Current progress on \(G_2\)-instantons over twisted connected sums. (English) Zbl 1447.53007
Karigiannis, Spiro (ed.) et al., Lectures and surveys on \(G_2\)-manifolds and related topics. Minischool and workshop on \(G_2\)-manifolds, Fields Institute, Toronto, Canada, August 19–25, 2017. New York, NY: Springer. Fields Inst. Commun. 84, 319-348 (2020).
Summary: We review a method to construct \(G_2\)-instantons over compact \(G_2\)-manifolds arising as the twisted connected sum of a matching pair of Calabi-Yau 3-folds with cylindrical end, based on the series of articles [M. Jardim et al., Bull. Lond. Math. Soc. 49, No. 1, 117–132 (2017; Zbl 1386.14063); the author, Geom. Topol. 19, No. 1, 61–111 (2015; Zbl 1312.53044); G. Menet, J. Nordström and the author, “Construction of \(G_2 \)-instantons viatwisted connected sums”, Preprint, arXiv:1510.03836; the author and T. Walpuski, Geom. Topol. 19, No. 3, 1263–1285 (2015; Zbl 1317.53034)] by the author and others. The construction is based on gluing \(G_2\)-instantons obtained from holomorphic bundles over such building blocks, subject to natural compatibility and transversality conditions. Explicit examples are obtained from matching pairs of semi-Fano 3-folds by an algorithmic procedure based on the Hartshorne-Serre correspondence.
For the entire collection see [Zbl 1445.53002].
For the entire collection see [Zbl 1445.53002].
MSC:
| 53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |
| 53C10 | \(G\)-structures |
| 53C38 | Calibrations and calibrated geometries |
References:
| [1] | Arrondo, E. (2007). A home-made Hartshorne-Serre correspondence. Revista Matematica Complutense, 20, 423-443. · Zbl 1133.14046 |
| [2] | Atiyah, M. (1988). New invariants of \(3\)- and \(4\)-dimensional manifolds, The mathematical heritage of Hermann Weyl (pp. 285-299). Durham, NC, 1987. · Zbl 0667.57018 |
| [3] | Bryant, R. (1985). Metrics with holonomy \({\rm{G}}_2\) or Spin(7). Lecture Notes in Mathematics, 1111, 269-277. · Zbl 0562.53034 · doi:10.1007/BFb0084595 |
| [4] | Bando, S., & Siu, Y. T. (1994). Stable sheaves and Einstein-Hermitian metrics. Geometry and Analysis on Complex Manifolds, 39, 39-50. · Zbl 0880.32004 · doi:10.1142/9789814350112_0002 |
| [5] | Corti, A., Haskins, M., Nordström, J., & Pacini, T. (2013). Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds. Geometry and Topology, 17(4), 1955-2059. · Zbl 1273.14081 · doi:10.2140/gt.2013.17.1955 |
| [6] | Corti, A., Haskins, M., Nordström, J., & Pacini, T. (2015). \({\rm{G}}_2 \)-manifolds and associative submanifolds via semi-Fano 3-folds. Duke Mathematical Journal, 164(10), 1971-2092. · Zbl 1343.53044 |
| [7] | Crowley, D., & Nordström, J. (2014). Exotic \(G_2\)-manifolds. arXiv:1411.0656 [math.AG]. |
| [8] | Donaldson, S. K. (2002). Floer homology groups in Yang-Mills theory. Cambridge Tracts in Mathematics (Vol. 147). Cambridge: Cambridge University Press. With the assistance of Furuta, M. & Kotschick, D. · Zbl 0998.53057 |
| [9] | Donaldson, S. K. (1985). Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proceedings of the London Mathematical Society, 50(1), 1-26. · Zbl 0529.53018 · doi:10.1112/plms/s3-50.1.1 |
| [10] | Donaldson, S. K. (1990). Polynomial invariants for smooth four-manifolds. Topology, 29(3), 257-315. · Zbl 0715.57007 · doi:10.1016/0040-9383(90)90001-Z |
| [11] | Donaldson, S. K., & Segal, E. (2011). Gauge theory in higher dimensions, II. Surveys in Differential Geometry, 16, 1-41. · Zbl 1256.53038 |
| [12] | Donaldson, S. K., & Thomas, R. P. (1996). Gauge theory in higher dimensions, The geometric universe Oxford, 1998, pp. 31-47. · Zbl 0926.58003 |
| [13] | Haskins, M., Hein, H.-J., & Nordström, J. (2015). Asymptotically cylindrical Calabi-Yau manifolds. Journal of Differential Geometry, 101(2), 213-265. · Zbl 1332.32028 |
| [14] | Huybrechts, D., & Lehn, M. (2010). The geometry of moduli spaces of sheaves (2nd ed.). Cambridge: Cambridge Mathematical Library. · Zbl 1206.14027 · doi:10.1017/CBO9780511711985 |
| [15] | Huybrechts, D., & Lehn, M. (1997). The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31, Friedr. Braunschweig: Vieweg & Sohn. · Zbl 0872.14002 |
