Generalized Monge-Ampère Equation

• Generalized Monge-Ampère equations are a family of fully nonlinear PDEs that extend the classical determinant-based structure by coupling top-degree forms with lower-degree terms in diverse geometric settings.
• They incorporate cohomological balance and positivity cone conditions to ensure solvability, employing weak and singular solution frameworks along with advanced continuity methods.
• These equations have wide applications in complex and Kähler geometry, jet-space formulations, and physical theories such as generalized Kähler geometry and Type II supergravity.

Searching arXiv for recent and foundational papers on generalized Monge–Ampère equations. Search query: "generalized Monge-Ampère equation" The generalized Monge–Ampère equation is not a single canonical PDE but a family of fully nonlinear equations and exterior differential systems that extend the classical Monge–Ampère paradigm by coupling top-degree determinants or wedge powers to lower-degree terms, mixed forms, or minor conditions on jet spaces. In complex geometry, one encounters equations of the form

$$\omega_\varphi^n-\sum_{k=1}^{n-1}c_k\,\chi^{\,n-k}\wedge\omega_\varphi^k=f\,\chi^n,$$

weak-current identities involving nonpluripolar products,

$$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$

and determinant relations for generalized Kähler potentials; in jet-space formulations, generalized Monge–Ampère equations are systems built from minors of prolongation matrices and are equivalent to special exterior differential systems (Datar et al., 2020, Murakami, 28 May 2026, Hull et al., 2010, Kawamata et al., 2020).

1. Families of equations and representative forms

A broad Kähler formulation appears in Pingali’s equation

$$a_0\wedge(\omega+i\partial\bar\partial\varphi)^n+a_1\wedge(\omega+i\partial\bar\partial\varphi)^{n-1}+\cdots+a_{n-1}\wedge(\omega+i\partial\bar\partial\varphi)=n\,\Omega,$$

with $\int_X\varphi\,\omega^n=0$ on a compact Kähler manifold (Pingali, 2012). In dimension three, Pingali also studies the torus equation

$$(\omega+dd^c\phi)^3-\alpha\wedge(\omega+dd^c\phi) = e^F\,(\omega^3-\alpha\wedge\omega),$$

for a flat complex $3$-torus and a constant-coefficient $(2,2)$-form $\alpha$ satisfying $\alpha-\omega^2>0$ and $\omega^3-\alpha\wedge\omega>0$ (Pingali, 2013).

On projective and Hermitian manifolds, generalized complex Monge–Ampère equations are written as mixed wedge equations. Datar–Pingali study

$$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$0

with a cone condition expressing admissibility, while Sun considers

$$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$1

on closed Hermitian manifolds (Datar et al., 2020, Sun, 2014).

The terminology also extends beyond scalar potentials. For forms of positive degree, the original Monge–Ampère-type equation of Dinew–Popovici seeks a real $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$2-form $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$3 satisfying

$$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$4

supplemented by a gauge condition, while generalized Kähler geometry yields a determinant identity for a single real generalized Kähler potential $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$5,

$$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$6

These formulations show that the terminology is structurally rather than syntactically uniform: what remains common is a fully nonlinear positivity-preserving relation generalizing the Monge–Ampère determinant (George, 22 Jun 2026, Hull et al., 2010).

2. Positivity, cohomological balance, and solvable regions

A recurrent feature is a cohomological or volume-normalization constraint. In Murakami’s Kähler setup, $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$7 is a compact Kähler manifold, $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$8 is a nef real $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$9 class, the constants $$a_0\wedge(\omega+i\partial\bar\partial\varphi)^n+a_1\wedge(\omega+i\partial\bar\partial\varphi)^{n-1}+\cdots+a_{n-1}\wedge(\omega+i\partial\bar\partial\varphi)=n\,\Omega,$$0 are nonnegative, and $$a_0\wedge(\omega+i\partial\bar\partial\varphi)^n+a_1\wedge(\omega+i\partial\bar\partial\varphi)^{n-1}+\cdots+a_{n-1}\wedge(\omega+i\partial\bar\partial\varphi)=n\,\Omega,$$1 is continuous, subject to

$$a_0\wedge(\omega+i\partial\bar\partial\varphi)^n+a_1\wedge(\omega+i\partial\bar\partial\varphi)^{n-1}+\cdots+a_{n-1}\wedge(\omega+i\partial\bar\partial\varphi)=n\,\Omega,$$2

