Monge-Ampère-type equation for forms of positive degree and Demailly's transcendental Morse inequality

Abstract: The Monge-Ampère-type equation for forms of positive degree was introduced by Dinew and Popovici to prove the qualitative part of an analytic version of Demailly's transcendental Morse inequality for higher cohomology classes, conditional on the solvability of this nonlinear PDE. In this paper, we show that this can be proven unconditionally. We first demonstrate that the originally proposed Laplacian trace condition, $Λ^{m-2}Δu = 0$, is analytically too rigid to permit solutions. To overcome this, we introduce a general gauge-fixing that leads to the $(a,b)$ Monge-Ampère-type equation. By deriving \emph{a priori} estimates, we establish the solvability of this class of equations for different parametric regimes $(a,b)$. As a geometric application, for $b=0$ this framework reduces to the classical complex Monge-Ampère equation, which yields the qualitative Demailly's inequality for higher-degree forms. Furthermore, we show that the uniqueness of these solutions can be significantly strengthened under the kernel condition $\bar{\partial}^* u = 0$.

• The paper proves that Dinew and Popovici’s original gauge is analytically over-constrained, introduces a two-parameter (a,b) replacement, and establishes solvability through elliptic estimates and continuity methods.
• The (a,b) equation has unconditional solvability when a+b and b share a sign, while mixed-sign parameters require a spectral condition and sufficiently small data to preserve uniform ellipticity.
• The paper removes the prior solvability assumption in the Dinew–Popovici program and proves the qualitative higher-degree Demailly Morse inequality for specified positive forms, with a Bott–Chern refinement when the data are closed.

This paper by Mathew George resolves, unconditionally, the analytic gap left in the program of Dinew and Popovici (Dinew et al., 31 Oct 2025) toward an analytic version of Demailly's transcendental Morse inequality for higher-degree cohomology classes. The key contributions are threefold: a proof that the gauge condition originally proposed in that program is analytically unsolvable except trivially; the introduction of a two-parameter family of gauge-fixings leading to what the author calls the $(a,b)$ Monge-Ampère-type equation; and a complete solvability theory for this family via a priori estimates, which in turn yields the qualitative part of the higher-degree Morse inequality.

Background and motivation

Demailly's transcendental Morse inequalities provide cohomological volume thresholds guaranteeing that a difference of nef classes contains a strictly positive Kähler current. For $(1,1)$ classes on a compact $n$-dimensional Kähler manifold, the qualitative statement — that $$\int_M \alpha^n - n\int_M \alpha^{n-1}\wedge\beta > 0$$ forces $[\alpha]-[\beta]$ to contain a strictly positive current — was established by Popovici [Popovici16], building on partial results in [BDPP13] and [Xiao2015]. This inequality is a crucial ingredient for extending the duality between the pseudo-effective cone and the cone of movable classes from projective to compact Kähler manifolds.

Dinew and Popovici proposed extending this to $(m,m)$-forms with $1 < m \leq n$ via the Monge-Ampère-type equation

$$\left[\star_{\omega}\left((\alpha + i\partial\bar{\partial}u)\wedge \omega_{n-m-1}\right)\right]^n = dV,$$

for a strongly positive closed $(m,m)$-form $\alpha$, subject to the gauge conditions $(1,1)$0, $(1,1)$1, and $(1,1)$2. Their bigness criterion was conditional on solvability of this system. The present paper shows this conditional hypothesis cannot be met as stated and replaces it with a workable one.

Rigidity of the original Laplacian trace condition

Writing the Lefschetz decomposition of $(1,1)$3 as $(1,1)$4 with primitive coordinates $(1,1)$5, the paper shows that the Monge-Ampère-type equation depends only on the first two primitives $(1,1)$6 and $(1,1)$7. The condition $(1,1)$8, combined with the trace identity relating $(1,1)$9 to $n$0 and $n$1, forces both $n$2 and $n$3 by uniqueness of the Lefschetz decomposition. Hence $n$4 is constant and the equation degenerates to $n$5, which cannot hold for arbitrary volume forms. The conclusion is stark: the original gauge is over-constraining and admits no solutions except when $n$6 is a constant multiple of $n$7. This negative result justifies the reformulation that follows.

