Deformed Hermitian Yang–Mills Connection

• Deformed Hermitian Yang–Mills connections are unitary Chern connections on holomorphic line bundles over compact Kähler manifolds whose curvature satisfies a fully nonlinear constant-phase condition.
• They are the complex-geometric mirror of the special Lagrangian equation in the SYZ program, featuring equivalent wedge-product and eigenvalue formulations.
• Existence proofs, numerical criteria, and flow methods establish solvability by linking analytic subsolutions and stability conditions with topological and algebraic invariants.

A deformed Hermitian Yang–Mills connection is a unitary Chern connection on a holomorphic line bundle over a compact Kähler manifold whose curvature has constant Lagrangian phase relative to a background Kähler form. In the standard formulation, if $L\to X$ is a holomorphic line bundle over a compact Kähler manifold $(X,\omega)$, $h$ is a Hermitian metric on $L$, and $F_h$ is the Chern curvature, then the deformed Hermitian–Yang–Mills equation requires

$$\operatorname{Im}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)=0,\qquad \operatorname{Re}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)>0,$$

with $\hat\theta$ determined topologically by $\hat\theta=\arg\int_X(\omega+iF_h)^n$. Equivalently, if $\lambda_i$ are the eigenvalues of $\omega^{-1}F_h$, then $(X,\omega)$0. The equation is a fully nonlinear complex Hessian equation, it is the complex-geometric mirror of the special Lagrangian equation in the Strominger–Yau–Zaslow program, and it also appears as the supersymmetry condition for B-model D-branes (Collins et al., 2017, Chu et al., 2021).

1. Foundational definition and equivalent formulations

Let $(X,\omega)$1 be a compact Kähler manifold of complex dimension $(X,\omega)$2, and let $(X,\omega)$3 be a holomorphic line bundle with Hermitian metric $(X,\omega)$4. Its unitary Chern connection has curvature $(X,\omega)$5, and one often writes $(X,\omega)$6 or $(X,\omega)$7, depending on convention. The deformed Hermitian–Yang–Mills equation is the requirement that the complex top form $(X,\omega)$8 have constant argument. In eigenvalue form,

$(X,\omega)$9

where $h$0 are the eigenvalues of $h$1. In arccot normalization, used in several papers, one instead writes

$h$2

and the supercritical phase becomes $h$3 (Chu et al., 2021, Murakami, 2024).

The same equation admits a wedge-product formulation. In one common convention,

$h$4

In the sign convention used by Schlitzer–Stoppa, with $h$5, the equation is written as

$h$6

or equivalently

$h$7

These formulations differ only by notation and branch conventions (Kawai et al., 2020, Schlitzer et al., 2019, Pingali, 2015).

The topological phase is fixed by cohomology: $h$8 On compact Hermitian manifolds, the corresponding constant-phase problem is naturally posed in Bott–Chern cohomology, and under the structural assumptions

$h$9

the phase $L$0 is constant on a fixed Bott–Chern class $L$1 (Lin, 2020). On almost Hermitian manifolds, by contrast, the phase $L$2 is not invariant with $L$3, and this non-invariance is a central analytic complication (Huang et al., 2020).

2. Phase ranges, admissibility, and subsolution conditions

Ellipticity is branch-dependent. In the standard Kähler theory, one commonly imposes

$L$4

to select the admissible branch. Several papers distinguish supercritical and hypercritical regimes. One standard supercritical interval is

$L$5

while the broader interval

$L$6

is used in a priori estimates and existence theorems of Collins–Jacob–Yau type (Chu et al., 2021, Collins et al., 2017). On Kähler surfaces, Murakami formulates the supercritical condition in arccot normalization as $L$7 (Murakami, 2024). On compact Kähler three-folds, Pingali proves existence for the full admissible phase range

$L$8

with the endpoints excluded because $L$9 degenerates the wedge formulation (Pingali, 2019).

