Good Wild Harmonic Bundles and Good Filtered Higgs Bundles

Takuro Mochizuki

Symmetry, integrability and geometry: methods and applications, Tome 17 (2021)

Résumé

We prove the Kobayashi–Hitchin correspondence between good wild harmonic bundles and polystable good filtered $\lambda$-flat bundles satisfying a vanishing condition. We also study the correspondence for good wild harmonic bundles with the homogeneity with respect to a group action, which is expected to provide another way to construct Frobenius manifolds.

Détail

Keywords: wild harmonic bundles, Higgs bundles, $\lambda$-flat bundles, Kobayashi–Hitchin correspondence.

Bibliographie

[1] Barannikov S., “Quantum periods. I. Semi-infinite variations of Hodge structures”, Int. Math. Res. Not., 2001 (2001), 1243–1264, arXiv: math.AG/0006193 | DOI | MR | Zbl

[2] Biquard O., “Fibrés de Higgs et connexions intégrables: le cas logarithmique (diviseur lisse)”, Ann. Sci. École Norm. Sup. (4), 30 (1997), 41–96 | DOI | MR | Zbl

[3] Biquard O., Boalch P., “Wild non-abelian Hodge theory on curves”, Compos. Math., 140 (2004), 179–204, arXiv: math.DG/0111098 | DOI | MR | Zbl

[4] Bogomolov F. A., “Holomorphic tensors and vector bundles on projective manifolds”, Math. USSR Izv., 13 (1979), 499–555 | DOI | MR | Zbl

[5] Borne N., “Fibrés paraboliques et champ des racines”, Int. Math. Res. Not., 2007 (2007), rnm049, 38 pp., arXiv: math.AG/0604458 | DOI | MR | Zbl

[6] Borne N., “Sur les représentations du groupe fondamental d'une variété privée d'un diviseur à croisements normaux simples”, Indiana Univ. Math. J., 58 (2009), 137–180, arXiv: 0704.1236 | DOI | MR | Zbl

[7] Cecotti S., Vafa C., “Topological–anti-topological fusion”, Nuclear Phys. B, 367 (1991), 359–461 | DOI | MR | Zbl

[8] Cecotti S., Vafa C., “On classification of $N=2$ supersymmetric theories”, Comm. Math. Phys., 158 (1993), 569–644, arXiv: hep-th/9211097 | DOI | MR | Zbl

[9] Coates T., Iritani H., Tseng H. H., “Wall-crossings in toric Gromov–Witten theory. I Crepant examples”, Geom. Topol., 13 (2009), 2675–2744, arXiv: math.AG/0611550 | DOI | MR | Zbl

[10] Collier B., Wentworth R., “Conformal limits and the Białynicki-Birula stratification of the space of $\lambda$-connections”, Adv. Math., 350 (2019), 1193–1225, arXiv: 1808.01622 | DOI | MR | Zbl

[11] Corlette K., “Flat $G$-bundles with canonical metrics”, J. Differential Geom., 28 (1988), 361–382 | DOI | MR | Zbl

[12] Cornalba M., Griffiths P., “Analytic cycles and vector bundles on non-compact algebraic varieties”, Invent. Math., 28 (1975), 1–106 | DOI | MR | Zbl

[13] Donagi R., Pantev T., “Geometric Langlands and non-abelian Hodge theory”, Geometry, Analysis, and Algebraic Geometry: Forty Years of the Journal of Differential Geometry, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009, 85–116 | DOI | MR | Zbl

[14] Donaldson S. K., “A new proof of a theorem of Narasimhan and Seshadri”, J. Differential Geom., 18 (1983), 269–277 | DOI | MR | Zbl

[15] Donaldson S. K., “Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles”, Proc. London Math. Soc., 50 (1985), 1–26 | DOI | MR | Zbl

[16] Donaldson S. K., “Infinite determinants, stable bundles and curvature”, Duke Math. J., 54 (1987), 231–247 | DOI | MR | Zbl

[17] Donaldson S. K., “Twisted harmonic maps and the self-duality equations”, Proc. London Math. Soc., 55 (1987), 127–131 | DOI | MR | Zbl

[18] Dubrovin B., “Geometry and integrability of topological-antitopological fusion”, Comm. Math. Phys., 152 (1993), 539–564, arXiv: hep-th/9206037 | DOI | MR | Zbl

[19] Gieseker D., “On a theorem of Bogomolov on Chern classes of stable bundles”, Amer. J. Math., 101 (1979), 77–85 | DOI | MR | Zbl

[20] Grothendieck A., “Techniques de construction et théorèmes d'existence en géométrie algébrique. IV Les schémas de Hilbert”, Séminaire Bourbaki, 6, no. 221, Soc. Math. France, Paris, 1961, 28 pp. | MR

[21] Hertling C., “$tt^*$ geometry, Frobenius manifolds, their connections, and the construction for singularities”, J. Reine Angew. Math., 555 (2003), 77–161, arXiv: math.AG/0203054 | DOI | MR | Zbl

