Supercritical dHYM Equation

• Supercritical dHYM is a fully nonlinear PDE on compact Kähler manifolds that sets the phase of a complex top wedge to be constant.
• The formulation uses distinct phase conventions (via arctan or arccot) critical for ensuring ellipticity and guiding subsolution methods.
• Recent research links dHYM solvability with numerical stability, mirror symmetry, and geometrically meaningful positivity criteria across dimensions.

The supercritical deformed Hermitian–Yang–Mills equation is a fully nonlinear geometric PDE on a compact Kähler manifold requiring the phase of a complexified top wedge to be constant. In one standard convention, for a real $(1,1)$-form $\chi$ with eigenvalues $\lambda_1,\dots,\lambda_n$ relative to a Kähler form $\omega$, the equation is

$$\operatorname{Im}\bigl(e^{-\sqrt{-1}\hat\theta}(\omega+\sqrt{-1}\chi)^n\bigr)=0, \qquad \sum_{i=1}^n \arctan(\lambda_i)=\hat\theta,$$

while in the $\arccot$-normalization used in a substantial part of the recent literature it is written as

$$\sum_{i=1}^n \operatorname{arccot}(\lambda_i)=\theta,$$

with $\hat\theta=\frac{n\pi}{2}-\theta$. The supercritical branch is therefore $$\hat\theta\in\bigl(\frac{(n-2)\pi}{2},\frac{n\pi}{2}\bigr)$$, equivalently $\theta\in(0,\pi)$. This is the branch in which the main existence, subsolution, and stability theories are formulated, and it is also the branch for which recent work has clarified both the strength and the limitations of numerical positivity criteria (Collins et al., 2017, Murakami, 28 May 2026, Zhang, 2023).

1. Phase conventions and defining equations

A standard geometric setup fixes a compact Kähler manifold $\chi$0, a holomorphic line bundle $\chi$1, and a Hermitian metric $\chi$2 on $\chi$3 with Chern curvature $\chi$4. One form of the equation used by Jacob–Yau and Pingali is

$\chi$5

or equivalently

$\chi$6

where $\chi$7 is the associated real $\chi$8-form. In local coordinates, if $\chi$9 are the eigenvalues of $\lambda_1,\dots,\lambda_n$0, then the equation becomes $\lambda_1,\dots,\lambda_n$1 for a suitable lifted branch of the argument (Pingali, 2015).

A parallel convention, used for example in the supercritical formulation of Chen, Sun, and Xu, fixes a background Kähler form $\lambda_1,\dots,\lambda_n$2, a closed real $\lambda_1,\dots,\lambda_n$3-form $\lambda_1,\dots,\lambda_n$4, and defines

$\lambda_1,\dots,\lambda_n$5

with $\lambda_1,\dots,\lambda_n$6. The supercritical dHYM equation is then

$\lambda_1,\dots,\lambda_n$7

where the constant phase is determined by the cohomology class through $\lambda_1,\dots,\lambda_n$8 (Zhang, 2023).

The literature repeatedly warns that sign and branch conventions vary. In particular, one paper explicitly states that it follows the sign convention of Collins–Yau and that, in most references, its equation corresponds to the dHYM equation for the inverse line bundle $\lambda_1,\dots,\lambda_n$9. As a result, comparisons across papers require care about whether one writes $\omega$0 or $\omega$1, whether one uses $\omega$2 or $\omega$3, and which lift of the phase is being fixed (Schlitzer et al., 2019).

The branch issue is not cosmetic. The same cohomological phase modulo $\omega$4 can correspond to different analytic branches, and later counterexamples show that this discrepancy can separate the numerically admissible locus from the actual solvable locus for the supercritical equation (Zhang, 2023).

2. Admissibility, subsolutions, and analytic structure

The supercritical branch is analytically distinguished because it is the branch on which the equation is elliptic on the relevant admissible cone and because the main a priori estimates are built from subsolution inequalities. A recurring criterion, quoted in several papers from Collins–Jacob–Yau, is that a $\omega$5-form $\omega$6 is a $\omega$7-subsolution if and only if

$\omega$8

or, in the equivalent wedge-form language,

$\omega$9

for every $$\operatorname{Im}\bigl(e^{-\sqrt{-1}\hat\theta}(\omega+\sqrt{-1}\chi)^n\bigr)=0, \qquad \sum_{i=1}^n \arctan(\lambda_i)=\hat\theta,$$0 and every nonzero simple positive $$\operatorname{Im}\bigl(e^{-\sqrt{-1}\hat\theta}(\omega+\sqrt{-1}\chi)^n\bigr)=0, \qquad \sum_{i=1}^n \arctan(\lambda_i)=\hat\theta,$$1-form $$\operatorname{Im}\bigl(e^{-\sqrt{-1}\hat\theta}(\omega+\sqrt{-1}\chi)^n\bigr)=0, \qquad \sum_{i=1}^n \arctan(\lambda_i)=\hat\theta,$$2 (Pingali, 2015).

