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A theory of generalized Donaldson–Thomas invariants
About this Title
Dominic Joyce, The Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, United Kingdom and Yinan Song
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 217, Number 1020
ISBNs: 978-0-8218-5279-8 (print); 978-0-8218-8752-3 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00630-1
Published electronically: July 18, 2011
Keywords: Donaldson–Thomas invariant,
Calabi–Yau 3-fold,
coherent sheaf,
vector bundle,
stability condition,
semistable,
Gieseker stability,
moduli space,
Artin stack
MSC: Primary 14N35; Secondary 14J32, 14F05, 14J60, 14D23
Table of Contents
Chapters
- 1. Introduction
- 2. Constructible functions and stack functions
- 3. Background material
- 4. Behrend functions and Donaldson–Thomas theory
- 5. Statements of main results
- 6. Examples, applications, and generalizations
- 7. Donaldson–Thomas theory for quivers with superpotentials
- 8. The proof of Theorem
- 9. The proofs of Theorems and
- 10. The proof of Theorem
- 11. The proof of Theorem
- 12. The proofs of Theorems , and
- 13. The proof of Theorem
Abstract
Donaldson–Thomas invariants $DT^\alpha (\tau )$ are integers which ‘count’ $\tau$-stable coherent sheaves with Chern character $\alpha$ on a Calabi–Yau 3-fold $X$, where $\tau$ denotes Gieseker stability for some ample line bundle on $X$. They are unchanged under deformations of $X$. The conventional definition works only for classes $\alpha$ containing no strictly $\tau$-semistable sheaves. Behrend showed that $DT^\alpha (\tau )$ can be written as a weighted Euler characteristic $\chi \bigl ({\mathcal M}_{\mathrm {st}}^\alpha (\tau ), \nu _{{\mathcal M}_{\mathrm {st}}^\alpha (\tau )}\bigr )$ of the stable moduli scheme ${\mathcal M}_{\mathrm {st}}^\alpha (\tau )$ by a constructible function $\nu _{{\mathcal M}_{\mathrm {st}}^\alpha (\tau )}$ we call the ‘Behrend function’.
This book studies generalized Donaldson–Thomas invariants $\bar {DT}{}^\alpha (\tau )$. They are rational numbers which ‘count’ both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern character $\alpha$ on $X$; strictly $\tau$-semistable sheaves must be counted with complicated rational weights. The $\bar {DT}{}^\alpha (\tau )$ are defined for all classes $\alpha$, and are equal to $DT^\alpha (\tau )$ when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $\tau$.
To prove all this we study the local structure of the moduli stack ${\mathfrak M}$ of coherent sheaves on $X$. We show that an atlas for ${\mathfrak M}$ may be written locally as $\textrm {Crit}(f)$ for $f:U\rightarrow {\mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $\nu _{\mathfrak M}$. We compute our invariants $\bar {DT}{}^\alpha (\tau )$ in examples, and make a conjecture about their integrality properties. We also extend the theory to abelian categories mod-$\mathbb {C}Q/I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$, and connect our ideas with Szendrői’s noncommutative Donaldson–Thomas invariants, and work by Reineke and others on invariants counting quiver representations. Our book is closely related to Kontsevich and Soibelman’s independent paper Stability structures, motivic Donaldson-Thomas invariants and cluster transformations.
- Allen B. Altman and Steven L. Kleiman, Compactifying the Picard scheme, Adv. in Math. 35 (1980), no. 1, 50–112. MR 555258, DOI 10.1016/0001-8708(80)90043-2
- M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189. MR 399094, DOI 10.1007/BF01390174
- Kai Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), no. 3, 1307–1338. MR 2600874, DOI 10.4007/annals.2009.170.1307
- K. Behrend, J. Bryan and B. Szendrői, Motivic degree zero Donaldson–Thomas invariants, arXiv:0909.5088, 2009.
- K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88. MR 1437495, DOI 10.1007/s002220050136
- Kai Behrend and Barbara Fantechi, Symmetric obstruction theories and Hilbert schemes of points on threefolds, Algebra Number Theory 2 (2008), no. 3, 313–345. MR 2407118, DOI 10.2140/ant.2008.2.313
- D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR 1110581
- Raf Bocklandt, Graded Calabi Yau algebras of dimension 3, J. Pure Appl. Algebra 212 (2008), no. 1, 14–32. MR 2355031, DOI 10.1016/j.jpaa.2007.03.009
- A. Bondal and D. Orlov, Semiorthogonal decompositions for algebraic varieties, alg-geom/9506012, 1995.
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012
- Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. MR 2373143, DOI 10.4007/annals.2007.166.317
- T. Bridgeland, Hall algebras and curve-counting invariants, arXiv:1002.4374, 2010.