[22] Hertling C., Sevenheck C., “Nilpotent orbits of a generalization of Hodge structures”, J. Reine Angew. Math., 609 (2007), 23–80, arXiv: math.AG/0603564 | DOI | MR | Zbl

[23] Hertling C., Sevenheck C., “Limits of families of Brieskorn lattices and compactified classifying spaces”, Adv. Math., 223 (2010), 1155–1224, arXiv: 0805.4777 | DOI | MR | Zbl

[24] Hitchin N., “The self-duality equations on a Riemann surface”, Proc. London Math. Soc., 55 (1987), 59–126 | DOI | MR | Zbl

[25] Hitchin N., “A note on vanishing theorems”, Geometry and Analysis on Manifolds, Progr. Math., 308, Birkhäuser/Springer, Cham, 2015, 373–382 | DOI | MR | Zbl

[26] Hu Z., Huang P., Simpson–Mochizuki correspondence for $\lambda$-flat bundles, arXiv: 1905.10765

[27] Huang P., “Non-Abelian Hodge theory and related topics”, SIGMA, 16 (2020), 029, 34 pp., arXiv: 1908.08348 | DOI | MR | Zbl

[28] Iritani H., $tt^{\ast}$-geometry in quantum cohomology, arXiv: 0906.1307

[29] Iyer J. N.N., Simpson C. T., “A relation between the parabolic Chern characters of the de Rham bundles”, Math. Ann., 338 (2007), 347–383, arXiv: math.AG/0603677 | DOI | MR | Zbl

[30] Jost J., Zuo K., “Harmonic maps of infinite energy and rigidity results for representations of fundamental groups of quasiprojective varieties”, J. Differential Geom., 47 (1997), 469–503 | DOI | MR | Zbl

[31] Kashiwara M., “Semisimple holonomic $\mathcal D$-modules”, Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progr. Math., 160, Birkhäuser Boston, Boston, MA, 1998, 267–271 | DOI | MR | Zbl

[32] Kobayashi S., “First Chern class and holomorphic tensor fields”, Nagoya Math. J., 77 (1980), 5–11 | DOI | MR | Zbl

[33] Kobayashi S., “Curvature and stability of vector bundles”, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 158–162 | DOI | MR

[34] Kobayashi S., Differential geometry of holomorphic vector bundles, Seminary Note in the University of Tokyo, 41, University of Tokyo, Japan, 1982 | MR

[35] Kobayashi S., Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, Princeton, NJ, 1987 | DOI | MR | Zbl

[36] Kotake T., Ochiai T. (eds.), Non-linear problems in geometry, Proceedings of the Sixth International Taniguchi Symposium, The Taniguchi Foundation, Tohoku University, Japan, 1979 | MR

[37] Li J., “Hermitian–Einstein metrics and Chern number inequalities on parabolic stable bundles over Kähler manifolds”, Comm. Anal. Geom., 8 (2000), 445–475 | DOI | MR | Zbl

[38] Li J., Narasimhan M. S., “Hermitian–Einstein metrics on parabolic stable bundles”, Acta Math. Sin. (Engl. Ser.), 15 (1999), 93–114 | DOI | MR | Zbl

[39] Lübke M., “Chernklassen von Hermite–Einstein–Vektorbündeln”, Math. Ann., 260 (1982), 133–141 | DOI | MR | Zbl

[40] Lübke M., “Stability of Einstein–Hermitian vector bundles”, Manuscripta Math., 42 (1983), 245–257 | DOI | MR | Zbl

[41] Lübke M., Teleman A., The universal Kobayashi–Hitchin correspondence on Hermitian manifolds, Mem. Amer. Math. Soc., 183, 2006, vi+97 pp., arXiv: math.DG/0402341 | DOI | MR

[42] Maruyama M., Yokogawa K., “Moduli of parabolic stable sheaves”, Math. Ann., 293 (1992), 77–99 | DOI | MR | Zbl

[43] Mehta V. B., Ramanathan A., “Semistable sheaves on projective varieties and their restriction to curves”, Math. Ann., 258 (1982), 213–224 | DOI | MR | Zbl

[44] Mehta V. B., Ramanathan A., “Restriction of stable sheaves and representations of the fundamental group”, Invent. Math., 77 (1984), 163–172 | DOI | MR | Zbl

[45] Mehta V. B., Seshadri C. S., “Moduli of vector bundles on curves with parabolic structures”, Math. Ann., 248 (1980), 205–239 | DOI | MR | Zbl

[46] Mochizuki T., Kobayashi–Hitchin correspondence for tame harmonic bundles and an application, Astérisque, 309, 2006, viii+117 pp. | MR | Zbl

[47] Mochizuki T., Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules, v. I, Mem. Amer. Math. Soc., 185, 2007, xii+324 pp. | DOI | MR

[48] Mochizuki T., Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules, v. II, Mem. Amer. Math. Soc., 185, 2007, xii+565 pp. | DOI | MR

[49] Mochizuki T., “Kobayashi–Hitchin correspondence for tame harmonic bundles. II”, Geom. Topol., 13 (2009), 359–455, arXiv: math.DG/0602266 | DOI | MR | Zbl

