Properties of Gromov–Witten invariants defined via global Kuranishi charts
Annales Henri Lebesgue, Volume 8 (2025), pp. 811-871

Metadata

Keywords Moduli space ,  Gromov–Witten invariants ,  Kuranishi chart ,  CohFT

Abstract

Using the global Kuranishi charts constructed by Hirschi–Swaminathan, we define gravitational descendants and equivariant Gromov–Witten invariants for general symplectic manifolds. We prove that these invariants satisfy the axioms of Kontsevich and Manin and their generalisations. A virtual localisation formula holds in this setting; we use it to derive an explicit formula for the equivariant Gromov–Witten invariants of Hamiltonian GKM manifolds. In particular, the symplectic Gromov–Witten invariants of smooth toric varieties agree with their algebro-geometric counterpart. In the semipositive case, the invariants studied here recover those of Ruan and Tian.


References

[AB84] Atiyah, Michael F.; Bott, Raoul The moment map and equivariant cohomology, Topology, Volume 23 (1984) no. 1, pp. 1-28 | DOI | MR | Zbl

[ACG11] Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip A. Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften, 268, Springer, 2011 (with a contribution by Joseph Daniel Harris) | DOI | MR | Zbl

[AMS21] Abouzaid, Mohammed; McLean, Mark; Smith, Ivan Complex cobordism, Hamiltonian loops and global Kuranishi charts (2021) | arXiv | Zbl

[AMS24] Abouzaid, Mohammed; McLean, Mark; Smith, Ivan Gromov–Witten invariants in complex and Morava-local K-theories, Geom. Funct. Anal., Volume 34 (2024) no. 6, pp. 1647-1733 | DOI | MR | Zbl

[Aue73] Auer, John W. Fiber integration in smooth bundles, Pac. J. Math., Volume 44 (1973), pp. 33-43 | DOI | MR | Zbl

[Beh99] Behrend, Kai A. The product formula for Gromov–Witten invariants, J. Algebr. Geom., Volume 8 (1999) no. 3, pp. 529-541 | MR | Zbl

[Beh04] Behrend, Kai A. Cohomology of stacks, School and conference on intersection theory and moduli (ICTP Lecture Notes), Volume 19, The Abdus Salam International Centre for Theoretical Physics, 2004, pp. 249-294 | Zbl

[BF97] Behrend, Kai A.; Fantechi, Barbara The intrinsic normal cone, Invent. Math., Volume 128 (1997) no. 1, pp. 45-88 | DOI | MR | Zbl

[Bre72] Bredon, Glen E. Introduction to compact transformation groups, Pure and Applied Mathematics, 46, Academic Press Inc., 1972 | MR | Zbl

[Bre97] Bredon, Glen E. Sheaf theory, Graduate Texts in Mathematics, 170, Springer, 1997 | DOI | MR | Zbl

[Bro62] Brown, Morton Locally flat imbeddings of topological manifolds, Ann. Math. (2), Volume 75 (1962), pp. 331-341 | DOI | Zbl

[Bro14] Brown, Jeff Gromov–Witten invariants of toric fibrations, Int. Math. Res. Not., Volume 2014 (2014) no. 19, pp. 5437-5482 | MR | DOI | Zbl

[BT82] Bott, Raoul; Tu, Loring W. Differential forms in algebraic topology, Graduate Texts in Mathematics, 82, Springer, 1982 | DOI | MR | Zbl

[CGMiRS02] Cieliebak, Kai; Gaio, A. Rita; Mundet i Riera, Ignasi; Salamon, Dietmar A. The symplectic vortex equations and invariants of Hamiltonian group actions, J. Symplectic Geom., Volume 1 (2002) no. 3, pp. 543-645 | DOI | MR | Zbl

[CM07] Cieliebak, Kai; Mohnke, Klaus Symplectic hypersurfaces and transversality in Gromov–Witten theory, J. Symplectic Geom., Volume 5 (2007) no. 3, pp. 281-356 | DOI | MR | Zbl

[CT10] Chen, Bohui; Tian, Gang Virtual manifolds and localization, Acta Math. Sin., Engl. Ser., Volume 26 (2010) no. 1, pp. 1-24 | MR | DOI | Zbl

[Del88] Delzant, Thomas Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. Fr., Volume 116 (1988) no. 3, pp. 315-339 | Numdam | DOI | MR | Zbl