- Tom Bridgeland, Alastair King, and Miles Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535–554. MR 1824990, DOI 10.1090/S0894-0347-01-00368-X
- Jim Bryan and Amin Gholampour, The quantum McKay correspondence for polyhedral singularities, Invent. Math. 178 (2009), no. 3, 655–681. MR 2551767, DOI 10.1007/s00222-009-0212-8
- Steven Bradlow, Georgios D. Daskalopoulos, Oscar García-Prada, and Richard Wentworth, Stable augmented bundles over Riemann surfaces, Vector bundles in algebraic geometry (Durham, 1993) London Math. Soc. Lecture Note Ser., vol. 208, Cambridge Univ. Press, Cambridge, 1995, pp. 15–67. MR 1338412, DOI 10.1017/CBO9780511569319.003
- Anneaux de Chow et applications, Séminaire C. Chevalley, 2e année, École Normale Supérieure, Secrétariat mathématique, Paris, 1958.
- Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky, Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14 (2008), no. 1, 59–119. MR 2480710, DOI 10.1007/s00029-008-0057-9
- Alexandru Dimca, Sheaves in topology, Universitext, Springer-Verlag, Berlin, 2004. MR 2050072
- Alexandru Dimca and Balázs Szendrői, The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on $\Bbb C^3$, Math. Res. Lett. 16 (2009), no. 6, 1037–1055. MR 2576692, DOI 10.4310/MRL.2009.v16.n6.a12
- S. K. Donaldson and R. P. Thomas, Gauge theory in higher dimensions, The geometric universe (Oxford, 1996) Oxford Univ. Press, Oxford, 1998, pp. 31–47. MR 1634503
- Adrien Douady, Le problème des modules pour les variétés analytiques complexes (d’après Masatake Kuranishi), Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 277, 7–13 (French). MR 1608786
- Adrien Douady, Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 1–95 (French). MR 203082
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
- Johannes Engel and Markus Reineke, Smooth models of quiver moduli, Math. Z. 262 (2009), no. 4, 817–848. MR 2511752, DOI 10.1007/s00209-008-0401-y
- O. Forster and K. Knorr, Über die Deformationen von Vektorraumbündeln auf kompakten komplexen Räumen, Math. Ann. 209 (1974), 291–346 (German). MR 374495, DOI 10.1007/BF01351725
- Sebastián Franco, Amihay Hanany, David Vegh, Brian Wecht, and Kristian D. Kennaway, Brane dimers and quiver gauge theories, J. High Energy Phys. 1 (2006), 096, 48. MR 2201227, DOI 10.1088/1126-6708/2006/01/096
- Robert Friedman and John W. Morgan, Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 27, Springer-Verlag, Berlin, 1994. MR 1288304
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620
- Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. MR 1950475
- V. Ginzburg, Calabi–Yau algebras, math.AG/0612139, 2006.
- Tomás L. Gómez, Algebraic stacks, Proc. Indian Acad. Sci. Math. Sci. 111 (2001), no. 1, 1–31. MR 1818418, DOI 10.1007/BF02829538
- R. Gopakumar and C. Vafa, M-theory and topological strings. II, hep-th/9812127, 1998.
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
- A. Grothendieck, Elements de Géométrie Algébrique, part I Publ. Math. IHES 4 (1960), part II Publ. Math. IHES 8 (1961), part III Publ. Math. IHES 11 (1960) and 17 (1963), and part IV Publ. Math. IHES 20 (1964), 24 (1965), 28 (1966) and 32 (1967).
- Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651
- Amihay Hanany, Christopher P. Herzog, and David Vegh, Brane tilings and exceptional collections, J. High Energy Phys. 7 (2006), 001, 44. MR 2240899, DOI 10.1088/1126-6708/2006/07/001
- A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149, 2005.
