and rational curves

Author:
Yujiro Kawamata

Journal:
J. Algebraic Geom. **32** (2023), 59-91

DOI:
https://doi.org/10.1090/jag/805

Published electronically:
May 17, 2022

Full-text PDF

Abstract |
References |
Additional Information

Abstract: We consider non-commutative deformations of sheaves on algebraic varieties. We develop some tools to determine parameter algebras of versal non-commutative deformations for partial simple collections and the structure sheaves of smooth rational curves. We apply them to universal flopping contractions of length $2$ and higher. We confirm Donovan-Wemyss conjecture in the case of deformations of Lauferβs flops.

References
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*Quivers from matrix factorizations*, Comm. Math. Phys. **313** (2012), no.Β 3, 607β633. MR **2945618**, DOI 10.1007/s00220-012-1520-1
- A. I. Bondal,
*Representations of associative algebras and coherent sheaves*, Izv. Akad. Nauk SSSR Ser. Mat. **53** (1989), no.Β 1, 25β44 (Russian); English transl., Math. USSR-Izv. **34** (1990), no.Β 1, 23β42. MR **992977**, DOI 10.1070/IM1990v034n01ABEH000583
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*Gopakumar-Vafa invariants do not determine flops*, Comm. Math. Phys. **361** (2018), no.Β 1, 143β154. MR **3825938**, DOI 10.1007/s00220-017-3038-z
- Tom Bridgeland,
*Flops and derived categories*, Invent. Math. **147** (2002), no.Β 3, 613β632. MR **1893007**, DOI 10.1007/s002220100185
- Carina Curto and David R. Morrison,
*Threefold flops via matrix factorization*, J. Algebraic Geom. **22** (2013), no.Β 4, 599β627. MR **3084719**, DOI 10.1090/S1056-3911-2013-00633-5
- Will Donovan and Michael Wemyss,
*Noncommutative deformations and flops*, Duke Math. J. **165** (2016), no.Β 8, 1397β1474. MR **3504176**, DOI 10.1215/00127094-3449887
- David Eisenbud,
*Homological algebra on a complete intersection, with an application to group representations*, Trans. Amer. Math. Soc. **260** (1980), no.Β 1, 35β64. MR **570778**, DOI 10.1090/S0002-9947-1980-0570778-7
- Zheng Hua,
*Contraction algebra and singularity of three-dimensional flopping contraction*, Math. Z. **290** (2018), no.Β 1-2, 431β443. MR **3848439**, DOI 10.1007/s00209-017-2024-7
- Zheng Hua and Yukinobu Toda,
*Contraction algebra and invariants of singularities*, Int. Math. Res. Not. IMRN **10** (2018), 3173β3198. MR **3805201**, DOI 10.1093/imrn/rnw333
- Joseph Karmazyn,
*The length classification of threefold flops via noncommutative algebras*, Adv. Math. **343** (2019), 393β447. MR **3883210**, DOI 10.1016/j.aim.2018.11.023
