Pseudoholomorphic curves relative to a normal crossings symplectic divisor: compactification

Farajzadeh-Tehrani, Mohammad¹

¹ Department of Mathematics, The University of Iowa, Iowa City, IA, United States

Geometry & topology, Tome 26 (2022) no. 3, pp. 989-1075.

Résumé

Inspired by the log Gromov–Witten (or GW) theory of Gross–Siebert/Abramovich–Chen, we introduce a geometric notion of log $J$–holomorphic curve relative to a simple normal crossings symplectic divisor defined by Tehrani–McLean–Zinger (2018). Every such moduli space is characterized by a second homology class, genus and contact data. For certain almost complex structures, we show that the moduli space of stable log $J$–holomorphic curves of any fixed type is compact and metrizable with respect to an enhancement of the Gromov topology. In the case of smooth symplectic divisors, our compactification is often smaller than the relative compactification and there is a projection map from the latter onto the former. The latter is constructed via expanded degenerations of the target. Our construction does not need any modification of (or any extra structure on) the target. Unlike the classical moduli spaces of stable maps, these log moduli spaces are often virtually singular. We describe an explicit toric model for the normal cone (ie the space of gluing parameters) to each stratum in terms of the defining combinatorial data of that stratum. In an earlier preprint, we introduced a natural set up for studying the deformation theory of log (and relative) curves and obtained a logarithmic analogue of the space of Ruan–Tian perturbations for these moduli spaces. In a forthcoming paper, we will prove a gluing theorem for smoothing log curves in the normal direction to each stratum. With some modifications to the theory of Kuranishi spaces, the latter will allow us to construct a virtual fundamental class for every such log moduli space, and define relative GW invariants without any restriction.

Détail

DOI : 10.2140/gt.2022.26.989

Classification : 14N35, 53D45
Keywords: Gromov-Witten theory, normal crossing divisor, pseudoholomorphic curves, relative compactification

Bibliographie

[1] D Abramovich, Q Chen, Stable logarithmic maps to Deligne–Faltings pairs, II, Asian J. Math. 18 (2014) 465 | DOI

[2] D Abramovich, Q Chen, D Gillam, Y Huang, M Olsson, M Satriano, S Sun, Logarithmic geometry and moduli, from: "Handbook of moduli, I" (editors G Farkas, I Morrison), Adv. Lect. Math. 24, International (2013) 1

[3] D Abramovich, Q Chen, M Gross, B Siebert, Punctured logarithmic curves, preprint (2019)

[4] D Abramovich, Q Chen, M Gross, B Siebert, Decomposition of degenerate Gromov–Witten invariants, Compos. Math. 156 (2020) 2020 | DOI

[5] D Abramovich, S Marcus, J Wise, Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations, Ann. Inst. Fourier Grenoble 64 (2014) 1611 | DOI

[6] D Auroux, Mirror symmetry and T–duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. 1 (2007) 51

[7] D A Cox, J B Little, H K Schenck, Toric varieties, 124, Amer. Math. Soc. (2011) | DOI

[8] S K Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996) 666

[9] D Eisenbud, B Sturmfels, Binomial ideals, Duke Math. J. 84 (1996) 1 | DOI

[10] Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, from: "Visions in mathematics" (editors N Alon, J Bourgain, A Connes, M Gromov, V Milman), Birkhäuser (= GAFA special volume) (2000) 560 | DOI

[11] M Farajzadeh-Tehrani, Deformation theory of log pseudoholomorphic curves and logarithmic Ruan–Tian perturbations, preprint (2019)

[12] M Farajzadeh-Tehrani, Pseudoholomorphic curves relative to a normal crossings symplectic divisor: VFC, work in progress (2021)

[13] K Fukaya, Y G Oh, H Ohta, K Ono, Exponential decay estimates and smoothness of the moduli space of pseudoholomorphic curves, preprint (2016)

[14] K Fukaya, K Ono, Arnold conjecture and Gromov–Witten invariant, Topology 38 (1999) 933 | DOI

[15] S Ganatra, D Pomerleano, A log PSS morphism with applications to Lagrangian embeddings, J. Topol. 14 (2021) 291 | DOI

[16] M Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307 | DOI

[17] M Gross, B Siebert, Logarithmic Gromov–Witten invariants, J. Amer. Math. Soc. 26 (2013) 451 | DOI

[18] H Hofer, K Wysocki, E Zehnder, Applications of polyfold theory, I : The polyfolds of Gromov–Witten theory, 1179, Amer. Math. Soc. (2017) | DOI