The analytic subsolution condition is central. In Collins–Jacob–Yau form, a smooth representative $F_h$0 is a $F_h$1-subsolution if, for each $F_h$2,

$F_h$3

where $F_h$4 are the eigenvalues of $F_h$5 (Collins et al., 2017). In the almost Hermitian setting, the paper defines a $F_h$6-subsolution by boundedness of the shifted level set

$F_h$7

and recalls the equivalent Collins–Jacob–Yau condition

$F_h$8

for each $F_h$9 (Huang et al., 2020). On compact Hermitian manifolds, Lin uses the same $$\operatorname{Im}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)=0,\qquad \operatorname{Re}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)>0,$$0-subsolution framework for the variable-phase equation

$$\operatorname{Im}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)=0,\qquad \operatorname{Re}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)>0,$$1

in the supercritical range $$\operatorname{Im}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)=0,\qquad \operatorname{Re}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)>0,$$2 (Lin, 2020).

In dimension three, Pingali uses the shifted form

$$\operatorname{Im}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)=0,\qquad \operatorname{Re}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)>0,$$3

and the subsolution condition

$$\operatorname{Im}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)=0,\qquad \operatorname{Re}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)>0,$$4

which is equivalent to the $$\operatorname{Im}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)=0,\qquad \operatorname{Re}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)>0,$$5-subsolution condition in that dimension (Pingali, 2019). In the projective supercritical theory, test-family stability is phrased in terms of inequalities of the form

$$\operatorname{Im}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)=0,\qquad \operatorname{Re}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)>0,$$6

with strict inequality for $$\operatorname{Im}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)=0,\qquad \operatorname{Re}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)>0,$$7, and this replaces an earlier “uniform” version with a positive margin (Chu et al., 2021).

A frequent misconception is that the topological phase alone determines solvability. The cited results do not assert this in general. Instead, they place decisive weight on cone conditions, $$\operatorname{Im}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)=0,\qquad \operatorname{Re}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)>0,$$8-subsolutions, or numerical positivity on subvarieties. This is explicit in the Collins–Jacob–Yau program, in Pingali’s three-fold theorem, in Lin’s Hermitian and almost Hermitian extensions, and in the projective Nakai–Moishezon-type criteria (Collins et al., 2017, Pingali, 2019, Lin, 2020, Chu et al., 2021).

3. Existence theorems and numerical criteria

The modern existence theory has several distinct forms. In the classical Kähler setting, Collins–Jacob–Yau prove smooth solvability under the existence of a subsolution in suitable phase ranges, and the uniqueness is by the strong maximum principle because $$\operatorname{Im}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)=0,\qquad \operatorname{Re}\big(e^{-i\hat\theta}(\omega+iF_h)^n\big)>0,$$9 is increasing (Collins et al., 2017). Pingali proves an existence theorem on compact Kähler three-folds for the full admissible range $\hat\theta$0, conditioned on the necessary subsolution condition, by rewriting dHYM as a generalized Monge–Ampère equation and introducing a new continuity path with mixed-sign coefficients (Pingali, 2019). In complex dimension four, the Collins–Jacob–Yau conjecture is confirmed in the regime where $\hat\theta$1 is close to $\hat\theta$2 from the right: the existence of a $\hat\theta$3-subsolution implies solvability (Lin, 2022).

Projective and numerical criteria form a second major branch. Chu–Lee–Takahashi prove a Nakai–Moishezon-type criterion in the supercritical phase without a uniform constant. Under the cohomological normalization

$\hat\theta$4

they show equivalence between existence of a unique dHYM solution, stability along test families, and positivity conditions along subvarieties. In the projective case, this confirms the Collins–Jacob–Yau conjecture in the supercritical phase (Chu et al., 2021). Pingali extends Gao Chen’s twisted theory on compact projective manifolds, allowing non-constant and slightly negative twisting in all dimensions under explicit numerical positivity assumptions on all subvarieties (Ballal, 2021).

