Geometry & Topology Volume 23, issue 1 (2019)

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Motivic hyper-Kähler resolution conjecture, I: Generalized Kummer varieties

Lie Fu, Zhiyu Tian and Charles Vial

Geometry & Topology 23 (2019) 427–492

DOI: 10.2140/gt.2019.23.427

Bibliography

1	D Abramovich, T Graber, A Vistoli, Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math. 130 (2008) 1337 MR2450211
2	D Abramovich, A Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002) 27 MR1862797
3	Y André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), 17, Soc. Mat. de France (2004) MR2115000
4	V V Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonical singularities, from: "Integrable systems and algebraic geometry" (editors M H Saito, Y Shimizu, K Ueno), World Sci. (1998) 1 MR1672108
5	V V Batyrev, D I Dais, Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry, Topology 35 (1996) 901 MR1404917
6	A Beauville, Quelques remarques sur la transformation de Fourier dans l’anneau de Chow d’une variété abélienne, from: "Algebraic geometry" (editors M Raynaud, T Shioda), Lecture Notes in Math. 1016, Springer (1983) 238 MR726428
7	A Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983) 755 MR730926
8	A Beauville, Sur l’anneau de Chow d’une variété abélienne, Math. Ann. 273 (1986) 647 MR826463
9	A Beauville, Symplectic singularities, Invent. Math. 139 (2000) 541 MR1738060
10	A Beauville, On the splitting of the Bloch–Beilinson filtration, from: "Algebraic cycles and motives, II", London Math. Soc. Lecture Note Ser. 344, Cambridge Univ. Press (2007) 38 MR2187148
11	A Beauville, C Voisin, On the Chow ring of a K3 surface, J. Algebraic Geom. 13 (2004) 417 MR2047674
12	J Briançon, Description de Hilbⁿℂ{x,y}, Invent. Math. 41 (1977) 45 MR0457432
13	M Brion, B Fu, Symplectic resolutions for conical symplectic varieties, Int. Math. Res. Not. 2015 (2015) 4335 MR3356756
14	M Britze, On the cohomology of generalized Kummer varieties, PhD thesis, Universität zu Köln (2002)
15	J Bryan, T Graber, The crepant resolution conjecture, from: "Algebraic geometry, I" (editors D Abramovich, A Bertram, L Katzarkov, R Pandharipande, M Thaddeus), Proc. Sympos. Pure Math. 80, Amer. Math. Soc. (2009) 23 MR2483931
16	M A A de Cataldo, L Migliorini, The Chow groups and the motive of the Hilbert scheme of points on a surface, J. Algebra 251 (2002) 824 MR1919155
17	M A A de Cataldo, L Migliorini, The Chow motive of semismall resolutions, Math. Res. Lett. 11 (2004) 151 MR2067464
18	W Chen, Y Ruan, Orbifold Gromov–Witten theory, from: "Orbifolds in mathematics and physics" (editors A Adem, J Morava, Y Ruan), Contemp. Math. 310, Amer. Math. Soc. (2002) 25 MR1950941
19	W Chen, Y Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004) 1 MR2104605
20	T Coates, Y Ruan, Quantum cohomology and crepant resolutions: a conjecture, Ann. Inst. Fourier (Grenoble) 63 (2013) 431 MR3112518
21	P Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Inst. Hautes Études Sci. Publ. Math. 35 (1968) 259 MR0244265
22	C Deninger, J Murre, Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math. 422 (1991) 201 MR1133323
23	L Dixon, J A Harvey, C Vafa, E Witten, Strings on orbifolds, Nuclear Phys. B 261 (1985) 678 MR818423
24	L Dixon, J Harvey, C Vafa, E Witten, Strings on orbifolds, II, Nuclear Phys. B 274 (1986) 285 MR851703
25	D Edidin, T J Jarvis, T Kimura, Logarithmic trace and orbifold products, Duke Math. J. 153 (2010) 427 MR2667422
26	B Fantechi, L Göttsche, Orbifold cohomology for global quotients, Duke Math. J. 117 (2003) 197 MR1971293
27	J Fogarty, Algebraic families on an algebraic surface, Amer. J. Math 90 (1968) 511 MR0237496
28	B Fu, Symplectic resolutions for nilpotent orbits, Invent. Math. 151 (2003) 167 MR1943745
29	L Fu, Beauville–Voisin conjecture for generalized Kummer varieties, Int. Math. Res. Not. 2015 (2015) 3878 MR3356741
30	L Fu, Z Tian, Motivic hyper-Kähler resolution conjecture, II : Hilbert schemes of K3 surfaces, preprint (2017)
31	L Fu, Z Tian, Motivic multiplicative McKay correspondence for surfaces, Manuscripta Mathematica (2018)
