Geometry & Topology Volume 30, issue 7 (2026)

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

Counting surfaces on Calabi–Yau $4$-folds, I: Foundations

Younghan Bae, Martijn Kool and Hyeonjun Park

Geometry & Topology 30 (2026) 2431–2564

DOI: 10.2140/gt.2026.30.2431

Bibliography

1	D Abramovich, M Olsson, A Vistoli, Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble) 58 (2008) 1057 MR2427954
2	V Alexeev, Moduli spaces M_g,n(W) for surfaces, from: "Higher-dimensional complex varieties" (editors M Andreatta, T Peternell), de Gruyter (1996) 1 MR1463171
3	J Alper, Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble) 63 (2013) 2349 MR3237451
4	J Alper, J Hall, D Rydh, A Luna étale slice theorem for algebraic stacks, Ann. of Math. 191 (2020) 675 MR4088350
5	D Aranha, P Pstragowski, The intrinsic normal cone for Artin stacks, Ann. Inst. Fourier (Grenoble) 74 (2024) 71 MR4748168
6	D Aranha, A A Khan, A Latyntsev, H Park, C Ravi, Localization theorems for algebraic stacks, preprint (2022) arXiv:2207.01652v1
7	D Arinkin, R Bezrukavnikov, Perverse coherent sheaves, Mosc. Math. J. 10 (2010) 3, 271 MR2668828
8	M F Atiyah, F Hirzebruch, Vector bundles and homogeneous spaces, from: "Proc. Sympos. Pure Math.", Amer. Math. Soc. (1961) 7 MR139181
9	Y Bae, H Park, Intersection theory of stacks, in preparation
10	Y Bae, M Kool, H Park, Counting surfaces on Calabi–Yau 4-folds, II : DT–PT₀ correspondence, preprint (2024) arXiv:2402.06526
11	R Bandiera, E Lepri, M Manetti, L_∞ liftings of semiregularity maps via Chern–Simons classes, Adv. Math. 435 (2023) MR4658349
12	A Bayer, Polynomial Bridgeland stability conditions and the large volume limit, Geom. Topol. 13 (2009) 2389 MR2515708
13	K Behrend, B Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45 MR1437495
14	O Ben-Bassat, C Brav, V Bussi, D Joyce, A “Darboux theorem” for shifted symplectic structures on derived Artin stacks, with applications, Geom. Topol. 19 (2015) 1287 MR3352237
15	D Ben-Zvi, D Nadler, Loop spaces and connections, J. Topol. 5 (2012) 377 MR2928082
16	D Ben-Zvi, J Francis, D Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010) 909 MR2669705
17	I Bena, J Blåbäck, M Graña, S Lüst, The tadpole problem, J. High Energy Phys. 2021 (2021) MR4368221
18	D Bergh, O M Schnürer, Decompositions of derived categories of gerbes and of families of Brauer–Severi varieties, Doc. Math. 26 (2021) 1465 MR4334847
19	A Blanc, Topological K-theory of complex noncommutative spaces, Compos. Math. 152 (2016) 489 MR3477639
20	S Bloch, Semi-regularity and deRham cohomology, Invent. Math. 17 (1972) 51 MR325613
21	F A Bogomolov, Hamiltonian Kählerian manifolds, Dokl. Akad. Nauk SSSR 243 (1978) 1101 MR514769
22	A Bojko, Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi–Yau 4-folds, preprint (2021) arXiv:2102.01056
23	P I Booth, R Brown, Spaces of partial maps, fibred mapping spaces and the compact–open topology, General Topology and Appl. 8 (1978) 181 MR482636
24	D Borisov, D Joyce, Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds, Geom. Topol. 21 (2017) 3231 MR3692967
25	E Bouaziz, I Grojnowski, A d-shifted Darboux theorem, preprint (2013) arXiv:1309.2197
26	A P Braun, R Valandro, G₄ flux, algebraic cycles and complex structure moduli stabilization, J. High Energy Phys. 2021 (2021) MR4257634
27	A P Braun, A Collinucci, R Valandro, G-flux in F-theory and algebraic cycles, Nuclear Phys. B 856 (2012) 129 MR2868605
28	C Brav, V Bussi, D Joyce, A Darboux theorem for derived schemes with shifted symplectic structure, J. Amer. Math. Soc. 32 (2019) 399 MR3904157