One influential analytic route rewrites dHYM as a generalized Monge–Ampère equation with nonconstant coefficients. Pingali studies

$$\operatorname{Im}\bigl(e^{-\sqrt{-1}\hat\theta}(\omega+\sqrt{-1}\chi)^n\bigr)=0, \qquad \sum_{i=1}^n \arctan(\lambda_i)=\hat\theta,$$3

proves $$\operatorname{Im}\bigl(e^{-\sqrt{-1}\hat\theta}(\omega+\sqrt{-1}\chi)^n\bigr)=0, \qquad \sum_{i=1}^n \arctan(\lambda_i)=\hat\theta,$$4 a priori estimates under the structural positivity condition

$$\operatorname{Im}\bigl(e^{-\sqrt{-1}\hat\theta}(\omega+\sqrt{-1}\chi)^n\bigr)=0, \qquad \sum_{i=1}^n \arctan(\lambda_i)=\hat\theta,$$5

and then specializes this framework to dHYM in the supercritical phase regime $$\operatorname{Im}\bigl(e^{-\sqrt{-1}\hat\theta}(\omega+\sqrt{-1}\chi)^n\bigr)=0, \qquad \sum_{i=1}^n \arctan(\lambda_i)=\hat\theta,$$6. In that formulation, the subsolution is encoded by positivity of a background form $$\operatorname{Im}\bigl(e^{-\sqrt{-1}\hat\theta}(\omega+\sqrt{-1}\chi)^n\bigr)=0, \qquad \sum_{i=1}^n \arctan(\lambda_i)=\hat\theta,$$7 together with a phase-dependent wedge inequality (Pingali, 2015).

On compact Hermitian manifolds, Lin studies the prescribed-phase equation

$$\operatorname{Im}\bigl(e^{-\sqrt{-1}\hat\theta}(\omega+\sqrt{-1}\chi)^n\bigr)=0, \qquad \sum_{i=1}^n \arctan(\lambda_i)=\hat\theta,$$8

for $$\operatorname{Im}\bigl(e^{-\sqrt{-1}\hat\theta}(\omega+\sqrt{-1}\chi)^n\bigr)=0, \qquad \sum_{i=1}^n \arctan(\lambda_i)=\hat\theta,$$9, proves $\arccot$0 estimates under a $\arccot$1-subsolution, and obtains constant-phase existence when $\arccot$2. A key structural lemma states that if $\arccot$3, then $\arccot$4, all lower elementary symmetric polynomials are nonnegative, and the level set $\arccot$5 is a smooth convex hypersurface (Lin, 2020).

On closed almost Hermitian manifolds, the hypercritical/supercritical theory becomes more rigid. Huang–Zhang–Zhang work in the range

$\arccot$6

prove $\arccot$7, gradient, Hessian, and $\arccot$8 estimates under a $\arccot$9-subsolution, and solve the equation up to an additive phase constant assuming also the existence of a supersolution. Their analysis emphasizes that in the almost Hermitian setting the phase is no longer cohomologically invariant in the way used in the Kähler continuity method (Huang et al., 2020).

3. Numerical criteria, stability, and the Collins–Jacob–Yau conjecture

The central conjectural theme of the subject is that supercritical dHYM solvability should be equivalent to positivity of central-charge-type integrals on proper analytic subvarieties. In one precise formulation, for a compact Kähler manifold $$\sum_{i=1}^n \operatorname{arccot}(\lambda_i)=\theta,$$0, a real closed $$\sum_{i=1}^n \operatorname{arccot}(\lambda_i)=\theta,$$1-form $$\sum_{i=1}^n \operatorname{arccot}(\lambda_i)=\theta,$$2, and $$\sum_{i=1}^n \operatorname{arccot}(\lambda_i)=\theta,$$3, Chen proves that the supercritical dHYM equation is solvable if and only if there exists a test family $$\sum_{i=1}^n \operatorname{arccot}(\lambda_i)=\theta,$$4 such that for every $$\sum_{i=1}^n \operatorname{arccot}(\lambda_i)=\theta,$$5-dimensional subvariety $$\sum_{i=1}^n \operatorname{arccot}(\lambda_i)=\theta,$$6,

$$\sum_{i=1}^n \operatorname{arccot}(\lambda_i)=\theta,$$7

uniformly in $$\sum_{i=1}^n \operatorname{arccot}(\lambda_i)=\theta,$$8. The paper states that this confirms the mirror version of the Thomas–Yau conjecture about special Lagrangian submanifolds on Calabi–Yau manifolds (Chen, 2020).