In Pingali’s torus equation one requires

$$a_0\wedge(\omega+i\partial\bar\partial\varphi)^n+a_1\wedge(\omega+i\partial\bar\partial\varphi)^{n-1}+\cdots+a_{n-1}\wedge(\omega+i\partial\bar\partial\varphi)=n\,\Omega,$$3

and on a closed almost Kähler surface the analogue is

$$a_0\wedge(\omega+i\partial\bar\partial\varphi)^n+a_1\wedge(\omega+i\partial\bar\partial\varphi)^{n-1}+\cdots+a_{n-1}\wedge(\omega+i\partial\bar\partial\varphi)=n\,\Omega,$$4

These balance conditions ensure compatibility between the nonlinear equation and the fixed background cohomology or volume form (Murakami, 28 May 2026, Pingali, 2013, Wang et al., 2024).

Admissibility is encoded by positivity cones and subsolution conditions. In Murakami’s interior region, the generalized Monge–Ampère equation has a classical $$a_0\wedge(\omega+i\partial\bar\partial\varphi)^n+a_1\wedge(\omega+i\partial\bar\partial\varphi)^{n-1}+\cdots+a_{n-1}\wedge(\omega+i\partial\bar\partial\varphi)=n\,\Omega,$$5-solution precisely when there exists a smooth $$a_0\wedge(\omega+i\partial\bar\partial\varphi)^n+a_1\wedge(\omega+i\partial\bar\partial\varphi)^{n-1}+\cdots+a_{n-1}\wedge(\omega+i\partial\bar\partial\varphi)=n\,\Omega,$$6-subsolution

$$a_0\wedge(\omega+i\partial\bar\partial\varphi)^n+a_1\wedge(\omega+i\partial\bar\partial\varphi)^{n-1}+\cdots+a_{n-1}\wedge(\omega+i\partial\bar\partial\varphi)=n\,\Omega,$$7

The boundary case is obtained by approximating $$a_0\wedge(\omega+i\partial\bar\partial\varphi)^n+a_1\wedge(\omega+i\partial\bar\partial\varphi)^{n-1}+\cdots+a_{n-1}\wedge(\omega+i\partial\bar\partial\varphi)=n\,\Omega,$$8, $$a_0\wedge(\omega+i\partial\bar\partial\varphi)^n+a_1\wedge(\omega+i\partial\bar\partial\varphi)^{n-1}+\cdots+a_{n-1}\wedge(\omega+i\partial\bar\partial\varphi)=n\,\Omega,$$9, and the coefficients $\int_X\varphi\,\omega^n=0$0 by nearby data for which the relevant positivity tests hold strictly (Murakami, 28 May 2026).

For projective manifolds, Datar–Pingali prove an exact numerical criterion. Solvability of the mixed equation, existence of a Kähler metric in the class satisfying the pointwise cone condition, and positivity of the intersection numbers

$\int_X\varphi\,\omega^n=0$1

for every irreducible subvariety $\int_X\varphi\,\omega^n=0$2 of codimension $\int_X\varphi\,\omega^n=0$3 are equivalent (Datar et al., 2020). In this form, solvability is characterized by a Nakai–Moishezon-type criterion.

3. Weak, degenerate, and singular solutions

Murakami develops a weak-current theory for boundary classes using nonpluripolar products. If $\int_X\varphi\,\omega^n=0$4 with $\int_X\varphi\,\omega^n=0$5, then $\int_X\varphi\,\omega^n=0$6 is a closed positive current, and $\int_X\varphi\,\omega^n=0$7 is called a weak solution when

$\int_X\varphi\,\omega^n=0$8

together with the positivity mass-inequalities for every $\int_X\varphi\,\omega^n=0$9. Under the boundary-case assumption, there exists a unique $$(\omega+dd^c\phi)^3-\alpha\wedge(\omega+dd^c\phi) = e^F\,(\omega^3-\alpha\wedge\omega),$$0-psh weak solution, and the associated mixed Hessian flow converges weakly as currents to that solution (Murakami, 28 May 2026).