[50] Mochizuki T., “Asymptotic behavior of variation of pure polarized TERP structure”, Publ. Res. Inst. Math. Sci., 47 (2011), 419–534, arXiv: 0811.1384 | DOI | MR | Zbl

[51] Mochizuki T., Wild harmonic bundles and wild pure twistor $D$-modules, Astérisque, 340, 2011, x+607 pp. | MR | Zbl

[52] Mochizuki T., “Harmonic bundles and Toda lattices with opposite sign I”, RIMS Kôkyûroku Bessatsu (to appear) , arXiv: 1301.1718 | MR

[53] Mochizuki T., “Harmonic bundles and Toda lattices with opposite sign II”, Comm. Math. Phys., 328 (2014), 1159–1198, arXiv: 1301.1718 | DOI | MR | Zbl

[54] Mochizuki T., “Wild harmonic bundles and twistor $\mathcal D$-modules”, Proceedings of the International Congress of Mathematician (Seoul, 2014), v. 1, Kyung Moon Sa, Seoul, 2014, 499–527 | MR | Zbl

[55] Mochizuki T., Mixed twistor $\mathcal D$-modules, Lecture Notes in Math., 2125, Springer, Cham, 2015 | DOI | MR | Zbl

[56] Mumford D., “Projective invariants of projective structures and applications”, Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, 526–530 | MR

[57] Mumford D., Lectures on curves on an algebraic surface, Annals of Mathematics Studies, 59, Princeton University Press, Princeton, N.J., 1966 | MR | Zbl

[58] Narasimhan M. S., Seshadri C. S., “Stable and unitary vector bundles on a compact Riemann surface”, Ann. of Math., 82 (1965), 540–567 | DOI | MR | Zbl

[59] Sabbah C., “Harmonic metrics and connections with irregular singularities”, Ann. Inst. Fourier (Grenoble), 49 (1999), 1265–1291, arXiv: math.AG/9905039 | DOI | MR | Zbl

[60] Sabbah C., Polarizable twistor $\mathcal D$-modules, Astérisque, 300, 2005, vi+208 pp., arXiv: math.AG/0503038 | MR | Zbl

[61] Saito K., Takahashi A., “From primitive forms to Frobenius manifolds”, From Hodge Theory to Integrability and TQFT $tt^*$-Geometry, Proc. Sympos. Pure Math., 78, Amer. Math. Soc., Providence, RI, 2008, 31–48 | DOI | MR | Zbl

[62] Simpson C. T., “Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization”, J. Amer. Math. Soc., 1 (1988), 867–918 | DOI | MR | Zbl

[63] Simpson C. T., “Harmonic bundles on noncompact curves”, J. Amer. Math. Soc., 3 (1990), 713–770 | DOI | MR | Zbl

[64] Simpson C. T., “Higgs bundles and local systems”, Inst. Hautes Études Sci. Publ. Math., 75, 1992, 5–95 | DOI | MR | Zbl

[65] Simpson C. T., “The Hodge filtration on nonabelian cohomology”, Algebraic Geometry, Santa Cruz, 1995, Proc. Sympos. Pure Math., 62, Amer. Math. Soc., Providence, RI, 1997, 217–281, arXiv: alg-geom/9604005 | DOI | MR | Zbl

[66] Simpson C. T., Mixed twistor structures, arXiv: alg-geom/9705006

[67] Simpson C. T., “Iterated destabilizing modifications for vector bundles with connection”, Vector Bundles and Complex Geometry, Contemp. Math., 522, Amer. Math. Soc., Providence, RI, 2010, 183–206, arXiv: 0812.3472 | DOI | MR | Zbl

[68] Siu Y. T., Techniques of extension of analytic objects, Lecture Notes in Pure and Applied Mathematics, 8, Marcel Dekker, Inc., New York, 1974 | MR | Zbl

[69] Siu Y. T., “The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds”, Ann. of Math., 112 (1980), 73–111 | DOI | MR | Zbl

[70] Steer B., Wren A., “The Donaldson–Hitchin–Kobayashi correspondence for parabolic bundles over orbifold surfaces”, Canad. J. Math., 53 (2001), 1309–1339 | DOI | MR | Zbl

[71] Takemoto F., “Stable vector bundles on algebraic surfaces”, Nagoya Math. J., 47 (1972), 29–48 | DOI | MR | Zbl

[72] Takemoto F., “Stable vector bundles on algebraic surfaces. II”, Nagoya Math. J., 52 (1973), 173–195 | DOI | MR | Zbl

[73] Uhlenbeck K., Yau S.-T., “On the existence of Hermitian–Yang–Mills connections in stable vector bundles”, Comm. Pure Appl. Math., 39 (1986), S257–S293 | DOI | MR | Zbl

[74] Uhlenbeck K. K., “Connections with $L^{p}$ bounds on curvature”, Comm. Math. Phys., 83 (1982), 31–42 | DOI | MR | Zbl