Takahashi proves a Nakai–Moishezon type criterion without a uniform constant. Under the cohomological compatibility condition

$$\sum_{i=1}^n \operatorname{arccot}(\lambda_i)=\theta,$$9

the paper shows that solvability is equivalent to stability along some, equivalently any, test family, and on projective manifolds it confirms the Collins–Jacob–Yau conjecture in the supercritical phase (Chu et al., 2021).

Chu extends the projective theory to the twisted supercritical equation

$\hat\theta=\frac{n\pi}{2}-\theta$0

and proves solvability for smooth $\hat\theta=\frac{n\pi}{2}-\theta$1 under explicit numerical positivity conditions on all subvarieties. The extension to non-constant and slightly negative twisting functions is particularly important in dimension $\hat\theta=\frac{n\pi}{2}-\theta$2, where it supplies the analytic input for the projective twisted existence theorem (Ballal, 2021).

The strongest general limitation is now equally clear. Xu defines

$\hat\theta=\frac{n\pi}{2}-\theta$3

and

$\hat\theta=\frac{n\pi}{2}-\theta$4

proves that $\hat\theta=\frac{n\pi}{2}-\theta$5 is both open and closed as a subset of $\hat\theta=\frac{n\pi}{2}-\theta$6, and shows by examples that $\hat\theta=\frac{n\pi}{2}-\theta$7 can occur for every $\hat\theta=\frac{n\pi}{2}-\theta$8. This disproves the Collins–Jacob–Yau conjecture on general compact Kähler manifolds and identifies the obstruction as a genuine branch/phase phenomenon: numerical subvariety inequalities can forget the analytic branch required by the PDE (Zhang, 2023).

4. Dimension-specific and highly symmetric solvability results

On compact Kähler surfaces, the supercritical regime is especially explicit. In the surface notation of Datar–Mete–Song, supercritical means

$\hat\theta=\frac{n\pi}{2}-\theta$9

and the relevant cohomology class is

$$\hat\theta\in\bigl(\frac{(n-2)\pi}{2},\frac{n\pi}{2}\bigr)$$0

The paper proves that dHYM solvability is equivalent to the Kählerness of this class, equivalently to the inequalities

$$\hat\theta\in\bigl(\frac{(n-2)\pi}{2},\frac{n\pi}{2}\bigr)$$1

for all irreducible curves $$\hat\theta\in\bigl(\frac{(n-2)\pi}{2},\frac{n\pi}{2}\bigr)$$2. More sharply, on every compact subset of $$\hat\theta\in\bigl(\frac{(n-2)\pi}{2},\frac{n\pi}{2}\bigr)$$3, only finitely many curves of negative self-intersection need be checked, and the boundary of the solvable locus is a real algebraic codimension-one wall. This yields a first PDE analogue of a locally finite wall-chamber decomposition in Bridgeland stability (Khalid et al., 2022).

In complex dimension three, Pingali proves existence for the full admissible phase range

$$\hat\theta\in\bigl(\frac{(n-2)\pi}{2},\frac{n\pi}{2}\bigr)$$4

on compact Kähler threefolds, conditioned on a necessary subsolution condition. The crucial threefold identity is the generalized Monge–Ampère reformulation

$$\hat\theta\in\bigl(\frac{(n-2)\pi}{2},\frac{n\pi}{2}\bigr)$$5

whose constant term has mixed sign when $$\hat\theta\in\bigl(\frac{(n-2)\pi}{2},\frac{n\pi}{2}\bigr)$$6. The paper introduces a new continuity path with explicit coefficient $$\hat\theta\in\bigl(\frac{(n-2)\pi}{2},\frac{n\pi}{2}\bigr)$$7, proves $$\hat\theta\in\bigl(\frac{(n-2)\pi}{2},\frac{n\pi}{2}\bigr)$$8, second-order, and $$\hat\theta\in\bigl(\frac{(n-2)\pi}{2},\frac{n\pi}{2}\bigr)$$9 estimates, and thereby extends threefold existence to the full admissible range (Pingali, 2019).

In complex dimension four, the delicate interval

$\theta\in(0,\pi)$0

becomes $\theta\in(0,\pi)$1. Huang proves that when $\theta\in(0,\pi)$2 is close to $\theta\in(0,\pi)$3 from the right, that is for $\theta\in(0,\pi)$4, the existence of a $\theta\in(0,\pi)$5-subsolution already implies solvability, even without assuming the subsolution is supercritical. The proof uses a new continuity path, semialgebraic $\theta\in(0,\pi)$6-cones, and Positivstellensatz positivity in dimension four (Lin, 2022).