[Dua03] Duan, Haibao The degree of a Schubert variety, Adv. Math., Volume 180 (2003) no. 1, pp. 112-133 | MR | DOI | Zbl

[Dub14] Dubrovin, Boris A. Gromov–Witten invariants and integrable hierarchies of topological type, Topology, geometry, integrable systems, and mathematical physics (American Mathematical Society Translations, Series 2), Volume 234, American Mathematical Society, 2014, pp. 141-171 | DOI | MR | Zbl

[EGH00] Eliashberg, Yakov M.; Givental, Alexander B.; Hofer, Helmut Introduction to symplectic field theory, GAFA 2000. Visions in mathematics – Towards 2000 (Tel Aviv, 1999). Part II (Geometric and Functional Analysis), Volume Special Volume, Birkhäuser, 2000, pp. 560-673 | DOI | MR | Zbl

[Flo88] Floer, Andreas Morse theory for Lagrangian intersections, J. Differ. Geom., Volume 28 (1988) no. 3, pp. 513-547 | MR | DOI | Zbl

[FO99] Fukaya, Kenji; Ono, Kaoru Arnold conjecture and Gromov–Witten invariant, Topology, Volume 38 (1999) no. 5, pp. 933-1048 | DOI | MR | Zbl

[FP00a] Faber, Carel F.; Pandharipande, Rahul Hodge integrals and Gromov–Witten theory, Invent. Math., Volume 139 (2000) no. 1, pp. 173-199 | DOI | MR | Zbl

[FP00b] Faber, Carel F.; Pandharipande, Rahul Logarithmic series and Hodge integrals in the tautological ring, Mich. Math. J., Volume 48 (2000), pp. 215-252 (with an appendix by Don Zagier, dedicated to William Fulton on the occasion of his 60th birthday) | DOI | MR | Zbl

[Ger13] Gerstenberger, Andreas Geometric transversality in higher genus Gromov–Witten theory (2013) | arXiv | Zbl

[Giv96] Givental, Alexander B. Equivariant Gromov–Witten invariants, Int. Math. Res. Not., Volume 1996 (1996) no. 13, pp. 613-663 | DOI | MR | Zbl

[Giv01a] Givental, Alexander B. Gromov–Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J., Volume 1 (2001) no. 4, pp. 551-568 (dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary) | Zbl | DOI | MR

[Giv01b] Givental, Alexander B. Semisimple Frobenius structures at higher genus, Int. Math. Res. Not., Volume 2001 (2001) no. 23, pp. 1265-1286 | DOI | MR | Zbl

[GK95] Givental, Alexander B.; Kim, Bumsig Quantum cohomology of flag manifolds and Toda lattices, Commun. Math. Phys., Volume 168 (1995) no. 3, pp. 609-641 | DOI | MR | Zbl

[GKM98] Goresky, Mark; Kottwitz, Robert; MacPherson, Robert Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math., Volume 131 (1998) no. 1, pp. 25-83 | DOI | MR | Zbl

[GKZ20a] Goertsches, Oliver; Konstantis, Panagiotis; Zoller, Leopold GKM theory and Hamiltonian non-Kähler actions in dimension 6, Adv. Math., Volume 368 (2020), 107141, 17 pages | DOI | MR | Zbl

[GKZ20b] Goertsches, Oliver; Konstantis, Panagiotis; Zoller, Leopold Symplectic and Kähler structures on biquotients, J. Symplectic Geom., Volume 18 (2020) no. 3, pp. 791-813 | MR | Zbl | DOI

[GKZ22a] Goertsches, Oliver; Konstantis, Panagiotis; Zoller, Leopold GKM manifolds are not rigid, Algebr. Geom. Topol., Volume 22 (2022) no. 7, pp. 3511-3532 | DOI | MR | Zbl

[GKZ22b] Goertsches, Oliver; Konstantis, Panagiotis; Zoller, Leopold Low-dimensional GKM theory (2022) | arXiv

[GKZ23] Goertsches, Oliver; Konstantis, Panagiotis; Zoller, Leopold Realization of GKM fibrations and new examples of Hamiltonian non-Kähler actions, Compos. Math., Volume 159 (2023) no. 10, pp. 2149-2190 | DOI | MR | Zbl