- Amihay Hanany and David Vegh, Quivers, tilings, branes and rhombi, J. High Energy Phys. 10 (2007), 029, 35. MR 2357949, DOI 10.1088/1126-6708/2007/10/029
- Robin Hartshorne, On the De Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 5–99. MR 432647
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Shinobu Hosono, Masa-Hiko Saito, and Atsushi Takahashi, Relative Lefschetz action and BPS state counting, Internat. Math. Res. Notices 15 (2001), 783–816. MR 1849482, DOI 10.1155/S107379280100040X
- D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR 2244106
- D. Huybrechts and M. Lehn, Framed modules and their moduli, Internat. J. Math. 6 (1995), no. 2, 297–324. MR 1316305, DOI 10.1142/S0129167X9500050X
- Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1450870
- Daniel Huybrechts and Richard P. Thomas, Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes, Math. Ann. 346 (2010), no. 3, 545–569. MR 2578562, DOI 10.1007/s00208-009-0397-6
- Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
- Luc Illusie, Cotangent complex and deformations of torsors and group schemes, Toposes, algebraic geometry and logic (Conf., Dalhousie Univ., Halifax, N.S., 1971) Springer, Berlin, 1972, pp. 159–189. Lecture Notes in Math., Vol. 274. MR 0491682
- Akira Ishii and Kazushi Ueda, On moduli spaces of quiver representations associated with dimer models, Higher dimensional algebraic varieties and vector bundles, RIMS Kôkyûroku Bessatsu, B9, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, pp. 127–141. MR 2509696
- Dominic Joyce, Constructible functions on Artin stacks, J. London Math. Soc. (2) 74 (2006), no. 3, 583–606. MR 2286434, DOI 10.1112/S0024610706023180
- Dominic Joyce, Motivic invariants of Artin stacks and ‘stack functions’, Q. J. Math. 58 (2007), no. 3, 345–392. MR 2354923, DOI 10.1093/qmath/ham019
- Dominic Joyce, Configurations in abelian categories. I. Basic properties and moduli stacks, Adv. Math. 203 (2006), no. 1, 194–255. MR 2231046, DOI 10.1016/j.aim.2005.04.008
- Dominic Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, Adv. Math. 210 (2007), no. 2, 635–706. MR 2303235, DOI 10.1016/j.aim.2006.07.006
- Dominic Joyce, Configurations in abelian categories. III. Stability conditions and identities, Adv. Math. 215 (2007), no. 1, 153–219. MR 2354988, DOI 10.1016/j.aim.2007.04.002
- Dominic Joyce, Configurations in abelian categories. IV. Invariants and changing stability conditions, Adv. Math. 217 (2008), no. 1, 125–204. MR 2357325, DOI 10.1016/j.aim.2007.06.011
- Dominic Joyce, Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds, Geom. Topol. 11 (2007), 667–725. MR 2302500, DOI 10.2140/gt.2007.11.667
- D. Joyce, Generalized Donaldson–Thomas invariants, arXiv:0910.0105, 2009.
- Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. MR 1074006
- Sheldon Katz, Genus zero Gopakumar-Vafa invariants of contractible curves, J. Differential Geom. 79 (2008), no. 2, 185–195. MR 2420017
- Bernhard Keller, Calabi-Yau triangulated categories, Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp. 467–489. MR 2484733, DOI 10.4171/062-1/11
- Gary Kennedy, MacPherson’s Chern classes of singular algebraic varieties, Comm. Algebra 18 (1990), no. 9, 2821–2839. MR 1063344, DOI 10.1080/00927879008824054
- A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. MR 1315461, DOI 10.1093/qmath/45.4.515
- Shoshichi Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987. Kanô Memorial Lectures, 5. MR 909698
- M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, arXiv:0811.2435, 2008.
- M. Kontsevich and Y. Soibelman, Motivic Donaldson–Thomas invariants: summary of results, arXiv:0910.4315, 2009.
- Siegmund Kosarew and Christian Okonek, Global moduli spaces and simple holomorphic bundles, Publ. Res. Inst. Math. Sci. 25 (1989), no. 1, 1–19. MR 999347, DOI 10.2977/prims/1195173759
- Serge Lang, Differential manifolds, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0431240
- Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927
- Lê Dũng Tráng, Some remarks on relative monodromy, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 397–403. MR 0476739
- J. Le Potier, Faisceaux semi-stables et systèmes cohérents, Vector bundles in algebraic geometry (Durham, 1993) London Math. Soc. Lecture Note Ser., vol. 208, Cambridge Univ. Press, Cambridge, 1995, pp. 179–239 (French, with French summary). MR 1338417
- M. Levine and R. Pandharipande, Algebraic cobordism revisited, math.AG/0605196.
- Jun Li, Zero dimensional Donaldson-Thomas invariants of threefolds, Geom. Topol. 10 (2006), 2117–2171. MR 2284053, DOI 10.2140/gt.2006.10.2117
- M. Lübke and C. Okonek, Moduli spaces of simple bundles and Hermitian-Einstein connections, Math. Ann. 276 (1987), no. 4, 663–674. MR 879544, DOI 10.1007/BF01456994
- Martin Lübke and Andrei Teleman, The Kobayashi-Hitchin correspondence, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR 1370660
- Domingo Luna, Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, pp. 81–105. Bull. Soc. Math. France, Paris, Mémoire 33 (French). MR 0342523, DOI 10.24033/msmf.110
- R. D. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432. MR 361141, DOI 10.2307/1971080
- D.B. Massey, Notes on perverse sheaves and vanishing cycles, arXiv:math/9908107, 1999.