29	P Bressler, R Nest, B Tsygan, Riemann–Roch theorems via deformation quantization, preprint (1997) arXiv:alg-geom/9705014
30	T Bridgeland, Stability conditions on triangulated categories, Ann. of Math. 166 (2007) 317 MR2373143
31	R O Buchweitz, H Flenner, A semiregularity map for modules and applications to deformations, Compositio Math. 137 (2003) 135 MR1985003
32	R O Buchweitz, H Flenner, The global decomposition theorem for Hochschild (co-)homology of singular spaces via the Atiyah–Chern character, Adv. Math. 217 (2008) 243 MR2357327
33	D Calaque, T Pantev, B Toën, M Vaquié, G Vezzosi, Shifted Poisson structures and deformation quantization, J. Topol. 10 (2017) 483 MR3653319
34	Y Cao, M Kool, Zero-dimensional Donaldson–Thomas invariants of Calabi–Yau 4-folds, Adv. Math. 338 (2018) 601 MR3861713
35	Y Cao, M Kool, Curve counting and DT/PT correspondence for Calabi–Yau 4-folds, Adv. Math. 375 (2020) MR4135421
36	Y Cao, N C Leung, Donaldson-Thomas theory for Calabi–Yau 4-folds, preprint (2014) arXiv:1407.7659
37	Y Cao, Y Toda, Gopakumar–Vafa type invariants on Calabi–Yau 4-folds via descendent insertions, Comm. Math. Phys. 383 (2021) 281 MR4236066
38	Y Cao, Y Toda, Curve counting via stable objects in derived categories of Calabi–Yau 4-folds, Adv. Math. 406 (2022) MR4440076
39	Y Cao, Y Toda, Tautological stable pair invariants of Calabi–Yau 4-folds, Adv. Math. 396 (2022) MR4362779
40	Y Cao, D Maulik, Y Toda, Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds, Adv. Math. 338 (2018) 41 MR3861701
41	Y Cao, J Gross, D Joyce, Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi–Yau 4-folds, Adv. Math. 368 (2020) MR4085139
42	Y Cao, M Kool, S Monavari, K-theoretic DT/PT correspondence for toric Calabi–Yau 4-folds, Comm. Math. Phys. 396 (2022) 225 MR4499016
43	Y Cao, D Maulik, Y Toda, Stable pairs and Gopakumar–Vafa type invariants for Calabi–Yau 4-folds, J. Eur. Math. Soc. 24 (2022) 527 MR4382478
44	Y Cao, G Oberdieck, Y Toda, Stable pairs and Gopakumar–Vafa type invariants on holomorphic symplectic 4-folds, Adv. Math. 408 (2022) MR4460281
45	Y Cao, G Oberdieck, Y Toda, Gopakumar–Vafa type invariants of holomorphic symplectic 4-folds, Comm. Math. Phys. 405 (2024) MR4698661
46	J Carlson, M Green, P Griffiths, Variations of Hodge structure considered as an exterior differential system : old and new results, Symmetry Integrability Geom. Methods Appl. 5 (2009) MR2559674
47	F Charles, C Schnell, Notes on absolute Hodge classes, from: "Hodge theory" (editors E Cattani, F El Zein, P A Griffiths, L D Tráng), Math. Notes 49, Princeton Univ. Press (2014) 469 MR3290133
48	C Ciliberto, M Mendes Lopes, X Roulleau, On Schoen surfaces, Comment. Math. Helv. 90 (2015) 59 MR3317333
49	A Căldăraru, The Mukai pairing, II : The Hochschild–Kostant–Rosenberg isomorphism, Adv. Math. 194 (2005) 34 MR2141853
50	P Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Inst. Hautes Études Sci. Publ. Math. 35 (1968) 259 MR244265
51	P Deligne, Théorie de Hodge, II, Inst. Hautes Études Sci. Publ. Math. 40 (1971) 5 MR498551
52	S K Donaldson, R P Thomas, Gauge theory in higher dimensions, from: "The geometric universe" (editors S A Huggett, L J Mason, K P Tod, S T Tsou, N M J Woodhouse), Oxford Univ. Press (1998) 31 MR1634503
53	D Edidin, W Graham, Characteristic classes and quadric bundles, Duke Math. J. 78 (1995) 277 MR1333501
54	D Edidin, W Graham, Equivariant intersection theory, Invent. Math. 131 (1998) 595 MR1614555