On compact Kähler surfaces, the equation becomes especially concrete. The dHYM equation reduces to the existence of a Kähler representative in the class

$\hat\theta$5

and existence is equivalent to curve positivity. Dervan–McCarthy–Sektnan show that for compact sets of initial data it suffices to test finitely many irreducible curves of negative self-intersection, with the number of test curves bounded in terms of the Picard number $\hat\theta$6 (Khalid et al., 2022). This gives a finite effective criterion and a wall–chamber description on surfaces.

Several special geometries admit stronger existence theorems. On rational homogeneous varieties $\hat\theta$7, equipped with any invariant Kähler metric, the dHYM equation always admits a solution. The phase is given explicitly by

$\hat\theta$8

and homogeneous supercritical and hypercritical solutions are characterized in Lie-theoretic terms (Correa, 2023). On the blowup $\hat\theta$9, the dHYM PDE reduces under Calabi symmetry to an exact ODE

$\hat\theta=\arg\int_X(\omega+iF_h)^n$0

and an algebraic stability condition is shown to suffice for solvability (Jacob et al., 2020). On tropical manifolds, the Leung–Yau–Zaslow correspondence is generalized beyond section-type Lagrangians: the dHYM condition on the mirror connection is equivalent to the same determinant phase condition that characterizes special Lagrangians on the original torus fibration (Yamamoto, 2017).

4. Flows, weak limits, and boundary phenomena

Parabolic methods form a parallel approach. Murakami studies the deformed Hermitian–Yang–Mills flow on compact Kähler surfaces: $\hat\theta=\arg\int_X(\omega+iF_h)^n$1 with $\hat\theta=\arg\int_X(\omega+iF_h)^n$2 preserved along the flow (Murakami, 2024). The $\hat\theta=\arg\int_X(\omega+iF_h)^n$3-functional adapted to dHYM is convex along the flow, and this yields $\hat\theta=\arg\int_X(\omega+iF_h)^n$4-control of the time derivative. In the boundary case where $\hat\theta=\arg\int_X(\omega+iF_h)^n$5 is nef and big, the flow converges in the sense of currents to the unique current $\hat\theta=\arg\int_X(\omega+iF_h)^n$6 such that $\hat\theta=\arg\int_X(\omega+iF_h)^n$7 is a positive current and

$\hat\theta=\arg\int_X(\omega+iF_h)^n$8

in the non-pluripolar sense. This produces a weak dHYM connection given by a singular Hermitian metric whose curvature current solves the equation in the viscosity/non-pluripolar sense (Murakami, 2024).

A different flow is developed for the LYZ form of the equation: $\hat\theta=\arg\int_X(\omega+iF_h)^n$9 Under $\lambda_i$0, long-time existence holds, and under a Collins–Jacob–Yau subsolution with $\lambda_i$1 and $\lambda_i$2, the longtime solution converges smoothly to a solution of $\lambda_i$3 (Fu et al., 2021). On compact Kähler surfaces, semi-subsolutions suffice to produce singular limits away from finitely many curves of negative self-intersection; the limit defines a Kähler current and is described as a boundary point of the moduli space of smooth LYZ connections (Fu et al., 2021).

These flow results clarify a point that is sometimes misstated: boundary classes need not admit smooth dHYM metrics, but they can still admit canonical weak limits. Murakami proves this in the nef-and-big case for the dHYM flow on surfaces, while the LYZ flow yields singular solutions away from a finite exceptional locus under semi-subsolution hypotheses (Murakami, 2024, Fu et al., 2021).

5. Moduli, coupled equations, and mirror-symmetry structures

Assuming nonemptiness, the moduli space of dHYM connections has a simple global structure. For a compact Kähler manifold $\lambda_i$4, a Hermitian line bundle $\lambda_i$5, and a fixed phase $\lambda_i$6, the moduli space

$\lambda_i$7

is homeomorphic to a $\lambda_i$8-dimensional torus: $\lambda_i$9 In particular it is connected and orientable (Kawai et al., 2020). The same paper proves deformation stability: under smooth deformations of the Kähler structure, Hermitian metric, and phase satisfying the cohomological conditions

$\omega^{-1}F_h$0

the nearby moduli spaces form a $\omega^{-1}F_h$1-bundle (Kawai et al., 2020).