On compact Hermitian manifolds, Nguyen proves existence and uniqueness of continuous solutions for exponential and degenerate equations. For $$(\omega+dd^c\phi)^3-\alpha\wedge(\omega+dd^c\phi) = e^F\,(\omega^3-\alpha\wedge\omega),$$1, $$(\omega+dd^c\phi)^3-\alpha\wedge(\omega+dd^c\phi) = e^F\,(\omega^3-\alpha\wedge\omega),$$2, one obtains a unique $$(\omega+dd^c\phi)^3-\alpha\wedge(\omega+dd^c\phi) = e^F\,(\omega^3-\alpha\wedge\omega),$$3 with $$(\omega+dd^c\phi)^3-\alpha\wedge(\omega+dd^c\phi) = e^F\,(\omega^3-\alpha\wedge\omega),$$4 solving

$$(\omega+dd^c\phi)^3-\alpha\wedge(\omega+dd^c\phi) = e^F\,(\omega^3-\alpha\wedge\omega),$$5

and likewise a unique continuous solution for semi-positive closed backgrounds $$(\omega+dd^c\phi)^3-\alpha\wedge(\omega+dd^c\phi) = e^F\,(\omega^3-\alpha\wedge\omega),$$6 with $$(\omega+dd^c\phi)^3-\alpha\wedge(\omega+dd^c\phi) = e^F\,(\omega^3-\alpha\wedge\omega),$$7 (Nguyen, 2015). Li–Lin–Zhou develop a complementary Hermitian theory for

$$(\omega+dd^c\phi)^3-\alpha\wedge(\omega+dd^c\phi) = e^F\,(\omega^3-\alpha\wedge\omega),$$8

in the relative full-mass class $$(\omega+dd^c\phi)^3-\alpha\wedge(\omega+dd^c\phi) = e^F\,(\omega^3-\alpha\wedge\omega),$$9, where $3$0 is a positive non-pluripolar Radon measure, $3$1, and $3$2 is a model potential; uniqueness follows from the comparison principle, and one moreover has

$3$3

so the solution inherits the Lelong-number profile of the model potential (Li et al., 2023).

Di Nezza–Lu treat singular right-hand sides on quasi-projective varieties. For

$3$4

with $3$5 smooth outside a divisor and satisfying the conditions $3$6 or, in the SNC setting, $3$7, there exists a unique normalized $3$8 that is smooth on $3$9. The asymptotics depend sharply on $(2,2)$0: if $(2,2)$1, $(2,2)$2 extends continuously; if $(2,2)$3, one gets double-logarithmic lower bounds; and if $(2,2)$4, the solution grows at a power rate in $(2,2)$5 (Nezza et al., 2014).

4. Analytic frameworks and proof strategies

Method-of-continuity arguments dominate the smooth theory. Pingali’s torus theorem uses openness via positivity of the linearized operator and closedness from uniform $(2,2)$6, $(2,2)$7, and $(2,2)$8 estimates, followed by Evans–Krylov and Schauder regularity. The $(2,2)$9 estimate uses Blocki’s device

$\alpha$0

and the $\alpha$1 estimate applies the maximum principle to $\alpha$2, with the mixed term $\alpha$3 controlled by convexity (Pingali, 2013). In the positive-degree setting, the originally proposed gauge $\alpha$4 is shown to be analytically too rigid, and the replacement is the $\alpha$5-gauge

$\alpha$6

which leads to a scalar $\alpha$7–Monge–Ampère equation with $\alpha$8, gradient, and $\alpha$9 estimates, and then Evans–Krylov–Schauder bootstrapping (George, 22 Jun 2026).

On closed Hermitian manifolds, Sun derives the a priori estimates needed for admissible solutions of

$\alpha-\omega^2>0$0

The $\alpha-\omega^2>0$1 bound comes from Moser iteration, the partial $\alpha-\omega^2>0$2 bound from a maximum principle applied to $\alpha-\omega^2>0$3, and the improved gradient estimate from a Blocki–Phong–Sturm–Guan–Sun argument (Sun, 2014). The parabolic counterpart evolves by

$\alpha-\omega^2>0$4

and under the cone condition the normalized flow has uniform $\alpha-\omega^2>0$5 estimates and converges in $\alpha-\omega^2>0$6 to the elliptic solution (Sun, 2015).

Murakami’s boundary-case theory combines viscosity and pluripotential methods. The limit of smooth approximants is first shown, via the CIL lemma and convolution, to be a viscosity subsolution; a pluripotential mass-comparison in the spirit of Bedford–Taylor and BEGZ upgrades the inequality to equality; and uniqueness follows from a viscosity/pluripotential maximum principle together with convexity of the operator and mass conservation (Murakami, 28 May 2026). In the projective setting, Datar–Pingali introduce a degenerate concentration of mass argument, producing weak limit currents with positive mass on ample divisors and then building barrier currents by resolution, induction on dimension, and Blocki–Kołodziej/Richberg gluing (Datar et al., 2020).