[GP99] Graber, Tom; Pandharipande, Rahul Localization of virtual classes, Invent. Math., Volume 135 (1999) no. 2, pp. 487-518 | DOI | MR | Zbl

[Gro85] Gromov, Mikhael L. Pseudo holomorphic curves in symplectic manifolds, Invent. Math., Volume 82 (1985) no. 2, pp. 307-347 | DOI | MR | Zbl

[GW22] González, Eduardo A.; Woodward, Christopher T. Quantum Kirwan for quantum K-theory, Facets of algebraic geometry. Vol. I (London Mathematical Society Lecture Note Series), Volume 472, Cambridge University Press, 2022, pp. 265-332 | DOI | MR | Zbl

[GZ00] Guillemin, Victor W.; Zara, Catalin Equivariant de Rham theory and graphs, Surveys in differential geometry (Surveys in Differential Geometry), Volume 7, International Press, 2000, pp. 221-257 | DOI | MR | Zbl

[GZ01] Guillemin, Victor W.; Zara, Catalin 1-skeleta, Betti numbers, and equivariant cohomology, Duke Math. J., Volume 107 (2001) no. 2, pp. 283-349 | DOI | MR | Zbl

[HKK + 03] Hori, Kentaro; Katz, Sheldon; Klemm, Albrecht; Pandharipande, Rahul; Thomas, Richard; Vafa, Cumrun; Vakil, Ravi; Zaslow, Eric Mirror symmetry, Clay Mathematics Monographs, 1, American Mathematical Society; Clay Mathematics Institute, 2003 (with a preface by Vafa) | MR | Zbl

[HS24] Hirschi, Amanda; Swaminathan, Mohan Global Kuranishi charts and a product formula in symplectic Gromov–Witten theory, Sel. Math., New Ser., Volume 30 (2024) no. 5, 87, 74 pages | DOI | MR | Zbl

[Hsi75] Hsiang, Wu-yi Cohomology theory of topological transformation groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 85, Springer, 1975 | DOI | MR | Zbl

[HW24] Hirschi, Amanda; Wang, Luya On Donaldson’s 4-6 question (2024) | arXiv

[HWZ17] Hofer, Helmut H. W.; Wysocki, Krzysztof; Zehnder, Eduard J. Applications of polyfold theory I: The polyfolds of Gromov–Witten theory, Memoirs of the American Mathematical Society, 1179, American Mathematical Society, 2017 | DOI | MR | Zbl

[HZ99] Hofer, Helmut H. W.; Zehnder, Eduard J. Pseudoholomorphic curves and dynamics, The Arnoldfest (Toronto, ON, 1997) (Fields Institute Communications), Volume 24, American Mathematical Society, 1999, pp. 225-239 | DOI | MR | Zbl

[IP13] Ionel, Eleny-Nicoleta; Parker, Thomas H. A natural Gromov–Witten virtual fundamental class (2013) | arXiv | Zbl

[IP19a] Ionel, Eleny-Nicoleta; Parker, Thomas H. Relating VFCs on thin compactifications, Math. Ann., Volume 375 (2019), pp. 845-893 | MR | DOI | Zbl

[IP19b] Ionel, Eleny-Nicoleta; Parker, Thomas H. Thin compactifications and relative fundamental classes, J. Symplectic Geom., Volume 17 (2019) no. 3, pp. 703-752 | DOI | MR | Zbl

[Ive86] Iversen, Birger Cohomology of sheaves, Universitext, Springer, 1986 | DOI | MR | Zbl

[Joy17] Joyce, Dominic Kuranishi spaces and Symplectic Geometry, 2017 https://people.maths.ox.ac.uk/joyce/kubook.pdf

[Kir84] Kirwan, Frances Clare Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, 31, Princeton University Press, 1984 | DOI | MR | Zbl

[KKP03] Kim, Bumsig; Kresch, Andrew; Pantev, Tony Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra, Volume 179 (2003) no. 1-2, pp. 127-136 | MR | DOI | Zbl

[KM94] Kontsevich, Maxim; Manin, Yuriĭ I. Gromov–Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys., Volume 164 (1994) no. 3, pp. 525-562 | DOI | MR | Zbl

[KM98] Kontsevich, Maxim; Manin, Yuriĭ I. Relations between the correlators of the topological sigma-model coupled to gravity, Commun. Math. Phys., Volume 196 (1998) no. 2, pp. 385-398 | MR | Zbl | DOI

[Kon95] Kontsevich, Maxim Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island, 1994) (Progress in Mathematics), Volume 129, Birkhäuser, 1995, pp. 335-368 | DOI | MR | Zbl

[KS90] Kashiwara, Masaki; Schapira, Pierre Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, 292, Springer, 1990 | DOI | MR | Zbl

[Kur09] Kuroki, Shintarô Introduction to GKM theory, 2009 https://mathsci.kaist.ac.kr/...

