A Moment Map for Twisted-Hamiltonian Vector Fields on Locally Conformally Kähler Manifolds

Transformation Groups https://doi.org/10.1007/s00031-023-09815-2 Transformation Groups ORIGINAL PAPER A Moment Map for Twisted-Hamiltonian Vector Fields on Locally Conformally Kähler Manifolds Daniele Angella1 · Simone Calamai1 · Francesco Pediconi2 Cristiano Spotti2 · Received: 24 December 2022 / Accepted: 5 July 2023 © The Author(s) 2023 Abstract We extend the classical Donaldson-Fujiki interpretation of the scalar curvature as moment map in Kähler geometry to the wider framework of locally conformally Kähler geometry. Keywords Locally conformally Kähler · Twisted Hamiltonian · Moment map · Donaldson-Fujiki · Chern connection · Weyl connection Mathematics Subject Classification (2010) 53B35 · 53C55 · 53D15 Introduction By the foundational work of Fujiki [11] and Donaldson [8], it is known that the scalar curvature of Kähler metrics arise as a moment map for an infinite-dimensional The first-named, second-named, and third-named authors are supported by project PRIN2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” (code 2017JZ2SW5) and by GNSAGA of INdAM. The third-named and fourth-named authors are supported by Villum Fonden Grant 0019098. B Francesco Pediconi ; Daniele Angella ; Simone Calamai ; Cristiano Spotti 1 Dipartimento Di Matematica E Informatica “Ulisse Dini”, Università Degli Studi di Firenze, viale Morgagni 67/a, 50134 Florence, Italy 2 Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark D. Angella et al. Hamiltonian action. More precisely, given a compact symplectic manifold (M, ωo ), the space Jalm (ωo ) of compatible almost complex structures can be endowed with a natural structure of infinite-dimensional Kähler manifold (J, ˙) that is invariant under the action of the automorphism group Aut(M, ωo ) by pullback. Here, an almost complex structure J is said to be compatible if ωo (_, J _) is a J -almost Hermitian (and hence almost Kähler) metric. This action preserves the analytic subset J (ωo ) ⊂ Jalm (ωo ) of integrable almost complex structures and the map scal : Jalm (ωo ) → C ∞ (M, R), that assigns to any J ∈ Jalm (ωo ) the scalar curvature of the metric ωo (_, J _), is smooth, Aut(M, ωo )-equivariant and it verifies  M d scal| J (v) h X ωon = −21 ˙ J (X ∗J , v) for any J ∈ J (ωo ) , v ∈ T J Jalm (ωo ) , X ∈ ham(M, ωo ) . Here, ham(M, ωo ) denotes the Lie subalgebra of Hamiltonian vector fields, i.e., those infinitesimal automorphisms X ∈ aut(M, ωo ) such that X ωo = dh X for some h X ∈ C ∞ (M, R), uniquely determined up to a constant, and X ∗ denotes the fundamental vector field associated to X . For this reason, we say that scal is a moment map for the action of Ham(M, ωo ) on J (ωo ), where Ham(M, ωo ) denotes the Hamiltonian diffeomorphism group. This fact has deep consequences in Kähler Geometry. In particular, the constant scalar curvature Kähler metrics arise as zeroes of a moment map equation. More precisely, the problem of finding constant scalar curvature Kähler metrics in a fixed Kähler class is reduced to finding a zero for the moment map in the orbit of the complex structure under the complexified action. Inspired by the Hilbert-Mumford criterion in geometric invariant theory in finite-dimension, this leads to the notion of K-stability; see, e.g., [32]. This paper is an attempt to generalize this moment map framework to the nonKähler setting. More precisely, we deal with a compact almost symplectic manifold (M, ω) with dω = 0, admitting compatible complex structures. We notice here that, in the literature, there appeared other works concerning this problem (see, e.g., [3, 12]) which are distinct from our approach. For other appearance of moment maps for group actions on locally conformally symplectic or Vaisman manifolds, see [17, 20, 31]. However, we stress that the group action that we consider lives in an infinitedimensional Kähler setting. Recently, García-Prada and Salamon [13] described a setting where the moment map is given by the Ricci form and studied moment map interpretations of the Kähler-Einstein condition. As the anonymous referee suggested, it would be interesting to try to extend this construction to our locally conformally Kähler setting. One of the first natural non-Kähler cases of study [19] is given by the so-called locally conformally symplectic condition, i.e., we ask that d ω = θ ∧ ω for a given closed 1-form θ . Notice that, by introducing the twisted exterior differential operator dθ := ∗ d - θ ∧, the previous condition can be written as dθ ω = 0. This nomenclature is due to the following characterization: ω is locally conformally symplectic if and only if, for any point x ∈ M, ω is locally conformal to a symplectic form in a neighborhood of x. This shows clearly that the locally conformally symplectic condition is actually a property for the whole conformal class [ω] of ω. Moreover, there is a further relation A Moment Map for Twisted-Hamiltonian Vector Fields... with the symplectic framework: indeed, (M, ω) admits a unique minimal covering  → M such that π ∗ ω is globally conformal to a symplectic form  ωo on map π : M    M and the deck transformation group acts by homotheties on ( M, ωo ) (see, e.g., [18, Section 2] and [6, Section 2.1]). Moreover, a vector field X ∈ X(M) lifts through π   ωo ) if and only if dθ (X ω) = 0 [6, to an infinitesimal automorphism π ∗ X ∈ aut( M, ∗  ωo ) if and only if X ω is dθ -exact [6, Corollary Proposition 3.3] and π X ∈ ham( M,  3.10]. Therefore, it is natural to look at the action of the generalized Lie transformation group Aut  (M, [ω]) generated by the Lie algebra of special conformal vector fields of (M, ω)   aut (M, [ω]) := X ∈ X(M) : dθ (X ω) = 0 , (see Sects. 2.1 and 2.2) on the space Jalm (ω) of compatible almost complex structures on (M, ω). As in the Kähler setting, one may expect the existence of a moment map, related to the scalar curvature and the 1-form θ , when one restricts aut (M, [ω]) to the Lie subalgebra of twisted-Hamiltonian vector fields   ham(M, [ω]) := X ∈ X(M) : X ω = dθ h for some h ∈ C ∞ (M, R) , and Jalm (ω) to the analytic subset J (ω) of integrable almost complex structures. However, two main difficulties occur in this picture. The first one is that the group Aut  (M, [ω]) preserves just the conformal class [ω], and so the map scal : Jalm (ω) → C ∞ (M, R) that associates with any J ∈ Jalm (ω) the Riemannian scalar curvature of ω(_, J _) turns out to be non-Aut  (M, [ω])-equivariant. The second one is that, during the linearization procedure of the map scal, the term D(θ  ) appears, where D denotes the Levi-Civita connection and _ the metric duality, and it seems to us that it cannot be handled via an absorption scheme by adding extra terms depending on θ itself. In order to overcome these issues, we impose a further symmetry. More precisely, we consider the symplectic dual of θ , i.e., the unique vector field V ∈ X(M) such that V (...truncated)