55	A Grothendieck, Éléments de géométrie algébrique, III : Étude cohomologique des faisceaux cohérents, I, Inst. Hautes Études Sci. Publ. Math. 11 (1961) 5 MR0163910
56	D Eisenbud, Commutative algebra: with a view toward algebraic geometry, 150, Springer (1995) MR1322960
57	G Ellingsrud, L Göttsche, M Lehn, On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom. 10 (2001) 81 MR1795551
58	I Emmanouil, The cyclic homology of affine algebras, Invent. Math. 121 (1995) 1 MR1345282
59	B L Feigin, B L Tsygan, Additive K-theory and crystalline cohomology, Funktsional. Anal. i Prilozhen. 19 (1985) 52, 96 MR800920
60	W Fulton, Intersection theory, 2, Springer (1998) MR1644323
61	S I Gelfand, Y I Manin, Methods of homological algebra, Springer (2003) MR1950475
62	A Gholampour, M Kool, Stable reflexive sheaves and localization, J. Pure Appl. Algebra 221 (2017) 1934 MR3623176
63	A Gholampour, Y Jiang, J Lo, Stable pairs of 2-dimensional sheaves on 4-folds, Pure Appl. Math. Q. 20 (2024) 2781 MR4829866
64	T G Goodwillie, Cyclic homology, derivations, and the free loopspace, Topology 24 (1985) 187 MR793184
65	T G Goodwillie, Relative algebraic K-theory and cyclic homology, Ann. of Math. 124 (1986) 347 MR855300
66	M Graña, T W Grimm, D van de Heisteeg, A Herraez, E Plauschinn, The tadpole conjecture in asymptotic limits, J. High Energy Phys. 2022 (2022) MR4473849
67	P A Griffiths, Periods of integrals on algebraic manifolds, I : Construction and properties of the modular varieties, Amer. J. Math. 90 (1968) 568 MR229641
68	P A Griffiths, Periods of integrals on algebraic manifolds, II : Local study of the period mapping, Amer. J. Math. 90 (1968) 805 MR233825
69	J Gross, D Joyce, Y Tanaka, Universal structures in ℂ-linear enumerative invariant theories, Symmetry Integrability Geom. Methods Appl. 18 (2022) MR4489089
70	A Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966) 95 MR199194
71	A Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique, IV : Les schémas de Hilbert, from: "Séminaire Bourbaki, 1960/1961", W. A. Benjamin (1966) MR197245
72	S Gukov, C Vafa, E Witten, CFT’s from Calabi–Yau four-folds, Nuclear Phys. B 584 (2000) 69 MR1786689
73	J Hall, D Rydh, Perfect complexes on algebraic stacks, Compos. Math. 153 (2017) 2318 MR3705292
74	D Halpern-Leistner, A Preygel, Mapping stacks and categorical notions of properness, Compos. Math. 159 (2023) 530 MR4560539
75	D Happel, I Reiten, S O Smalø, Tilting in abelian categories and quasitilted algebras, 575, Amer. Math. Soc. (1996) MR1327209
76	R Hartshorne, On the De Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975) 5 MR432647
77	B Hassett, S J Kovács, Reflexive pull-backs and base extension, J. Algebraic Geom. 13 (2004) 233 MR2047697
78	G Hochschild, B Kostant, A Rosenberg, Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962) 383 MR142598
79	M Hoyois, The homotopy fixed points of the circle action on Hochschild homology, preprint (2015) arXiv:1506.07123
80	M Hoyois, P Safronov, S Scherotzke, N Sibilla, The categorified Grothendieck–Riemann–Roch theorem, Compos. Math. 157 (2021) 154 MR4219217
81	D Huybrechts, M Lehn, The geometry of moduli spaces of sheaves, Cambridge Univ. Press (2010) MR2665168
82	D Huybrechts, R P Thomas, Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes, Math. Ann. 346 (2010) 545 MR2578562
83	D Huybrechts, C Xu, Lagrangian fibrations of hyperkähler fourfolds, J. Inst. Math. Jussieu 21 (2022) 921 MR4404129
84	J M Hwang, Base manifolds for fibrations of projective irreducible symplectic manifolds, Invent. Math. 174 (2008) 625 MR2453602