[Las79] Lashof, Richard Stable G-smoothing, Algebraic topology, Waterloo, 1978 (Lecture Notes in Mathematics), Volume 741, Springer, 1979, pp. 283-306 | DOI | MR | Zbl

[Liu13] Liu, Chiu-Chu Melissa Localization in Gromov–Witten theory and orbifold Gromov–Witten theory, Handbook of moduli. Vol. II (Advanced Lectures in Mathematics), Volume 25, International Press, 2013, pp. 353-425 | MR | Zbl

[LMP99] Lalonde, François; McDuff, Dusa; Polterovich, Leonid Topological rigidity of Hamiltonian loops and quantum homology, Invent. Math., Volume 135 (1999) no. 2, pp. 369-385 | DOI | MR | Zbl

[LP21] Lindsay, Nicholas; Panov, Dmitri Symplectic and Kähler structures on ℂℙ 1 -bundles over ℂℙ 2 , Sel. Math., New Ser., Volume 27 (2021) no. 5, 93, 24 pages | Zbl | DOI | MR

[LS17] Liu, Chiu-Chu Melissa; Sheshmani, Artan Equivariant Gromov–Witten invariants of algebraic GKM manifolds, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 13 (2017), 048, 21 pages | DOI | MR | Zbl

[LT98] Li, Jun; Tian, Gang Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds, Topics in symplectic 4-manifolds (Irvine, CA, 1996) (First International Press Lecture Series), Volume 1, International Press, 1998, pp. 47-83 | MR | Zbl

[LT99] Li, Jun; Tian, Gang Comparison of algebraic and symplectic Gromov–Witten invariants, Asian J. Math., Volume 3 (1999) no. 3, pp. 689-728 | DOI | MR | Zbl

[Man99] Manin, Yuri I. Frobenius manifolds, quantum cohomology, and moduli spaces, Colloquium Publications, 47, American Mathematical Society, 1999 | DOI | MR | Zbl

[McD09] McDuff, Dusa Hamiltonian S 1 -manifolds are uniruled, Duke Math. J., Volume 146 (2009) no. 3, pp. 449-507 | DOI | MR | Zbl

[Mei98] Meinrenken, Eckhard Symplectic surgery and the Spin c -Dirac operator, Adv. Math., Volume 134 (1998) no. 2, pp. 240-277 | DOI | MR | Zbl

[MS12] McDuff, Dusa; Salamon, Dietmar A. J-holomorphic curves and symplectic topology, Colloquium Publications, 52, American Mathematical Society, 2012 | MR | Zbl

[Mum83] Mumford, David Towards an enumerative geometry of the moduli space of curves, Arithmetic and Geometry: Papers dedicated to IR Shafarevich on the Occasion of His Sixtieth Birthday. Volume II: Geometry, Springer, 1983, pp. 271-328 | Zbl

[Mun01] Mundet I Riera, Ignasi Lifts of smooth group actions to line bundles, Bull. Lond. Math. Soc., Volume 33 (2001) no. 3, pp. 351-361 | MR | DOI | Zbl

[Mun03] Mundet I Riera, Ignasi Hamiltonian Gromov–Witten invariants, Topology, Volume 42 (2003) no. 3, pp. 525-553 | Zbl | DOI

[MW17] McDuff, Dusa; Wehrheim, Katrin Smooth Kuranishi atlases with isotropy, Geom. Topol., Volume 21 (2017) no. 5, pp. 2725-2809 | DOI | MR | Zbl

[NWZ14] Nguyen, Khoa Lu; Woodward, Christopher T.; Ziltener, Fabian Morphisms of CohFT algebras and quantization of the Kirwan map, Symplectic, Poisson, and noncommutative geometry (Mathematical Sciences Research Institute Publications), Volume 62, Cambridge University Press, 2014, pp. 131-170 | DOI | MR | Zbl