The kernel condition and strengthened uniqueness

The paper develops the structure theory of forms satisfying $n$8. Using the Kähler commutation relations, the kernel condition translates into a cascade of first-order relations among successive primitives:

$n$9

A notable corollary is that if $$\int_M \alpha^n - n\int_M \alpha^{n-1}\wedge\beta > 0$$0 satisfies $$\int_M \alpha^n - n\int_M \alpha^{n-1}\wedge\beta > 0$$1, then $$\int_M \alpha^n - n\int_M \alpha^{n-1}\wedge\beta > 0$$2 for all $$\int_M \alpha^n - n\int_M \alpha^{n-1}\wedge\beta > 0$$3 with $$\int_M \alpha^n - n\int_M \alpha^{n-1}\wedge\beta > 0$$4. Combined with a Dolbeault Green's operator argument, this yields a rigidity claim: if all intermediate primitives are $$\int_M \alpha^n - n\int_M \alpha^{n-1}\wedge\beta > 0$$5-exact and the top primitive vanishes, then all intermediate primitives vanish identically and $$\int_M \alpha^n - n\int_M \alpha^{n-1}\wedge\beta > 0$$6 is constant.

On this basis the paper proves a strengthened uniqueness theorem for solutions lying in the kernel: under exactness assumptions on the differences of primitives of orders 2 through $$\int_M \alpha^n - n\int_M \alpha^{n-1}\wedge\beta > 0$$7, any two solutions agree up to $$\int_M \alpha^n - n\int_M \alpha^{n-1}\wedge\beta > 0$$8 plus harmonic primitive components. The proof reduces the problem to an elliptic equation for $$\int_M \alpha^n - n\int_M \alpha^{n-1}\wedge\beta > 0$$9 constructed from the binomial expansion of the two Monge-Ampère expressions, to which the maximum principle applies. This improves on the uniqueness-up-to-$[\alpha]-[\beta]$0 obtained in [DP25] under stronger hypotheses.

The $[\alpha]-[\beta]$1 Monge-Ampère-type equation

The replacement gauge takes the form

$[\alpha]-[\beta]$2

with real constants satisfying the necessary constraint $[\alpha]-[\beta]$3 (obtained by tracing). Under this gauge, together with the kernel condition, the full form equation collapses to a scalar fully nonlinear PDE for $[\alpha]-[\beta]$4: the $[\alpha]-[\beta]$5 Monge-Ampère-type equation

$[\alpha]-[\beta]$6

where $[\alpha]-[\beta]$7. The choice $[\alpha]-[\beta]$8, $[\alpha]-[\beta]$9 recovers the classical complex Monge-Ampère equation, which is precisely the regime needed for the geometric application.

Solvability regimes

The main analytical theorem classifies solvability into two parametric regimes:

Regime	Parameter conditions	Solvability
Regime 1	$(m,m)$0 (or both reversed)	Unconditional, unique up to constants
Regime 2	$(m,m)$1 or $(m,m)$2	Conditional: requires $(m,m)$3 and small data $(m,m)$4

In Regime 1, the symbol $(m,m)$5 satisfies a uniform lower bound $(m,m)$6, so uniform ellipticity is automatic. The paper derives independent second-order estimates (via the standard test function $(m,m)$7 maximum-principle argument) and gradient estimates (via $(m,m)$8), then obtains $(m,m)$9 bounds from the ABP maximum principle as in Székelyhidi's framework, closing existence by continuity method with Evans-Krylov and Schauder regularity.

Regime 2 is more delicate because the lower bound on the smallest symbol eigenvalue involves the factor $1 < m \leq n$0, which can degenerate. The paper introduces the cone $1 < m \leq n$1 and proves a quantitative implicit function theorem via Banach contraction: solutions satisfy $1 < m \leq n$2 and $1 < m \leq n$3. An explicit smallness threshold,

$1 < m \leq n$4

keeps the solution metric strictly inside the elliptic cone along the continuity path, giving uniform ellipticity and hence existence at $1 < m \leq n$5. The author states plainly that removing the small-data assumption in Regime 2 remains open: standard maximum-principle arguments on $1 < m \leq n$6 do not appear to work, and it is unclear how to maintain uniform ellipticity along alternative paths.