| [16] | Jardim, M., Menet, G., Prata, D. M., & Sá Earp, H. N. (2017). Holomorphic bundles for higher dimensional gauge theory. Bulletin of the London Mathematical Society, 49(1), 117-132. · Zbl 1386.14063 · doi:10.1112/blms.12017 |
| [17] | Joyce, D. D. (2000). Compact manifolds with special holonomy. Oxford Mathematical Monographs. Oxford: Oxford University Press. · Zbl 1027.53052 |
| [18] | Joyce, D. D. (1996) Compact Riemannian 7-manifolds with holonomy \({\rm{G}}_2 \). I, II. Journal Differential Geometry, 43(2), 291-328, 329-375. · Zbl 0861.53023 |
| [19] | Jacob, A. & Walpuski, T. (2016). Hermitian Yang-Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds. arXiv:1603.07702v1 [math.DG]. · Zbl 1516.53029 |
| [20] | Kovalev, A., & Lee, N.-H. (2011). K3 surfaces with non-symplectic involution and compact irreducible \({\rm{G}}_2 \)-manifolds. Mathematical Proceedings of the Cambridge Philosophical Society, 151(2), 193-218. · Zbl 1228.53064 |
| [21] | Kovalev, A. (2003). Twisted connected sums and special Riemannian holonomy. Journal für die reine und angewandte Mathematik, 565, 125-160. · Zbl 1043.53041 |
| [22] | Lockhart, R. B., & McOwen, R. C. (1985). Elliptic differential operators on noncompact manifolds. Scuola Normale Superiore di Pisa, Classe di Scienze (4), 12(3), 409-447. · Zbl 0615.58048 |
| [23] | Maruyama, M. (1978). Moduli of stable sheaves ii. Journal of Mathematics of Kyoto University, 18, 557-614. · Zbl 0395.14006 · doi:10.1215/kjm/1250522511 |
| [24] | Menet, G., Nordström, J., & Sá Earp, H.N. (2017). Construction of \({\rm{G}}_2 \)-instantons via twisted connected sums. arXiv:1510.03836 [math.AG]. To appear in Mathematical Research Letters. |
| [25] | Maz’ya, V. G., & Plamenevskiĭ, B. A. (1978). Estimates in \(L_p\) and in Hölder classes, and the Miranda-Agmon maximum principle for the solutions of elliptic boundary value problems in domains with singular points on the boundary. Mathematische Nachrichten, 81, 25-82. |
| [26] | Milnor, J. W., & Stasheff, J. D. (1974). Characteristic classes. Princeton: Princeton University Press. · Zbl 0298.57008 · doi:10.1515/9781400881826 |
| [27] | Pacini, T. (2012). Desingularizing isolated conical singularities: Uniform estimates via weighted sobolev spaces. arXiv:1005.3511v3. To appear in Communications in Analysis and Geometry. · Zbl 1284.46031 |
| [28] | Salur, S. (2006). Asymptotically cylindrical ricci-flat manifolds. Proceedings of the American Mathematical Society, 134(10), 3049-3056. · Zbl 1095.53024 · doi:10.1090/S0002-9939-06-08313-4 |
| [29] | Salamon, S. (1989). Riemannian geometry and holonomy groups. Pitman Research Notes in Mathematics Series (Vol. 201). · Zbl 0685.53001 |
| [30] | Sá Earp, H. N. (2009). Instantons on \({\rm{G}}_2 \)-manifolds. Ph.D. Thesis. |
| [31] | Sá Earp, H. N. (2014). Generalised Chern-Simons theory and \({\rm{G}}_2 \)-instantons over associative fibrations. SIGMA - Symmetry, Integrability and Geometry: Methods and Applications, 10(083). · Zbl 1296.53052 |
| [32] | Sá Earp, H. N. (2015). \({\rm{G}}_2 \)-instantons over asymptotically cylindrical manifolds. Geometry and Topology, 19(1), 61-111. · Zbl 1312.53044 · doi:10.2140/gt.2015.19.61 |
| [33] | Sá Earp, H. N., & Walpuski, T. (2015). \( \rm{G}_2-\) instantons over twisted connected sums. Geometry & Topology, 19(3), 1263-1285. · Zbl 1317.53034 |
| [34] | Thomas, R. (1997). Gauge theories on Calabi-Yau manifolds. D. Phil. thesis, University of Oxford. |
| [35] | Tian, G. (2000). Gauge theory and calibrated geometry. I. Annals of Mathematics (1), 151(1), 193-268. · Zbl 0957.58013 · doi:10.2307/121116 |
| [36] | Tyurin, A. (2012) Vector bundles, Universitätsverlag Göttingen. In Fedor Bogomolov, Alexey Gorodentsev, Victor Pidstrigach, Miles Reid and Nikolay Tyurin (Eds.), Collected works (Vol. I, p. 330). |
| [37] | Uhlenbeck, K. K., & Yau, S-T. (1986). On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Communications on Pure and Applied Mathematics, 39(S1), S257-S293. · Zbl 0615.58045 |
| [38] | Walpuski, T. (2013). \({\rm{G}}_2 \)-instantons on generalised Kummer constructions. Geometry and Topology, 17(4), 2345-2388. · Zbl 1278.53051 · doi:10.2140/gt.2013.17.2345 |
| [39] | Walpuski, T. · Zbl 1350.53032 · doi:10.4310/MRL.2016.v23.n2.a11 |
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