- Sheldon Katz,
*Genus zero Gopakumar-Vafa invariants of contractible curves*, J. Differential Geom. **79** (2008), no.Β 2, 185β195. MR **2420017**
- Sheldon Katz and David R. Morrison,
*Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups*, J. Algebraic Geom. **1** (1992), no.Β 3, 449β530. MR **1158626**
- Yujiro Kawamata,
*General hyperplane sections of nonsingular flops in dimension $3$*, Math. Res. Lett. **1** (1994), no.Β 1, 49β52. MR **1258489**, DOI 10.4310/MRL.1994.v1.n1.a6
- Yujiro Kawamata,
*On multi-pointed non-commutative deformations and Calabi-Yau threefolds*, Compos. Math. **154** (2018), no.Β 9, 1815β1842. MR **3867285**, DOI 10.1112/s0010437x18007248
- Yujiro Kawamata,
*Non-commutative deformations of simple objects in a category of perverse coherent sheaves*, Selecta Math. (N.S.) **26** (2020), no.Β 3, Paper No. 43, 22. MR **4117994**, DOI 10.1007/s00029-020-00570-w
- Yujiro Kawamata,
*On non-commutative formal deformations of coherent sheaves on an algebraic variety*, EMS Surv. Math. Sci. **8** (2021), no.Β 1-2, 237β263. MR **4307209**, DOI 10.4171/emss/49
- Henry B. Laufer,
*On $\textbf {C}P^{1}$ as an exceptional set*, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp.Β 261β275. MR **627762**
- John N. Mather and Stephen S. T. Yau,
*Classification of isolated hypersurface singularities by their moduli algebras*, Invent. Math. **69** (1982), no.Β 2, 243β251. MR **674404**, DOI 10.1007/BF01399504
- Henry C. Pinkham,
*Factorization of birational maps in dimension $3$*, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp.Β 343β371. MR **713260**
- Miles Reid,
*Minimal models of canonical $3$-folds*, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp.Β 131β180. MR **715649**, DOI 10.2969/aspm/00110131
- Jeremy Rickard,
*Morita theory for derived categories*, J. London Math. Soc. (2) **39** (1989), no.Β 3, 436β456. MR **1002456**, DOI 10.1112/jlms/s2-39.3.436
- Yukinobu Toda,
*Non-commutative width and Gopakumar-Vafa invariants*, Manuscripta Math. **148** (2015), no.Β 3-4, 521β533. MR **3414491**, DOI 10.1007/s00229-015-0760-8
- Michel Van den Bergh,
*Three-dimensional flops and noncommutative rings*, Duke Math. J. **122** (2004), no.Β 3, 423β455. MR **2057015**, DOI 10.1215/S0012-7094-04-12231-6
- Michel Van den Bergh,
*Calabi-Yau algebras and superpotentials*, Selecta Math. (N.S.) **21** (2015), no.Β 2, 555β603. MR **3338683**, DOI 10.1007/s00029-014-0166-6
- O. van Garderen,
*Donaldson-Thomas invariants of length 2 flops*, arXiv:2008.02591, 2020.