85	D Iacono, M Manetti, Semiregularity and obstructions of complete intersections, Adv. Math. 235 (2013) 92 MR3010052
86	L Illusie, Complexe cotangent et déformations, I, 239, Springer (1971) MR491680
87	L Illusie, Complexe cotangent et déformations, II, 283, Springer (1972) MR491681
88	M a Inaba, Toward a definition of moduli of complexes of coherent sheaves on a projective scheme, J. Math. Kyoto Univ. 42 (2002) 317 MR1966840
89	Y Jiang, The virtual fundamental class for the moduli space of surfaces of general type, preprint (2022) arXiv:2206.00575
90	D Joyce, Ringel–Hall style vertex algebra and Lie algebra structures on the homology of moduli spaces, preprint (2018)
91	D Joyce, Enumerative invariants and wall-crossing formulae in abelian categories, preprint (2021) arXiv:2111.04694
92	D Joyce, Y Tanaka, M Upmeier, On orientations for gauge-theoretic moduli spaces, Adv. Math. 362 (2020) MR4050581
93	N M Katz, T Oda, On the differentiation of de Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ. 8 (1968) 199 MR237510
94	B Keller, On the cyclic homology of ringed spaces and schemes, Doc. Math. 3 (1998) 231 MR1647519
95	A A Khan, Virtual fundamental classes of derived stacks, I, preprint (2019) arXiv:1909.01332
96	Y H Kiem, J Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013) 1025 MR3073883
97	Y H Kiem, H Park, Virtual intersection theories, Adv. Math. 388 (2021) MR4281772
98	Y H Kiem, H Park, Localizing virtual cycles for Donaldson–Thomas invariants of Calabi–Yau 4-folds, J. Algebraic Geom. 32 (2023) 585 MR4652574
99	A Klemm, R Pandharipande, Enumerative geometry of Calabi–Yau 4-folds, Comm. Math. Phys. 281 (2008) 621 MR2415462
100	J Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990) 235 MR1064874
101	J Kollár, Families of varieties of general type, 231, Cambridge Univ. Press (2023) MR4566297
102	M Kontsevich, Enumeration of rational curves via torus actions, from: "The moduli space of curves" (editors R Dijkgraaf, C Faber, G van der Geer), Progr. Math. 129, Birkhäuser (1995) 335 MR1363062
103	M Kool, J V Rennemo, Proof of a magnficent conjecture, preprint (2025) arXiv:2507.02852
104	M Kool, R Thomas, Reduced classes and curve counting on surfaces, I : Theory, Algebr. Geom. 1 (2014) 334 MR3238154
105	A Kresch, Canonical rational equivalence of intersections of divisors, Invent. Math. 136 (1999) 483 MR1695204
106	A Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999) 495 MR1719823
107	G Laumon, L Moret-Bailly, Champs algébriques, 39, Springer (2000) MR1771927
108	J Le Potier, Systèmes cohérents et structures de niveau, 214, Soc. Math. France (1993) 143 MR1244404
109	J Le Potier, Faisceaux semi-stables et systèmes cohérents, from: "Vector bundles in algebraic geometry" (editors N J Hitchin, P E Newstead, W M Oxbury), London Math. Soc. Lecture Note Ser. 208, Cambridge Univ. Press (1995) 179 MR1338417
110	J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998) 119 MR1467172
111	M Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006) 175 MR2177199
112	J Lo, Moduli of PT-semistable objects, I, J. Algebra 339 (2011) 203 MR2811320
113	J L Loday, Cyclic homology, 301, Springer (1992) MR1217970
114	J Lurie, Higher topos theory, 170, Princeton Univ. Press (2009) MR2522659
115	J Lurie, Higher algebra, book project (2017)
116	S Lüst, C Vafa, M Wiesner, K Xu, Holography and the KKLT scenario, J. High Energy Phys. 2022 (2022) MR4505668
117	C Manolache, Virtual pull-backs, J. Algebraic Geom. 21 (2012) 201 MR2877433