[19] C Hummel, Gromov’s compactness theorem for pseudoholomorphic curves, 151, Birkhäuser (1997) | DOI

[20] E N Ionel, GW invariants relative to normal crossing divisors, Adv. Math. 281 (2015) 40 | DOI

[21] E N Ionel, T H Parker, Relative Gromov–Witten invariants, Ann. of Math. 157 (2003) 45 | DOI

[22] E N Ionel, T H Parker, The symplectic sum formula for Gromov–Witten invariants, Ann. of Math. 159 (2004) 935 | DOI

[23] A M Li, Y Ruan, Symplectic surgery and Gromov–Witten invariants of Calabi–Yau 3–folds, Invent. Math. 145 (2001) 151 | DOI

[24] J Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001) 509

[25] J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998) 119 | DOI

[26] T J Li, C Y Mak, Geometry of symplectic log Calabi–Yau pairs, ICCM Not. 6 (2018) 42 | DOI

[27] D Mcduff, D Salamon, J–holomorphic curves and quantum cohomology, 6, Amer. Math. Soc. (1994) | DOI

[28] D Mcduff, D Salamon, Introduction to symplectic topology, Oxford Univ. Press (1998)

[29] D Mcduff, D Salamon, J–holomorphic curves and symplectic topology, 52, Amer. Math. Soc. (2004) | DOI

[30] D Mcduff, K Wehrheim, Smooth Kuranishi atlases with isotropy, Geom. Topol. 21 (2017) 2725 | DOI

[31] M Mclean, Reeb orbits and the minimal discrepancy of an isolated singularity, Invent. Math. 204 (2016) 505 | DOI

[32] P Pansu, Compactness, from: "Holomorphic curves in symplectic geometry" (editors M Audin, J Lafontaine), Progr. Math. 117, Birkhäuser (1994) 233 | DOI

[33] J Pardon, An algebraic approach to virtual fundamental cycles on moduli spaces of pseudoholomorphic curves, Geom. Topol. 20 (2016) 779 | DOI

[34] B Parker, Exploded manifolds, Adv. Math. 229 (2012) 3256 | DOI

[35] B Parker, On the value of thinking tropically to understand Ionel’s GW invariants relative normal crossing divisors, preprint (2014)

[36] B Parker, Holomorphic curves in exploded manifolds: compactness, Adv. Math. 283 (2015) 377 | DOI

[37] B Parker, Tropical gluing formulae for Gromov–Witten invariants, preprint (2017)

[38] B Parker, De Rham theory of exploded manifolds, Geom. Topol. 22 (2018) 1 | DOI

[39] B Parker, Holomorphic curves in exploded manifolds: regularity, Geom. Topol. 23 (2019) 1621 | DOI

[40] B Parker, Holomorphic curves in exploded manifolds: virtual fundamental class, Geom. Topol. 23 (2019) 1877 | DOI

[41] J W Robbin, D A Salamon, A construction of the Deligne–Mumford orbifold, J. Eur. Math. Soc. 8 (2006) 611 | DOI

[42] Y Ruan, G Tian, Higher genus symplectic invariants and sigma models coupled with gravity, Invent. Math. 130 (1997) 455 | DOI

[43] M F Tehrani, Towards a degeneration formula for the Gromov–Witten invariants of symplectic manifolds, preprint (2017)

[44] M F Tehrani, K Fukaya, Gromov–Witten theory via Kuranishi structures, from: "Virtual fundamental cycles in symplectic topology" (editor J W Morgan), Math. Surveys Monogr. 237, Amer. Math. Soc. (2019) 111 | DOI

[45] M F Tehrani, M Mclean, A Zinger, Normal crossings singularities for symplectic topology, Adv. Math. 339 (2018) 672 | DOI

[46] M F Tehrani, M Mclean, A Zinger, Singularities and semistable degenerations for symplectic topology, C. R. Math. Acad. Sci. Paris 356 (2018) 420 | DOI

[47] M F Tehrani, M Mclean, A Zinger, Normal crossings singularities for symplectic topology, II, preprint (2019)

[48] M F Tehrani, A Zinger, On symplectic sum formulas in Gromov–Witten theory, preprint (2014)

[49] R Ye, Gromov’s compactness theorem for pseudoholomorphic curves, Trans. Amer. Math. Soc. 342 (1994) 671 | DOI

[50] A Zinger, Basic Riemannian geometry and Sobolev estimates used in symplectic topology, preprint (2010)