References
- Paul S. Aspinwall and David R. Morrison,
*Quivers from matrix factorizations*, Comm. Math. Phys. **313** (2012), no. 3, 607β633. MR **2945618**, DOI 10.1007/s00220-012-1520-1
- A. I. Bondal,
*Representations of associative algebras and coherent sheaves*, Izv. Akad. Nauk SSSR Ser. Mat. **53** (1989), no. 1, 25β44 (Russian); English transl., Math. USSR-Izv. **34** (1990), no. 1, 23β42. MR **992977**, DOI 10.1070/IM1990v034n01ABEH000583
- Gavin Brown and Michael Wemyss,
*Gopakumar-Vafa invariants do not determine flops*, Comm. Math. Phys. **361** (2018), no. 1, 143β154. MR **3825938**, DOI 10.1007/s00220-017-3038-z
- Tom Bridgeland,
*Flops and derived categories*, Invent. Math. **147** (2002), no. 3, 613β632. MR **1893007**, DOI 10.1007/s002220100185
- Carina Curto and David R. Morrison,
*Threefold flops via matrix factorization*, J. Algebraic Geom. **22** (2013), no. 4, 599β627. MR **3084719**, DOI 10.1090/S1056-3911-2013-00633-5
- Will Donovan and Michael Wemyss,
*Noncommutative deformations and flops*, Duke Math. J. **165** (2016), no. 8, 1397β1474. MR **3504176**, DOI 10.1215/00127094-3449887
- David Eisenbud,
*Homological algebra on a complete intersection, with an application to group representations*, Trans. Amer. Math. Soc. **260** (1980), no. 1, 35β64. MR **570778**, DOI 10.2307/1999875
- Zheng Hua,
*Contraction algebra and singularity of three-dimensional flopping contraction*, Math. Z. **290** (2018), no. 1-2, 431β443. MR **3848439**, DOI 10.1007/s00209-017-2024-7
- Zheng Hua and Yukinobu Toda,
*Contraction algebra and invariants of singularities*, Int. Math. Res. Not. IMRN **10** (2018), 3173β3198. MR **3805201**, DOI 10.1093/imrn/rnw333
- Joseph Karmazyn,
*The length classification of threefold flops via noncommutative algebras*, Adv. Math. **343** (2019), 393β447. MR **3883210**, DOI 10.1016/j.aim.2018.11.023
- Sheldon Katz,
*Genus zero Gopakumar-Vafa invariants of contractible curves*, J. Differential Geom. **79** (2008), no. 2, 185β195. MR **2420017**
- Sheldon Katz and David R. Morrison,
*Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups*, J. Algebraic Geom. **1** (1992), no. 3, 449β530. MR **1158626**
- Yujiro Kawamata,
*General hyperplane sections of nonsingular flops in dimension $3$*, Math. Res. Lett. **1** (1994), no. 1, 49β52. MR **1258489**, DOI 10.4310/MRL.1994.v1.n1.a6
- Yujiro Kawamata,
*On multi-pointed non-commutative deformations and Calabi-Yau threefolds*, Compos. Math. **154** (2018), no. 9, 1815β1842. MR **3867285**, DOI 10.1112/s0010437x18007248
- Yujiro Kawamata,
*Non-commutative deformations of simple objects in a category of perverse coherent sheaves*, Selecta Math. (N.S.) **26** (2020), no. 3, Paper No. 43, 22. MR **4117994**, DOI 10.1007/s00029-020-00570-w
- Yujiro Kawamata,
*On non-commutative formal deformations of coherent sheaves on an algebraic variety*, EMS Surv. Math. Sci. **8** (2021), no. 1-2, 237β263. MR **4307209**, DOI 10.4171/emss/49
- Henry B. Laufer,
*On ${\mathbf C}P^{1}$ as an exceptional set*, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 261β275. MR **627762**
- John N. Mather and Stephen S. T. Yau,
*Classification of isolated hypersurface singularities by their moduli algebras*, Invent. Math. **69** (1982), no. 2, 243β251. MR **674404**, DOI 10.1007/BF01399504
- Henry C. Pinkham,
*Factorization of birational maps in dimension $3$*, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 343β371. MR **713260**
- Miles Reid,
*Minimal models of canonical $3$-folds*, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 131β180. MR **715649**, DOI 10.2969/aspm/00110131
- Jeremy Rickard,
*Morita theory for derived categories*, J. London Math. Soc. (2) **39** (1989), no. 3, 436β456. MR **1002456**, DOI 10.1112/jlms/s2-39.3.436
- Yukinobu Toda,
*Non-commutative width and Gopakumar-Vafa invariants*, Manuscripta Math. **148** (2015), no. 3-4, 521β533. MR **3414491**, DOI 10.1007/s00229-015-0760-8
- Michel Van den Bergh,
*Three-dimensional flops and noncommutative rings*, Duke Math. J. **122** (2004), no. 3, 423β455. MR **2057015**, DOI 10.1215/S0012-7094-04-12231-6
- Michel Van den Bergh,
*Calabi-Yau algebras and superpotentials*, Selecta Math. (N.S.) **21** (2015), no. 2, 555β603. MR **3338683**, DOI 10.1007/s00029-014-0166-6
- O. van Garderen,
*Donaldson-Thomas invariants of length 2 flops*, arXiv:2008.02591, 2020.

Additional Information

**Yujiro Kawamata**

Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153-8914 Japan; Morningside Center of Mathematics, Chinese Academy of Sciences, Haidian District, Beijing 100190, Peopleβs Republic of China; Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Korea; and National Center for Theoretical Sciences, Mathematics Division, National Taiwan University, Taipei 106, Taiwan

MR Author ID:
99410

Email:
kawamata@ms.u-tokyo.ac.jp

Received by editor(s):
June 17, 2020

Published electronically:
May 17, 2022

Additional Notes:
This work was partly supported by JSPS Grant-in-Aid 16H02141.

Article copyright:
© Copyright 2022
University Press, Inc.