32	W Fulton, Intersection theory, 2, Springer (1998) MR1644323
33	M Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001) 941 MR1839919
34	Y Ito, I Nakamura, McKay correspondence and Hilbert schemes, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996) 135 MR1420598
35	U Jannsen, Motivic sheaves and filtrations on Chow groups, from: "Motives, I" (editors U Jannsen, S Kleiman, J P Serre), Proc. Sympos. Pure Math. 55, Amer. Math. Soc. (1994) 245 MR1265533
36	T J Jarvis, R Kaufmann, T Kimura, Stringy K–theory and the Chern character, Invent. Math. 168 (2007) 23 MR2285746
37	Z Jiang, Q Yin, On the Chow ring of certain rational cohomology tori, C. R. Math. Acad. Sci. Paris 355 (2017) 571 MR3650385
38	T Kimura, Orbifold cohomology reloaded, from: "Toric topology" (editors M Harada, Y Karshon, M Masuda, T Panov), Contemp. Math. 460, Amer. Math. Soc. (2008) 231 MR2428359
39	G Kings, Higher regulators, Hilbert modular surfaces, and special values of L–functions, Duke Math. J. 92 (1998) 61 MR1609325
40	M Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999) 157 MR1681097
41	M Lehn, C Sorger, The cup product of Hilbert schemes for K3 surfaces, Invent. Math. 152 (2003) 305 MR1974889
42	E Lupercio, M Poddar, The global McKay–Ruan correspondence via motivic integration, Bull. London Math. Soc. 36 (2004) 509 MR2069013
43	D Mumford, Towards an enumerative geometry of the moduli space of curves, from: "Arithmetic and geometry, II" (editors M Artin, J Tate), Progr. Math. 36, Birkhäuser (1983) 271 MR717614
44	H Nakajima, Lectures on Hilbert schemes of points on surfaces, 18, Amer. Math. Soc. (1999) MR1711344
45	I Nakamura, Hilbert schemes of abelian group orbits, J. Algebraic Geom. 10 (2001) 757 MR1838978
46	Y Namikawa, Extension of 2–forms and symplectic varieties, J. Reine Angew. Math. 539 (2001) 123 MR1863856
47	M A Nieper-Wisskirchen, Twisted cohomology of the Hilbert schemes of points on surfaces, Doc. Math. 14 (2009) 749 MR2578804
48	P O’Sullivan, Algebraic cycles on an abelian variety, J. Reine Angew. Math. 654 (2011) 1 MR2795752
49	M Reid, La correspondance de McKay, from: "Séminaire Bourbaki, 1999/2000", Astérisque 276, Soc. Mat. de France (2002) 53
50	U Rieß, On Beauville’s conjectural weak splitting property, Int. Math. Res. Not. 2016 (2016) 6133 MR3579961
51	Y Ruan, Stringy geometry and topology of orbifolds, from: "Symposium in honor of C H Clemens" (editors A Bertram, J A Carlson, H Kley), Contemp. Math. 312, Amer. Math. Soc. (2002) 187 MR1941583
52	Y Ruan, The cohomology ring of crepant resolutions of orbifolds, from: "Gromov–Witten theory of spin curves and orbifolds" (editors T J Jarvis, T Kimura, A Vaintrob), Contemp. Math. 403, Amer. Math. Soc. (2006) 117 MR2234886
53	M Shen, C Vial, The Fourier transform for certain hyperKähler fourfolds, 1139, Amer. Math. Soc. (2016) MR3460114
54	B Uribe, Orbifold cohomology of the symmetric product, Comm. Anal. Geom. 13 (2005) 113 MR2154668
55	C Vial, On the motive of some hyperKähler varieties, J. Reine Angew. Math. 725 (2017) 235 MR3630123
56	C Vial, Remarks on motives of abelian type, Tohoku Math. J. 69 (2017) 195 MR3682163
57	C Voisin, Hodge theory and complex algebraic geometry, II, 77, Cambridge Univ. Press (2003) MR1997577
58	C Voisin, On the Chow ring of certain algebraic hyper-Kähler manifolds, Pure Appl. Math. Q. 4 (2008) 613 MR2435839
59	C Voisin, Chow rings and decomposition theorems for families of K3 surfaces and Calabi–Yau hypersurfaces, Geom. Topol. 16 (2012) 433 MR2916291
60	C Voisin, Some new results on modified diagonals, Geom. Topol. 19 (2015) 3307 MR3447105
61	C A Weibel, The K–book : an introduction to algebraic K–theory, 145, Amer. Math. Soc. (2013) MR3076731
62	Z Xu, Algebraic cycles on a generalized Kummer variety, Int. Math. Res. Not. 2018 (2018) 932 MR3801451
63	T Yasuda, Twisted jets, motivic measures and orbifold cohomology, Compos. Math. 140 (2004) 396 MR2027195
64	Q Yin, Finite-dimensionality and cycles on powers of K3 surfaces, Comment. Math. Helv. 90 (2015) 503 MR3351754