118	N Markarian, The Atiyah class, Hochschild cohomology and the Riemann–Roch theorem, J. Lond. Math. Soc. 79 (2009) 129 MR2472137
119	D Matsushita, Higher direct images of dualizing sheaves of Lagrangian fibrations, Amer. J. Math. 127 (2005) 243 MR2130616
120	D Matsushita, On deformations of Lagrangian fibrations, from: "K3 surfaces and their moduli" (editors C Faber, G Farkas, G van der Geer), Progr. Math. 315, Springer (2016) 237 MR3524170
121	D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory, I, Compos. Math. 142 (2006) 1263 MR2264664
122	D Maulik, R Pandharipande, R P Thomas, Curves on K3 surfaces and modular forms, J. Topol. 3 (2010) 937 MR2746343
123	R McCarthy, The cyclic homology of an exact category, J. Pure Appl. Algebra 93 (1994) 251 MR1275967
124	H Movasati, A course in Hodge theory : with emphasis on multiple integrals, International Press (2021) MR4290021
125	D Mumford, J Fogarty, F Kirwan, Geometric invariant theory, 34, Springer (1994) MR1304906
126	N Nekrasov, BPS/CFT correspondence : non-perturbative Dyson–Schwinger equations and qq-characters, J. High Energ. Phys. 2016 (2016)
127	N Nekrasov, BPS/CFT correspondence, II : Instantons at crossroads, moduli and compactness theorem, Adv. Theor. Math. Phys. 21 (2017) 503 MR3678005
128	N Nekrasov, BPS/CFT correspondence, V : BPZ and KZ equations from qq-characters, preprint (2017) arXiv:1711.11582
129	N Nekrasov, BPS/CFT correspondence, III : Gauge origami partition function and qq-characters, Comm. Math. Phys. 358 (2018) 863 MR3778345
130	N Nekrasov, BPS/CFT correspondence, IV : Sigma models and defects in gauge theory, Lett. Math. Phys. 109 (2019) 579 MR3910136
131	N Nekrasov, Magnificent four, Ann. Inst. Henri Poincaré D 7 (2020) 505 MR4182774
132	N Nekrasov, N Piazzalunga, Magnificent four with colors, Comm. Math. Phys. 372 (2019) 573 MR4032874
133	J Oh, R P Thomas, Counting sheaves on Calabi–Yau 4-folds, I, Duke Math. J. 172 (2023) 1333 MR4583653
134	J Oh, R P Thomas, Counting sheaves on Calabi–Yau 4-folds, II, in preparation
135	M Olsson, Sheaves on Artin stacks, J. Reine Angew. Math. 603 (2007) 55 MR2312554
136	R Pandharipande, R P Thomas, The 3-fold vertex via stable pairs, Geom. Topol. 13 (2009) 1835 MR2497313
137	R Pandharipande, R P Thomas, Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009) 407 MR2545686
138	R Pandharipande, R P Thomas, Stable pairs and BPS invariants, J. Amer. Math. Soc. 23 (2010) 267 MR2552254
139	R Pandharipande, R P Thomas, 13/2 ways of counting curves, from: "Moduli spaces" (editors L Brambila-Paz, O García-Prada, P Newstead, R P Thomas), London Math. Soc. Lecture Note Ser. 411, Cambridge Univ. Press (2014) 282 MR3221298
140	T Pantev, B Toën, M Vaquié, G Vezzosi, Shifted symplectic structures, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 271 MR3090262
141	H Park, Virtual pullbacks in Donaldson–Thomas theory of Calabi–Yau 4-folds, preprint (2021) arXiv:2110.03631
142	H Park, Shifted symplectic pushforwards, preprint (2024) arXiv:2406.19192
143	J P Pridham, Semiregularity as a consequence of Goodwillie’s theorem, Forum Math. Sigma 12 (2024) MR4842403
144	F Qu, Virtual pullbacks in K-theory, Ann. Inst. Fourier (Grenoble) 68 (2018) 1609 MR3887429
145	M Savvas, Cosection localization and vanishing for virtual fundamental classes of d-manifolds, Adv. Math. 398 (2022) MR4383010
146	C Schoen, A family of surfaces constructed from genus 2 curves, Internat. J. Math. 18 (2007) 585 MR2331080