Application to Demailly's transcendental Morse inequality

Taking $1 < m \leq n$7 and $1 < m \leq n$8, the solvability theorem supplies the solution required by the Dinew–Popovici scheme, and the qualitative higher-degree Morse inequality follows unconditionally. Specifically, for $1 < m \leq n$9-closed strongly positive $$\left[\star_{\omega}\left((\alpha + i\partial\bar{\partial}u)\wedge \omega_{n-m-1}\right)\right]^n = dV,$$0 with $$\left[\star_{\omega}\left((\alpha + i\partial\bar{\partial}u)\wedge \omega_{n-m-1}\right)\right]^n = dV,$$1, and $$\left[\star_{\omega}\left((\alpha + i\partial\bar{\partial}u)\wedge \omega_{n-m-1}\right)\right]^n = dV,$$2 with $$\left[\star_{\omega}\left((\alpha + i\partial\bar{\partial}u)\wedge \omega_{n-m-1}\right)\right]^n = dV,$$3, the volume threshold

$$\left[\star_{\omega}\left((\alpha + i\partial\bar{\partial}u)\wedge \omega_{n-m-1}\right)\right]^n = dV,$$4

implies the Aeppli class $$\left[\star_{\omega}\left((\alpha + i\partial\bar{\partial}u)\wedge \omega_{n-m-1}\right)\right]^n = dV,$$5 contains a real $$\left[\star_{\omega}\left((\alpha + i\partial\bar{\partial}u)\wedge \omega_{n-m-1}\right)\right]^n = dV,$$6 current $$\left[\star_{\omega}\left((\alpha + i\partial\bar{\partial}u)\wedge \omega_{n-m-1}\right)\right]^n = dV,$$7 with $$\left[\star_{\omega}\left((\alpha + i\partial\bar{\partial}u)\wedge \omega_{n-m-1}\right)\right]^n = dV,$$8 for some $$\left[\star_{\omega}\left((\alpha + i\partial\bar{\partial}u)\wedge \omega_{n-m-1}\right)\right]^n = dV,$$9; if additionally $(m,m)$0, then $(m,m)$1 can be chosen $(m,m)$2-closed in the Bott-Chern class $(m,m)$3.

The proof follows the Lamari duality strategy: assuming a contradicting sequence $(m,m)$4 normalized by $(m,m)$5, one solves the Monge-Ampère-type equation with right-hand side $(m,m)$6, applies a Hölder-type pointwise inequality, and uses Stokes' theorem — exploiting $(m,m)$7-closedness of $(m,m)$8 and $(m,m)$9 and harmonicity of $\alpha$0 — to show the integrals involving the deformed form $\alpha$1 reduce to those of $\alpha$2. The contradiction with assumption 3 follows directly. The unconditional nature of this result is its principal advance over [DP25], where the same conclusion required assuming solvability of the now-discredited gauge.

Limitations and open questions

Several qualifications bear directly on the strength of the results. First, uniqueness of solutions to the general $\alpha$3 system is not guaranteed; only the lowest-order primitives $\alpha$4 and $\alpha$5 are unique up to harmonic forms, with further uniqueness requiring the additional exactness hypotheses of the strengthened uniqueness theorem. Second, Regime 2 solvability depends on both a spectral condition on the background data ($\alpha$6) and a small-data assumption on the target volume form; whether these can be removed is explicitly posed as open, with the author noting that the usual maximum-principle route fails and suggesting that structural properties preventing the solution metric from reaching the boundary of the elliptic cone may be needed. Third, the geometric application assumes $\alpha$7 and a uniform lower bound $\alpha$8, so the inequality is proven for this restricted class of data rather than in full generality.

Conclusion

The paper converts a conditional analytic program into an unconditional theorem by diagnosing the failure of the original gauge, introducing a flexible two-parameter replacement, and developing the requisite a priori estimates across the natural parameter regimes. The resulting unconditional qualitative Demailly inequality for $\alpha$9-classes removes the principal obstruction identified by Dinew and Popovici, while the $(1,1)$00 equation itself constitutes a new family of fully nonlinear equations interpolating between the complex Monge-Ampère equation ($(1,1)$01) and the $(1,1)$02-plurisubharmonic Monge-Ampère equation ($(1,1)$03). The main outstanding problem is large-data solvability in Regime 2, where uniform ellipticity along the continuity path is not currently controllable.