Seiberg–Witten and Gromov invariants for self-dual harmonic 2–forms

Gerig, Chris¹

¹ Department of Mathematics, UC Berkeley, Berkeley, CA, United States

Geometry & topology, Tome 26 (2022) no. 8, pp. 3307-3365.

Résumé

Our earlier work (Geom. Topol. 24 (2020) 1791–1839) gives an extension of Taubes’ “$\mathrm{SW}=\mathrm{Gr}$” theorem to nonsymplectic $4$–manifolds. The main result of this sequel asserts the following: whenever the Seiberg–Witten invariants are defined over a closed minimal $4$–manifold $X$, they are equivalent modulo 2 to “near-symplectic” Gromov invariants in the presence of certain self-dual harmonic $2$–forms on $X$. A version for nonminimal $4$–manifolds is also proved. A corollary to Morse theory on $3$–manifolds is also announced, recovering a result of Hutchings, Lee, and Turaev about the $3$–dimensional Seiberg–Witten invariants.

Détail

Classification : 53D42, 57R57
Keywords: near-symplectic, Seiberg–Witten, Gromov, pseudoholomorphic, ECH, monopole Floer

Bibliographie

[1] E Calabi, An intrinsic characterization of harmonic one-forms, from: "Global analysis" (editors D C Spencer, S Iyanaga), Princeton Math. Ser. 29, Princeton Univ. Press (1969) 101 | DOI

[2] D Cristofaro-Gardiner, The absolute gradings on embedded contact homology and Seiberg–Witten Floer cohomology, Algebr. Geom. Topol. 13 (2013) 2239 | DOI

[3] C Gerig, Circle-valued Morse theory and Seiberg–Witten invariants of 3–manifolds, in preparation

[4] C Gerig, Taming the pseudoholomorphic beasts in R × (S1 × S2), Geom. Topol. 24 (2020) 1791 | DOI

[5] R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148 (1998) 619 | DOI

[6] K Honda, A note on Morse theory of harmonic 1–forms, Topology 38 (1999) 223 | DOI

[7] K Honda, Transversality theorems for harmonic forms, Rocky Mountain J. Math. 34 (2004) 629 | DOI

[8] M Hutchings, Embedded contact homology as a (symplectic) field theory, in preparation

[9] M Hutchings, Reidemeister torsion in generalized Morse theory, Forum Math. 14 (2002) 209 | DOI

[10] M Hutchings, The embedded contact homology index revisited, from: "New perspectives and challenges in symplectic field theory" (editors M Abreu, F Lalonde, L Polterovich), CRM Proc. Lecture Notes 49, Amer. Math. Soc. (2009) 263 | DOI

[11] M Hutchings, Y J Lee, Circle-valued Morse theory and Reidemeister torsion, Geom. Topol. 3 (1999) 369 | DOI

[12] M Hutchings, Y J Lee, Circle-valued Morse theory, Reidemeister torsion, and Seiberg–Witten invariants of 3–manifolds, Topology 38 (1999) 861 | DOI

[13] M Hutchings, C H Taubes, Proof of the Arnold chord conjecture in three dimensions, II, Geom. Topol. 17 (2013) 2601 | DOI

[14] A Jaffe, C Taubes, Vortices and monopoles: structure of static gauge theories, 2, Birkhäuser (1980)

[15] P Kronheimer, T Mrowka, Monopoles and three-manifolds, 10, Cambridge Univ. Press (2007) | DOI

[16] P Kronheimer, T Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries, Ann. of Math. 165 (2007) 457 | DOI

[17] Ç Kutluhan, Y J Lee, C H Taubes, HF = HM, V : Seiberg–Witten Floer homology and handle additions, Geom. Topol. 24 (2020) 3471 | DOI

[18] C Lebrun, Yamabe constants and the perturbed Seiberg–Witten equations, Comm. Anal. Geom. 5 (1997) 535 | DOI

[19] T J Li, A K Liu, The equivalence between SW and Gr in the case where b+ = 1, Int. Math. Res. Not. 1999 (1999) 335 | DOI

[20] K Luttinger, C Simpson, A normal form for the birth/flight of closed selfdual 2–form degeneracies, unpublished (1996)

[21] N Manton, P Sutcliffe, Topological solitons, Cambridge Univ. Press (2004) | DOI

[22] C Okonek, A Teleman, 3–dimensional Seiberg–Witten invariants and non-Kählerian geometry, Math. Ann. 312 (1998) 261 | DOI

[23] P Ozsváth, Z Szabó, The symplectic Thom conjecture, Ann. of Math. 151 (2000) 93 | DOI

[24] D A Salamon, Seiberg–Witten invariants of mapping tori, symplectic fixed points, and Lefschetz numbers, Turkish J. Math. 23 (1999) 117

[25] C H Taubes, Arbitrary N–vortex solutions to the first order Ginzburg–Landau equations, Comm. Math. Phys. 72 (1980) 277 | DOI

[26] C H Taubes, Counting pseudo-holomorphic submanifolds in dimension 4, J. Differential Geom. 44 (1996) 818

[27] C H Taubes, SW ⇒ Gr : from the Seiberg–Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996) 845 | DOI

[28] C H Taubes, Gr ⇒ SW : from pseudo-holomorphic curves to Seiberg–Witten solutions, J. Differential Geom. 51 (1999) 203

[29] C H Taubes, Gr = SW : counting curves and connections, J. Differential Geom. 52 (1999) 453

[30] C H Taubes, Seiberg–Witten and Gromov invariants for symplectic 4–manifolds, 2, International (2000)

[31] C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, I, Geom. Topol. 14 (2010) 2497 | DOI

[32] C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, II, Geom. Topol. 14 (2010) 2583 | DOI

[33] C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, III, Geom. Topol. 14 (2010) 2721 | DOI

[34] C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, IV, Geom. Topol. 14 (2010) 2819 | DOI

[35] C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, V, Geom. Topol. 14 (2010) 2961 | DOI

[36] V Turaev, Torsion invariants of Spinc–structures on 3–manifolds, Math. Res. Lett. 4 (1997) 679 | DOI

[37] V Turaev, A combinatorial formulation for the Seiberg–Witten invariants of 3–manifolds, Math. Res. Lett. 5 (1998) 583 | DOI

[38] C Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, Comment. Math. Helv. 85 (2010) 347 | DOI