## Stable pairs and BPS invariants

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- by R. Pandharipande and R. P. Thomas
- J. Amer. Math. Soc.
**23**(2010), 267-297 - DOI: https://doi.org/10.1090/S0894-0347-09-00646-8
- Published electronically: October 1, 2009
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Previous version: Original version posted September 24, 2009

Corrected version: Current version corrects publisher's error in rendering authors' coding \curly H\!om and removes stray coding in Contents

## Abstract:

We define the BPS invariants of Gopakumar-Vafa in the case of irreducible curve classes on Calabi-Yau 3-folds. The main tools are the theory of stable pairs in the derived category and Behrend’s constructible function approach to the virtual class. For irreducible curve classes, we prove that the stable pairs’ generating function satisfies the strong BPS rationality conjectures.

We define the contribution of each curve $C$ to the BPS invariants and show that the contributions lie between the geometric genus and arithmetic genus of $C$. Complete formulae are derived for nonsingular and nodal curves.

A discussion of primitive classes on $K3$ surfaces from the point of view of stable pairs is given in the Appendix via calculations of Kawai-Yoshioka. A proof of the Yau-Zaslow formula for rational curve counts is obtained. A connection is made to the Katz-Klemm-Vafa formula for BPS counts in all genera.

## References

- A. Bayer,
*Polynomial Bridgeland stability conditions and the large volume limit*, arXiv:0712.1083. - A. Beauville,
*Counting rational curves on $K3$ surfaces*, arXiv:alg-geom/9701019. - K. Behrend,
*Donaldson-Thomas invariants via microlocal geometry,*arXiv:alg-geom/ 0507523. - K. Behrend and B. Fantechi,
*The intrinsic normal cone*, Invent. Math.**128**(1997), no. 1, 45–88. MR**1437495**, DOI 10.1007/s002220050136 - Jim Bryan and Naichung Conan Leung,
*The enumerative geometry of $K3$ surfaces and modular forms*, J. Amer. Math. Soc.**13**(2000), no. 2, 371–410. MR**1750955**, DOI 10.1090/S0894-0347-00-00326-X - X. Chen,
*Singularities of rational curves on $K3$ surfaces,*arXiv:alg-geom/9812050. - R. Gopakumar and C. Vafa,
*M-theory and topological strings–I,*arXiv:hep-th/9809187. - R. Gopakumar and C. Vafa,
*M-theory and topological strings–II,*arXiv:hep-th/9812127. - Min He,
*Espaces de modules de systèmes cohérents*, Internat. J. Math.**9**(1998), no. 5, 545–598 (French). MR**1644040**, DOI 10.1142/S0129167X98000257 - 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 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**, DOI 10.1007/978-3-663-11624-0 - D. Huybrechts and R. P. Thomas,
*Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes*, arXiv:0805.3527. - D. Joyce,
*Configurations in abelian categories. IV. Invariants and changing stability conditions,*arXiv:alg-geom/0410268. - S. Katz,
*Genus zero Gopakumar-Vafa invariants of contractible curves,*arXiv:alg-geom/ 0601193. - Sheldon Katz, Albrecht Klemm, and Cumrun Vafa,
*M-theory, topological strings and spinning black holes*, Adv. Theor. Math. Phys.**3**(1999), no. 5, 1445–1537. MR**1796683**, DOI 10.4310/ATMP.1999.v3.n5.a6 - Toshiya Kawai and K\B{o}ta Yoshioka,
*String partition functions and infinite products*, Adv. Theor. Math. Phys.**4**(2000), no. 2, 397–485. MR**1838446**, DOI 10.4310/ATMP.2000.v4.n2.a7 - A. Klemm, D. Maulik, R. Pandharipande, and E. Scheidegger,
*Noether-Lefschetz theory and the Yau-Zaslow conjecture*, arXiv:0807.2477. - M. Kontsevich and Y. Soibelman,
*Stability structures, motivic Donaldson-Thomas invariants and cluster transformations*, in preparation. - Joseph Le Potier,
*Systèmes cohérents et structures de niveau*, Astérisque**214**(1993), 143 (French, with English and French summaries). MR**1244404** - 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** - Jun Li and Gang Tian,
*Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties*, J. Amer. Math. Soc.**11**(1998), no. 1, 119–174. MR**1467172**, DOI 10.1090/S0894-0347-98-00250-1 - 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 and R. Pandharipande,
*Gromov-Witten theory and Noether-Lefschetz theory,*arxiv:0705.1653. - Shigeru Mukai,
*Symplectic structure of the moduli space of sheaves on an abelian or $K3$ surface*, Invent. Math.**77**(1984), no. 1, 101–116. MR**751133**, DOI 10.1007/BF01389137 - R. Pandharipande and R. P. Thomas,
*Curve counting via stable pairs in the derived category*, arXiv:0707.2348. - Rahul Pandharipande and Richard P. Thomas,
*The 3-fold vertex via stable pairs*, Geom. Topol.**13**(2009), no. 4, 1835–1876. MR**2497313**, DOI 10.2140/gt.2009.13.1835 - R. Pandharipande and R. P. Thomas,
*in preparation*. - Alexander H. W. Schmitt,
*Geometric invariant theory and decorated principal bundles*, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR**2437660**, DOI 10.4171/065 - A. Schwarz and I. Shapiro,
*Some remarks on Gopakumar-Vafa invariants*, Pure Appl. Math. Q.**1**(2005), no. 4, Special Issue: In memory of Armand Borel., 817–826. MR**2201001**, DOI 10.4310/PAMQ.2005.v1.n4.a5 - 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**, DOI 10.4310/jdg/1214341649 - Y. Toda,
*Birational Calabi-Yau $3$-folds and BPS state counting,*arXiv:0707.1643. - Y. Toda,
*Limit stable objects on Calabi-Yau $3$-folds,*arXiv:0803.2356. - Shing-Tung Yau and Eric Zaslow,
*BPS states, string duality, and nodal curves on $K3$*, Nuclear Phys. B**471**(1996), no. 3, 503–512. MR**1398633**, DOI 10.1016/0550-3213(96)00176-9

## Bibliographic Information

**R. Pandharipande**- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 357813
- Email: rahulp@math.princeton.edu
**R. P. Thomas**- Affiliation: Department of Mathematics, Imperial College, London, England
- MR Author ID: 636321
- Email: rpwt@imperial.ac.uk
- Received by editor(s): October 14, 2008
- Published electronically: October 1, 2009
- Additional Notes: The first author was partially supported by NSF grant DMS-0500187 and a Packard foundation fellowship

The second author was partially supported by a Royal Society University Research Fellowship. He also thanks the Leverhulme Trust and Columbia University for a visit to New York in the spring of 2007 when the project was started. Many of the results presented here were found during a visit to Lisbon in the summer of 2007. - © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**23**(2010), 267-297 - MSC (2000): Primary 14N35
- DOI: https://doi.org/10.1090/S0894-0347-09-00646-8
- MathSciNet review: 2552254