nLab lambda-connection

Idea

Deligne’s $\lambda$-connection is a notion interpolating between flat connections and Higgs fields (see Higgs bundle). Parameter $\lambda\in\mathbf{P}^1$.

Literature

The notion has been intrduced in a series of letters of Pierre Deligne to Carlos Simpson, realising the Higgs bundles as a degeneration of vector bundles with connections. Simpson further developed the notion and related moduli space of $\lambda$-connection which he called Hodge moduli space.

• C. T. Simpson, The Hodge filtration on nonabelian cohomology, in Algebraic geometry-Santa Cruz 1995, 217–281

• Carlos T. Simpson, A weight two phenomenon for the moduli space of rank one local systems on open varieties arXiv:0710.2800

• C. T. Simpson, Iterated destability modifications for vector bundles with connection, Contemp. Math. 522 (2010) 183–206

• Carlos Simpson, On the notion of lambda-connection, slides

• Zhi Hu, Pengfei Huang, Simpson-Mochizuki correspondence for $\lambda$-flat bundles, Journal de Mathématiques Pures et Appliquées 2022 doi arXiv:1905.10765

The notion of flat λ-connections as the interpolation of usual flat connections and Higgs fields was suggested by Deligne and further studied by Simpson. Mochizuki established the Kobayashi–Hitchin-type theorem for λ-flat bundles (λ≠0), which is called the Mochizuki correspondence. In this paper, on the one hand, we generalize Mochizuki’s result to the case when the base being a compact balanced manifold, more precisely, we prove the existence of harmonic metrics on stable λ-flat bundles (λ≠0). On the other hand, we study two applications of the Simpson–Mochizuki correspondence to moduli spaces. More concretely, we show this correspondence provides a homeomorphism between the moduli space of (semi)stable λ-flat bundles over a complex projective manifold and the Dolbeault moduli space, and also provides dynamical systems with two parameters on the latter moduli space. We investigate such dynamical systems, in particular, we calculate the first variation, the fixed points and discuss the asymptotic behaviour.

• Anoop Singh, On the Moduli space of $\lambda$-connections, (2020) RG
• C. C. Liu, S. Rayan, Y. Tanaka, The Kapustin–Witten equations and nonabelian Hodge theory Eur. J. Math. 8 (Suppl 1) 23–41 (2022) doi

Simpson introduced shapes of a (smooth projective complex) curve $X$, and in particular Betti shape $X_B$, de Rham shape $X_{dR}$ and Dolbeaut shape $X_{Dol}$ of a scheme. Coherent sheaves on them correspond to local systems, flat vector bundles and Higgs sheaves. So called Deligne’s shape $X_{Del}$ interpolates between de Rham and Dolbeaut shapes.

• Mauro Porta, Francesco Sala, Simpson’s shapes of schemes and stacks, draft pdf

This was used to obtain a categorified Hall algebra version of nonabelian Hodge correspondence in

• Mauro Porta, Francesco Salla, Two-dimensional categorified Hall algebras, J. Eur. Math. Soc. 25:3 (2023) 1113-1205 doi

category: algebraic geometry, analytic geometry