5. Geometric and physical applications

Several generalized Monge–Ampère equations are motivated by geometric PDE problems. Murakami treats the generalized Monge–Ampère equation and the supercritical deformed Hermitian–Yang–Mills equation simultaneously; under the same boundary hypotheses, the mixed Hessian flow converges weakly to the unique weak generalized Monge–Ampère solution, and the dHYM flow satisfies

$\alpha-\omega^2>0$7

hence converges in currents to the weak dHYM solution (Murakami, 28 May 2026).

Pingali’s flat-torus equation has a vector-bundle application. If $\alpha-\omega^2>0$8 is a conformal change of a Hermitian metric on a holomorphic vector bundle, then

$\alpha-\omega^2>0$9

and in complex dimension three the condition $\omega^3-\alpha\wedge\omega>0$0 becomes exactly the generalized Monge–Ampère equation after identifying

$\omega^3-\alpha\wedge\omega>0$1

Thus prescribed top Chern character forms can be realized by the Chern connection of a conformally modified metric under the stated positivity hypotheses (Pingali, 2013).

For forms of positive degree, the $\omega^3-\alpha\wedge\omega>0$2 framework yields the qualitative part of Demailly’s transcendental Morse inequality unconditionally. When $\omega^3-\alpha\wedge\omega>0$3, the equation reduces to the classical complex Monge–Ampère equation, and Yau’s theorem produces a strictly positive Kähler current in the Bott–Chern class $\omega^3-\alpha\wedge\omega>0$4 under the integral positivity hypothesis stated in the paper (George, 22 Jun 2026).

In generalized Kähler geometry, the determinant equation for the generalized Kähler potential is exactly the local condition for a generalized Kähler manifold to be a generalized Calabi–Yau metric. Its solutions yield target-space fields $\omega^3-\alpha\wedge\omega>0$5 solving the Type II supergravity equations with zero Ramond–Ramond fluxes; in the Kähler limit it reduces to the standard complex Monge–Ampère equation $\omega^3-\alpha\wedge\omega>0$6 (Hull et al., 2010). On closed almost Kähler surfaces, the equation

$\omega^3-\alpha\wedge\omega>0$7

has a unique smooth solution with full $\omega^3-\alpha\wedge\omega>0$8 estimates, and this is used to verify Donaldson’s conjecture for tamed almost complex $\omega^3-\alpha\wedge\omega>0$9-manifolds under the hypothesis $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$00 (Wang et al., 2024).

6. Jet-space and exterior-differential-system generalizations

A distinct line of work interprets generalized Monge–Ampère equations as special exterior differential systems on jet spaces. On the $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$01-jet space $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$02 with canonical contact system $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$03, Kawamata–Shibuya define a generalized Monge–Ampère system

$$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$04

generated by the contact forms and a finite collection of differential forms $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$05. The associated generalized Monge–Ampère equation is obtained from linear combinations of minors of the prolongation matrix $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$06, and Theorem 1.1 states that integral manifolds of the GMAS correspond bijectively to solutions of the GMAE; conversely, every such GMAE arises from a uniquely determined GMAS (Kawamata et al., 2020).

This framework recovers the classical second-order Monge–Ampère equation when $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$07, and it also accommodates examples not usually classified under the Monge–Ampère label. Kawamata–Shibuya verify that the Korteweg–de Vries equation

$$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$08

and the Cauchy–Riemann system

$$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$09

are both generalized Monge–Ampère equations in this sense (Kawamata et al., 2020).

Kawamata’s later study of second-order generalized Monge–Ampère equations on the plane encodes systems such as

$$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$10

as generalized Monge–Ampère systems on $$\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,$$11, reduces them via Cauchy characteristics, and constructs geometric singular solutions. The resulting wave fronts are classified by standard criteria into cuspidal edge, swallowtail, butterfly, and, in the non-generic case, beaks singularities (Kawamata, 2021). A plausible implication is that the expression “generalized Monge–Ampère equation” now names a mathematically coherent but genuinely plural class of nonlinear structures, unified less by a single formula than by determinant-like nonlinearity, positivity cones, and geometric origin.