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
- Kimio Miyajima, Kuranishi family of vector bundles and algebraic description of the moduli space of Einstein-Hermitian connections, Publ. Res. Inst. Math. Sci. 25 (1989), no. 2, 301–320. MR 1003790, DOI 10.2977/prims/1195173613
- D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. I, Compos. Math. 142 (2006), no. 5, 1263–1285. MR 2264664, DOI 10.1112/S0010437X06002302
- D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. II, Compos. Math. 142 (2006), no. 5, 1286–1304. MR 2264665, DOI 10.1112/S0010437X06002314
- Sergey Mozgovoy and Markus Reineke, On the noncommutative Donaldson-Thomas invariants arising from brane tilings, Adv. Math. 223 (2010), no. 5, 1521–1544. MR 2592501, DOI 10.1016/j.aim.2009.10.001
- K. Nagao, Derived categories of small toric Calabi–Yau $3$-folds and curve counting invariants, arXiv:0809.2994, 2008.
- K. Nagao and H. Nakajima, Counting invariant of perverse coherent sheaves and its wall-crossing, arXiv:0809.2992, 2008.
- Hiraku Nakajima, Varieties associated with quivers, Representation theory of algebras and related topics (Mexico City, 1994) CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 139–157. MR 1388562
- R. Pandharipande and R. P. Thomas, Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009), no. 2, 407–447. MR 2545686, DOI 10.1007/s00222-009-0203-9
- Adam Parusiński and Piotr Pragacz, Characteristic classes of hypersurfaces and characteristic cycles, J. Algebraic Geom. 10 (2001), no. 1, 63–79. MR 1795550
- Markus Reineke, Cohomology of noncommutative Hilbert schemes, Algebr. Represent. Theory 8 (2005), no. 4, 541–561. MR 2199209, DOI 10.1007/s10468-005-8762-y
- Markus Reineke, Framed quiver moduli, cohomology, and quantum groups, J. Algebra 320 (2008), no. 1, 94–115. MR 2417980, DOI 10.1016/j.jalgebra.2008.01.025
- M. Reineke, Cohomology of quiver moduli, functional equations, and integrality of Donaldson–Thomas type invariants, arXiv:0903.0261, 2009.
- C. Sabbah, Quelques remarques sur la géométrie des espaces conormaux, Astérisque 130 (1985), 161–192 (French). Differential systems and singularities (Luminy, 1983). MR 804052
- Morihiko Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333. MR 1047415, DOI 10.2977/prims/1195171082
- Jörg Schürmann, Topology of singular spaces and constructible sheaves, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 63, Birkhäuser Verlag, Basel, 2003. MR 2031639
- H.W. Schuster, Formale Deformationstheorien, Habilitationsschrift, München, 1971.
- Ed Segal, The $A_\infty$ deformation theory of a point and the derived categories of local Calabi-Yaus, J. Algebra 320 (2008), no. 8, 3232–3268. MR 2450725, DOI 10.1016/j.jalgebra.2008.06.019
- Paul Seidel and Richard Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37–108. MR 1831820, DOI 10.1215/S0012-7094-01-10812-0
- Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 1–42 (French). MR 82175
- J. Stoppa and R.P. Thomas, Hilbert schemes and stable pairs: GIT and derived category wall crossings, arXiv:0903.1444, 2009.
- Balázs Szendrői, Non-commutative Donaldson-Thomas invariants and the conifold, Geom. Topol. 12 (2008), no. 2, 1171–1202. MR 2403807, DOI 10.2140/gt.2008.12.1171
- R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000), no. 2, 367–438. MR 1818182
- Y. Toda, Curve counting theories via stable objects I. DT/PT correspondence, arXiv:0902.4371, 2009.
- J.-L Verdier, Spécialisation des classes de Chern, The Euler-Poincaré characteristic (French), Astérisque, vol. 82, Soc. Math. France, Paris, 1981, pp. 149–159 (French). MR 629126
- Claire Voisin, On integral Hodge classes on uniruled or Calabi-Yau threefolds, Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math., vol. 45, Math. Soc. Japan, Tokyo, 2006, pp. 43–73. MR 2306166, DOI 10.2969/aspm/04510043
- Ben Young, Computing a pyramid partition generating function with dimer shuffling, J. Combin. Theory Ser. A 116 (2009), no. 2, 334–350. MR 2475021, DOI 10.1016/j.jcta.2008.06.006
- Benjamin Young, Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds, Duke Math. J. 152 (2010), no. 1, 115–153. With an appendix by Jim Bryan. MR 2643058, DOI 10.1215/00127094-2010-009