Mathematical Research Letters

Volume 28 (2021)

Number 2

Construction of $\mathrm{G}_2$-instantons via twisted connected sums

Pages: 471 – 509



Grégoire Menet (Institut Fourier, Université Grenoble Alpes, Gières, France)

Johannes Nordström (Department of Mathematical Sciences, University of Bath, United Kingdom)

Henrique N. Sá Earp (Institute of Mathematics, Statistics and Scientific Computing University of Campinas (Unicamp), Campinas, S.P., Brazil)


We propose a method to construct $\mathrm{G}_2$–instantons over a compact twisted connected sum $\mathrm{G}_2$–manifold, applying a gluing result of Sá Earp and Walpuski to instantons over a pair of $7$-manifolds with a tubular end. In our example, the moduli spaces of the ingredient instantons are non-trivial, and their images in the moduli space over the asymptotic cross-section K3 surface intersect transversely. Such a pair of asymptotically stable holomorphic bundles is obtained using a twisted version of the Hartshorne–Serre construction, which can be adapted to produce other examples. Moreover, their deformation theory and asymptotic behaviour are explicitly understood, results which may be of independent interest.

G.M. is supported by grant 2014/05733-9, São Paulo Research Foundation (Fapesp), and the Marco Brunella Grant of Burgundy University.

H.S.E. is supported by grant 2014/24727-0, São Paulo Research Foundation (Fapesp), and the Brazilian National Council for Scientific and Technological Development (CNPq) Productivity Grant 312390/2014-9.

J.N. is supported by the Simons Foundation under the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics (grant #488631, Johannes Nordström).

Received 3 May 2019

Accepted 14 October 2019

Published 13 May 2021