Highly symmetric geometries provide exact models. On rational homogeneous varieties $\theta\in(0,\pi)$7, every invariant Kähler class and every real $\theta\in(0,\pi)$8-class admit a homogeneous dHYM solution, and the lifted phase is given explicitly by

$\theta\in(0,\pi)$9

Supercritical and hypercritical homogeneous solutions are therefore characterized directly by Cartan data (Correa, 2023).

Two further symmetric constructions sharpen the geometric picture. On the blow-up of $\chi$00 at a point, Collins–Xie–Yau prove the supercritical conjectural criterion in that model and show more generally that solvability is governed by a lifted phase together with divisor inequalities, thereby emphasizing the importance of branch selection beyond the supercritical case (Jacob et al., 2020). On the Calabi-symmetric projective bundles

$\chi$01

Jacob–Yau–Zhang reduce dHYM to an ODE and then to a level-set problem for an explicit harmonic polynomial; in that family, dHYM solvability is equivalent to exact lifted-phase inequalities for the charges of $\chi$02, $\chi$03, and $\chi$04 (Jacob, 2022).

5. Boundary cases, weak solutions, and singular flow limits

A separate line of work studies what happens when the strict supercritical inequalities degenerate. Sun formulates the boundary case by replacing the strict $\chi$05-subsolution condition

$\chi$06

with the non-strict condition

$\chi$07

solves strictly supercritical approximating equations with nef and big perturbations, proves the envelope-relative estimate

$\chi$08

and constructs bounded pluripotential solutions under phase- and dimension-dependent hypotheses. In dimension $\chi$09, the argument splits into the hypercritical range $\chi$10 and the remaining supercritical range $\chi$11, with an additional $\chi$12 assumption in the latter case (Sun, 2023).

Murakami pushes this boundary theory to the closure of the solvable region. For $\chi$13, $\chi$14, and an approximating sequence of stable interior data, the paper proves existence and uniqueness of a quasi-psh potential $\chi$15 solving

$\chi$16

where $\chi$17 denotes the non-pluripolar product. It also proves the analogous twisted weak existence theorem and shows that the dHYM flow converges to the weak solution in the sense of currents. In the standard $\chi$18-phase notation, this is exactly the boundary theory for the usual supercritical interval $\chi$19 (Murakami, 28 May 2026).

Explicit singularity formation is now known in higher dimension with symmetry. On

$\chi$20

with $\chi$21 and $\chi$22, the dHYM cotangent flow exists for all time but converges only weakly to a current $\chi$23 with divisorial singularity along the exceptional divisor $\chi$24. The limiting current satisfies the singular dHYM equation in the weak sense, with a new slope $\chi$25, and provides explicit evidence that unstable supercritical classes may still admit canonical singular limits rather than smooth solutions (Mete, 2024).

6. Mirror symmetry, coupled theories, and remaining problems

From its beginning, dHYM has been understood as the complex-geometric mirror of the special Lagrangian equation. The phase of $\chi$26 is the complex analogue of the Lagrangian angle, and the central charge

$\chi$27

packages the corresponding cohomological phase. In the supercritical branch, the equation is therefore tied simultaneously to calibrated geometry, mirror symmetry, and stability-type inequalities on analytic subvarieties (Collins et al., 2017).

This stability picture extends beyond the fixed-metric problem. Schlitzer–Stoppa introduce a coupled system

$\chi$28

obtained from the extended gauge group by coupling the dHYM moment-map picture to scalar curvature as a moment map. The paper explicitly states that one expects solutions to satisfy a mixture of $\chi$29-stability and Bridgeland-type stability, and on abelian surfaces it proves existence results in the branch singled out by

$\chi$30

under its sign convention (Schlitzer et al., 2019).

Several problems remain open in the supplied literature. In dimension four, the theorem near $\chi$31 does not cover the full interval $\chi$32, and higher dimensions in the delicate supercritical range remain unresolved (Lin, 2022). In the boundary weak theory, the bounded-envelope hypothesis, the extra $\chi$33 assumption in dimension $\chi$34 and non-hypercritical phase, and a sharper numerical characterization of the boundary solvability condition are identified as open issues (Sun, 2023). At a conceptual level, Xu’s counterexamples show that subvariety inequalities alone do not encode the analytic branch, and the paper concludes that on general Kähler manifolds numerical criteria for dHYM likely require more refined data, such as the test family introduced by G. Chen, rather than only the Collins–Jacob–Yau inequalities on subvarieties (Zhang, 2023).

The resulting picture is structurally coherent but no longer naive. In the interior of the supercritical region, the equation is governed by admissible cones, $\chi$35-subsolutions, and stability-type positivity. In projective and highly symmetric settings, this can lead to sharp numerical characterizations. On general compact Kähler manifolds, however, the branch choice is an essential part of the problem, and at the boundary of the solvable region the natural objects are weak or singular canonical solutions rather than smooth ones.