Schlitzer–Stoppa place dHYM in a larger moment-map framework by allowing the Kähler metric to vary and coupling the phase equation to scalar curvature. The coupled system is

$\omega^{-1}F_h$2

and it arises from the extended gauge group by combining the Collins–Yau moment map for dHYM with the Donaldson–Fujiki moment map for scalar curvature (Schlitzer et al., 2019). In the large-radius limit, this recovers the Kähler–Yang–Mills system

$\omega^{-1}F_h$3

and in the small-radius limit it yields

$\omega^{-1}F_h$4

which is a coupled J-equation/scalar-curvature system (Schlitzer et al., 2019).

The mirror-symmetry interpretation is equally structural. The dHYM equation is the complex-geometric mirror of the special Lagrangian equation in the SYZ picture (Chu et al., 2021, Yamamoto, 2017). In the tropical semi-flat setting, the same phase condition

$\omega^{-1}F_h$5

simultaneously characterizes a special Lagrangian on one side and a dHYM connection on the mirror complex submanifold on the other (Yamamoto, 2017). This suggests that the torus moduli result, the moment-map formulation, and the central-charge inequalities are part of a single geometric package rather than isolated PDE facts.

6. Rigidity, explicit geometries, and effective destabilization

Rigidity phenomena occur under strong background curvature hypotheses. On a compact Kähler manifold with non-negative orthogonal bisectional curvature, if a dHYM metric satisfies the uniform bound

$\omega^{-1}F_h$6

then the curvature is parallel: $\omega^{-1}F_h$7 If the orthogonal bisectional curvature is strictly positive at some point, then

$\omega^{-1}F_h$8

for a constant $\omega^{-1}F_h$9 (Han et al., 2019). The same paper proves a Liouville-type rigidity for self-shrinkers of the parabolic dHYM flow on $(X,\omega)$00: every smooth entire self-shrinker is a quadratic polynomial (Han et al., 2019).

Surface geometry also exhibits effective destabilization. For compact Kähler surfaces, the obstructing curves for dHYM can be reduced to finitely many negative curves, and the corresponding walls are codimension-one real algebraic loci of the form

$(X,\omega)$01

Optimally destabilizing curves for dHYM and Donaldson’s J-equation coincide and are characterized by orthogonality to a boundary nef class $(X,\omega)$02 (Khalid et al., 2022). This yields what the paper describes as a first PDE analogue of the locally finite wall–chamber decomposition in Bridgeland stability.

Calabi-symmetric families permit especially explicit criteria. On

$(X,\omega)$03

the dHYM equation reduces to the harmonic-polynomial level-set problem

$(X,\omega)$04

with $(X,\omega)$05. The existence of a smooth graph solution is equivalent to an explicit stability criterion in terms of a counting function defined from the Cauchy index, and equivalently to inequalities involving the lifted phases of the charges $(X,\omega)$06, $(X,\omega)$07, and $(X,\omega)$08 (Jacob, 2022). On rational homogeneous varieties, by contrast, the homogeneous geometry trivializes the existence problem: the phase is computed Lie-theoretically and every invariant class admits a solution (Correa, 2023).

Current limitations are also explicit in the literature. Several arguments depend crucially on the supercritical phase, on projectivity, or on positivity properties such as nef-and-big or semi-subsolution assumptions (Chu et al., 2021, Murakami, 2024, Ballal, 2021). Extending the strongest existence and weak-limit results to non-supercritical phases, singular ambient spaces, or higher-dimensional boundary cases is repeatedly identified as requiring new ideas. A plausible implication is that the subject is now divided less by the formal equation itself than by the geometry of the admissible cone and the available positivity mechanisms in each class of manifolds.