[Par16] Pardon, John An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves, Geom. Topol., Volume 20 (2016) no. 2, pp. 779-1034 | DOI | MR | Zbl

[Par22] Pardon, John Enough vector bundles on orbispaces, Compos. Math., Volume 158 (2022) no. 11, pp. 2046-2081 | MR | DOI | Zbl

[Pix13] Pixton, Aaron The tautological ring of the moduli space of curves, Ph. D. Thesis, Princeton University, Princeton, USA (2013) | MR

[RRS08] Robbin, Joel W.; Ruan, Yongbin; Salamon, Dietmar A. The moduli space of regular stable maps, Math. Z., Volume 259 (2008) no. 3, pp. 525-574 | DOI | MR | Zbl

[RT95] Ruan, Yongbin; Tian, Gang A mathematical theory of quantum cohomology, J. Differ. Geom., Volume 42 (1995) no. 2, pp. 259-367 | DOI | MR | Zbl

[RT97] Ruan, Yongbin; Tian, Gang Higher genus symplectic invariants and sigma models coupled with gravity, Invent. Math., Volume 130 (1997) no. 3, pp. 455-516 | MR | Zbl | DOI

[Sal99] Salamon, Dietmar A. Lectures on Floer homology (IAS/Park City Mathematics Series), Volume 7, American Mathematical Society, 1999, pp. 143-229 | DOI | MR | Zbl

[Sch23] Schmaltz, Wolfgang Pseudocycle Gromov–Witten invariants are a strict subset of polyfold Gromov–Witten invariants (2023) | arXiv | Zbl

[Sei99] Seidel, Paul On the group of symplectic automorphisms of CP m ×CP n , Northern California Symplectic Geometry Seminar (American Mathematical Society Translations, Series 2), Volume 196, American Mathematical Society, 1999, pp. 237-250 | DOI | MR | Zbl

[Sie99] Siebert, Bernd Algebraic and symplectic Gromov–Witten invariants coincide, Ann. Inst. Fourier, Volume 49 (1999) no. 6, pp. 1743-1795 | Numdam | DOI | MR | Zbl

[Spi99] Spielberg, Holger The Gromov–Witten invariants of symplectic toric manifolds, and their quantum cohomology ring, C. R. Acad. Sci., Paris, Sér. I, Math., Volume 329 (1999) no. 8, pp. 699-704 | DOI | MR | Zbl

[Swa21] Swaminathan, Mohan Rel-C structures on Gromov–Witten moduli spaces, J. Symplectic Geom., Volume 19 (2021) no. 2, pp. 413-473 | DOI | MR | Zbl

[Tel12] Teleman, Constantin The structure of 2D semi-simple field theories, Invent. Math., Volume 188 (2012) no. 3, pp. 525-588 | DOI | MR | Zbl

[Tol98] Tolman, Susan Examples of non-Kähler Hamiltonian torus actions, Invent. Math., Volume 131 (1998) no. 2, pp. 299-310 | DOI | MR | Zbl

[TX17] Tian, Gang; Xu, Guangbo The symplectic approach of gauged linear σ-model, Proceedings of the Gökova Geometry-Topology Conference 2016, International Press; Gökova Geometry/Topology Conference (GGT) (2017), pp. 86-111 | MR | Zbl

[Ush11] Usher, Michael Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms, Geom. Topol., Volume 15 (2011) no. 3, pp. 1313-1417 | DOI | MR | Zbl

[Wit91] Witten, Edward Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry. Vol. I: Proceedings of the conference on geometry and topology, (Cambridge, MA, 1990), American Mathematical Society; Lehigh University, 1991, pp. 243-310 | MR | Zbl

[Woo98] Woodward, Christopher T. Multiplicity-free Hamiltonian actions need not be Kähler, Invent. Math., Volume 131 (1998) no. 2, pp. 311-319 | DOI | MR | Zbl

[Xu24] Xu, Guangbo Quantum Kirwan map and quantum Steenrod operation (2024) | arXiv | Zbl

[Zin08] Zinger, Aleksey Pseudocycles and integral homology, Trans. Am. Math. Soc., Volume 360 (2008) no. 5, pp. 2741-2765 | MR | DOI | Zbl

[Zin17] Zinger, Aleksey Real Ruan–Tian Perturbations (2017) | arXiv | Zbl