Twisted Fourier-Mukai number of a 𝐾3 surface

Ma, Shouhei

doi:10.1090/S0002-9947-09-04963-0

Twisted Fourier-Mukai number of a $K3$ surface
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Trans. Amer. Math. Soc. 362 (2010), 537-552 Request permission

Abstract:

We give a counting formula for the twisted Fourier-Mukai partners of a projective $K3$ surface. As an application, we describe all twisted Fourier-Mukai partners of a projective $K3$ surface of Picard number $1$.

References

Căldăraru, A, Derived Categories of Twisted Sheaves on Calabi-Yau Manifolds. Ph.D. thesis, Cornell University, 2000.
Andrei Căldăraru, Nonfine moduli spaces of sheaves on $K3$ surfaces, Int. Math. Res. Not. 20 (2002), 1027–1056. MR 1902629, DOI 10.1155/S1073792802109093
Alberto Canonaco and Paolo Stellari, Twisted Fourier-Mukai functors, Adv. Math. 212 (2007), no. 2, 484–503. MR 2329310, DOI 10.1016/j.aim.2006.10.010
Shinobu Hosono, Bong H. Lian, Keiji Oguiso, and Shing-Tung Yau, Fourier-Mukai partners of a $K3$ surface of Picard number one, Vector bundles and representation theory (Columbia, MO, 2002) Contemp. Math., vol. 322, Amer. Math. Soc., Providence, RI, 2003, pp. 43–55. MR 1987738, DOI 10.1090/conm/322/05678
Shinobu Hosono, Bong H. Lian, Keiji Oguiso, and Shing-Tung Yau, Fourier-Mukai number of a K3 surface, Algebraic structures and moduli spaces, CRM Proc. Lecture Notes, vol. 38, Amer. Math. Soc., Providence, RI, 2004, pp. 177–192. MR 2096145, DOI 10.1090/crmp/038/08
Daniel Huybrechts and Paolo Stellari, Equivalences of twisted $K3$ surfaces, Math. Ann. 332 (2005), no. 4, 901–936. MR 2179782, DOI 10.1007/s00208-005-0662-2
Daniel Huybrechts and Paolo Stellari, Proof of Căldăraru’s conjecture. Appendix: “Moduli spaces of twisted sheaves on a projective variety” [in Moduli spaces and arithmetic geometry, 1–30, Math. Soc. Japan, Tokyo, 2006; MR2306170] by K. Yoshioka, Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math., vol. 45, Math. Soc. Japan, Tokyo, 2006, pp. 31–42. MR 2310257, DOI 10.2969/aspm/04510031
S. Mukai, On the moduli space of bundles on $K3$ surfaces. I, Vector bundles on algebraic varieties (Bombay, 1984) Tata Inst. Fund. Res. Stud. Math., vol. 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 341–413. MR 893604
V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
D. O. Orlov, Equivalences of derived categories and $K3$ surfaces, J. Math. Sci. (New York) 84 (1997), no. 5, 1361–1381. Algebraic geometry, 7. MR 1465519, DOI 10.1007/BF02399195
I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli’s theorem for algebraic surfaces of type $\textrm {K}3$, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572 (Russian). MR 0284440
Paolo Stellari, Some remarks about the FM-partners of $K3$ surfaces with Picard numbers 1 and 2, Geom. Dedicata 108 (2004), 1–13. MR 2112663, DOI 10.1007/s10711-004-9291-7
Andrei N. Todorov, Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of $K3$ surfaces, Invent. Math. 61 (1980), no. 3, 251–265. MR 592693, DOI 10.1007/BF01390067
K\B{o}ta Yoshioka, Moduli spaces of twisted sheaves on a projective variety, Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math., vol. 45, Math. Soc. Japan, Tokyo, 2006, pp. 1–30. MR 2306170, DOI 10.2969/aspm/04510001
Géométrie des surfaces $K3$: modules et périodes, Société Mathématique de France, Paris, 1985 (French). Papers from the seminar held in Palaiseau, October 1981–January 1982; Astérisque No. 126 (1985) (1985). MR 785216