147	T Schürg, B Toën, G Vezzosi, Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes, J. Reine Angew. Math. 702 (2015) 1 MR3341464
148	A Grothendieck, Geometrie alebgrique et geometrie analytique, 224, Springer (1971) MR0354651
149	P Berthelot, Imersions régulières et caclul du K^∙ d’un schéma eclaté, from: "Théorie des intersections et théorème de Riemann–Roch" (editors A Grothendieck, P Berthelot, L Illusie), Lecture Notes in Math. 225, Springer (1971) MR0354655
150	J Shen, Q Yin, Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds, Duke Math. J. 171 (2022) 209 MR4366204
151	C Simpson, The topological realization of a simplicial presheaf, preprint (1996) arXiv:q-alg/9609004
152	, The Stacks project, electronic reference (2005–)
153	J H M Steenbrink, Some remarks about the Hodge conjecture, from: "Hodge theory" (editors E Cattani, F Guillén, A Kaplan, F Puerta), Lecture Notes in Math. 1246, Springer (1987) 165 MR894051
154	J Stoppa, R P Thomas, Hilbert schemes and stable pairs : GIT and derived category wall crossings, Bull. Soc. Math. France 139 (2011) 297 MR2869309
155	R P Thomas, A holomorphic Casson invariant for Calabi–Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom. 54 (2000) 367 MR1818182
156	R W Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. in Math. 65 (1987) 16 MR893468
157	G Tian, Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric, from: "Mathematical aspects of string theory" (editor S T Yau), Adv. Ser. Math. Phys. 1, World Sci. (1987) 629 MR915841
158	A N Todorov, The Weil–Petersson geometry of the moduli space of SU(n ≥ 3) (Calabi–Yau) manifolds, I, Comm. Math. Phys. 126 (1989) 325 MR1027500
159	B Toën, M Vaquié, Moduli of objects in dg-categories, Ann. Sci. École Norm. Sup. 40 (2007) 387 MR2493386
160	B Toën, G Vezzosi, Homotopical algebraic geometry, II : Geometric stacks and applications, 902, Amer. Math. Soc. (2008) MR2394633
161	B Toën, G Vezzosi, Algèbres simpliciales S¹-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs, Compos. Math. 147 (2011) 1979 MR2862069
162	B Toën, G Vezzosi, Caractères de Chern, traces équivariantes et géométrie algébrique dérivée, Selecta Math. 21 (2015) 449 MR3338682
163	B Totaro, The Chow ring of a classifying space, from: "Algebraic K-theory" (editors W Raskind, C Weibel), Proc. Sympos. Pure Math. 67, Amer. Math. Soc. (1999) 249 MR1743244
164	E Viehweg, Quasi-projective moduli for polarized manifolds, 30, Springer (1995) MR1368632
165	A Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989) 613 MR1005008
166	C Vizman, Induced differential forms on manifolds of functions, Arch. Math. (Brno) 47 (2011) 201 MR2852381
167	C Voisin, Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes, from: "Complex projective geometry" (editors G Ellingsrud, C Peskine, G Sacchiero, S A Strømme), London Math. Soc. Lecture Note Ser. 179, Cambridge Univ. Press (1992) 294 MR1201391
168	C Voisin, Hodge loci, from: "Handbook of moduli, III" (editors G Farkas, I Morrison), Adv. Lect. Math. 26, International Press (2013) 507 MR3135443
169	C A Weibel, An introduction to homological algebra, 38, Cambridge Univ. Press (1994) MR1269324
170	C Weibel, Cyclic homology for schemes, Proc. Amer. Math. Soc. 124 (1996) 1655 MR1277141
171	C Weibel, The Hodge filtration and cyclic homology, K-Theory 12 (1997) 145 MR1469140
172	C A Weibel, S C Geller, Étale descent for Hochschild and cyclic homology, Comment. Math. Helv. 66 (1991) 368 MR1120653
173	A Yekutieli, The continuous Hochschild cochain complex of a scheme, Canad. J. Math. 54 (2002) 1319 MR1940241