Generalized Hermitian Scalar Curvature

• Generalized Hermitian scalar curvature is a scalar invariant extending Kähler and Chern curvatures to almost Hermitian manifolds with non-integrable structures.
• It unifies canonical connections like Lichnerowicz, Chern, and Bismut, relating curvature invariants to torsion, Lee forms, and underlying geometry.
• Applications include variational problems and PDE formulations that offer insights into existence, stability, and moduli of complex geometric structures.

A generalized Hermitian scalar curvature is a scalar invariant associated with a Hermitian or almost Hermitian manifold, designed to extend the notion of scalar curvature beyond the integrable (Kähler) case and, more generally, beyond the canonical Chern connection. These curvatures are intimately related to variational problems, geometric flows, canonical metrics, and stability conditions in complex and almost complex geometry. Modern developments have established a family of generalized scalar curvatures depending on both the Hermitian connection and the underlying almost Hermitian structure, unifying and extending classical results such as those of Kähler scalar curvature, Chern scalar curvature, and Bismut scalar curvature.

1. Canonical Connections and Scalar Curvatures in the Hermitian Setting

In Hermitian geometry, the metric $g$ and almost complex structure $J$ define a fundamental two-form $\omega(X, Y) = g(JX, Y)$. Canonical Hermitian connections interpolate between the Lichnerowicz ($D^0$), Chern ($D^1$), and Bismut ($D^{-1}$) connections. For any parameter $t \in \mathbb{R}$, the family is given by $D^t = (1-t) D^0 + t D^1$. The corresponding curvature tensor $K^t$ produces scalar invariants through traces in local unitary frames: $$s_1(t) = \sum_{i,j=1}^n K^t(u_i, \bar u_i, u_j, \bar u_j), \qquad s_2(t) = \sum_{i,j=1}^n K^t(\bar u_i, u_j, u_i, \bar u_j)$$ These generalize the classical Chern scalar curvature, which corresponds to $J$0 (Chern connection). Explicit formulas relate $J$1 to the Riemannian scalar curvature $J$2 and the Gray-Hervella types—encapsulating the non-Kähler (torsion) geometry—along with the Lee form and its codifferential. On a Kähler manifold, all these curvatures coincide and reduce to the usual scalar curvature. In general, the difference from Kähler geometry is measured via torsion and Lee form components (Fu et al., 2019, Li et al., 2022).

2. Generalized Hermitian Scalar Curvature in the Almost-Kähler and Symplectic Setting

In the almost-Kähler context—where $J$3 admits a compatible (possibly non-integrable) almost complex structure $J$4—the Levi-Civita connection $J$5 and its derived Hermitian (Chern) connection $J$6 yield a closed curvature form $J$7 on the anti-canonical bundle. The Hermitian scalar curvature $J$8 is defined by

$J$9

This definition generalizes the Kähler scalar curvature, reducing to the classical case when $\omega(X, Y) = g(JX, Y)$0 is integrable. The geometric analysis developed by Keller–Lejmi introduces an asymptotic Futaki-type invariant as an obstruction to the existence of constant Hermitian scalar curvature metrics, enabling lower bounds on the $\omega(X, Y) = g(JX, Y)$1-norm of $\omega(X, Y) = g(JX, Y)$2 via representation-theoretic and symplectic techniques that extend Donaldson's Kähler case results (Keller et al., 2017).

3. Scalar Curvature in Generalized Kähler Geometry

Scalar curvature in generalized Kähler (and more broadly, Courant algebroid) geometry transcends the usual Hermitian framework, relying on structures such as pure spinors and biHermitian pairs $\omega(X, Y) = g(JX, Y)$3. Goto’s definition employs the Chevalley pairing and the generalized Ricci form $\omega(X, Y) = g(JX, Y)$4, yielding a “generalized scalar curvature”

$\omega(X, Y) = g(JX, Y)$5

This invariant, initially formulated in the abstract setting of spinors, can be entirely expressed in terms of underlying biHermitian data via Chern–Bismut connection curvature. In toric generalized Kähler geometry, explicit moment-map formulas (e.g., Boulanger’s definition)

$\omega(X, Y) = g(JX, Y)$6

coincide with Goto’s pure spinor scalar curvature, thereby both unifying the two perspectives and confirming the meaningfulness of generalized scalar curvature invariants in broader geometries (Wang, 2019).

4. Prescribed and Variational Problems: Gauduchon and Yamabe-Type Equations

Generalized Hermitian scalar curvatures naturally appear in nonlinear PDEs prescribing scalar curvature within a Hermitian or almost Hermitian conformal class. For Gauduchon’s canonical scalars, the conformal transformation yields semi-linear PDEs of the form

$\omega(X, Y) = g(JX, Y)$7

where $\omega(X, Y) = g(JX, Y)$8 is the conformal factor and $\omega(X, Y) = g(JX, Y)$9 is the Lee form. The sign of the Gauduchon degree $D^0$0 governs existence theory via super- and sub-solution techniques: for negative degree, full necessary and sufficient criteria are available, yielding a unified approach encompassing the Chern–Yamabe and Bismut–Yamabe problems (Li et al., 2022). Analogous Yamabe-type results are established for other scalar curvature types—second Chern, Bismut—using continuity and variational methods (Zhang, 28 Jan 2026).

5. Curvature Proportionality, Obstructions, and Special Cases

In compact Hermitian geometry, the classical identity $D^0$1 characterizes the Kähler condition: proportionality between Riemannian and (Chern) Hermitian scalar curvature can only occur in the Kähler case, via vanishing of torsion and Lee form. On noncompact or highly symmetric (e.g., $D^0$2-invariant) backgrounds, solutions exist for more general proportionality relationships, producing a moduli of non-Kähler “Kähler-like scalar curvature” (Klsc) metrics (Dabkowski et al., 2015, Dabkowski et al., 2015). For certain generalized scalar curvatures, similar proportionality or Einstein-type equations characterize special geometric structures, such as “weak second Hermitian–Einstein” metrics, which under additional hypotheses (pluriclosed, Gauduchon) even force Kähler–Einstein metrics (Zhang, 28 Jan 2026).

6. Analytic and Geometric Consequences

Generalized Hermitian scalar curvature invariants admit explicit decompositions in terms of geometric quantities—Riemannian scalar curvature, Gray–Hervella torsion types, Lee form, and their codifferentials. Their integral and pointwise inequalities yield rigidity and classification results for balanced, locally conformally Kähler, and $D^0$3-Gauduchon metrics (Fu et al., 2019). The curvature invariants control elliptic PDE behavior under geometric flows, such as the La Nave–Tian continuity equation applied to Hermitian settings, where blow-up or convergence phenomena depend critically on the behavior of Chern or generalized scalar curvature (Liang et al., 2023).

7. Research Directions and Open Problems

Open problems include full characterization of existence domains for generalized constant scalar curvature metrics outside the Kähler regime, particularly in positive/indefinite Gauduchon or Bismut degree settings; construction of nontrivial solutions to generalized Hermitian Yamabe problems on arbitrary backgrounds; and the exploration of stability conditions, moduli, and singularities in non-integrable or noncompact geometries. The interplay between higher-order curvature invariants, complex geometric flows, and analytic techniques remains an active area, with particular attention being paid to the geometric meaning and moduli of solutions to curvature prescription problems for various choices of canonical Hermitian connections (Zhang, 28 Jan 2026, Dabkowski et al., 2015, Li et al., 2022, Wang, 2019, Keller et al., 2017).