Pith. sign in

REVIEW 3 major objections 5 minor 66 references

Complex Hessian equations with prescribed singularity on compact K\"ahler manifolds

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves complex Hessian equations with prescribed singularity are solvable for every non-m-polar measure of the right total mass, with a unique normalized solution.

desk verdict Genuinely new monotonicity and subextension results for complex Hessian equations, but Theorem 1.2 rests on an unproved Cegrell reduction that the text itself contradicts. read the letter →

arxiv 1909.02469 v1 pith:VV2LBVYP submitted 2019-09-05 math.CV math.DG

Chinh H. Lu , Van-Dong Nguyen This is my paper · ORCID
classification math.CVmath.DG MSC 32W2032U0532Q15
keywords compactKählermanifoldcomplexHessianequationprescribedsingularityω-m-subharmonicfunctionsmeasurefiniteenergyclassmodelpotentialHodgeindexinequality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper brings the theory of complex Hessian equations on compact Kähler manifolds from the full-mass class to classes with prescribed singularities. The first main theorem states that the total mass of the mixed Hessian measure is monotone under singularity type: if each $u_p$ is more singular than $v_p$, then $\int_X H_m(u_1,\ldots,u_m) \le \int_X H_m(v_1,\ldots,v_m)$. The second main theorem solves $H_m(u)=\mu$ in the relative finite-energy class $E_\varphi$ under the natural conditions that $\varphi$ is a model potential, $\mu$ does not charge $m$-polar sets, and the total masses agree. The third gives a Hodge-index type inequality for mixed Hessian masses. If correct, the paper provides a relative potential theory for Hessian operators, parallel to the classical theory in big cohomology classes, and opens the same equations to measures with heavy singularities.

What carries the argument

The load-bearing object is the mixed complex Hessian measure $H_m(u_1,\ldots,u_m)=(\omega+dd^c u_1)\wedge\cdots\wedge(\omega+dd^c u_m)\wedge\omega^{n-m}$, defined by the non-$m$-polar product for unbounded $\omega$-$m$-subharmonic functions. The argument proceeds through four interconnected tools: a slope formula for the Hessian energy $E(\max(u,-s))$ that yields the mass monotonicity theorem; relative potential theory in which the envelope $P[\varphi]$ defines model potentials and the class $E_\varphi$; a metric $d$ on the finite-energy class $E^1$, defined through the rooftop envelope $P(u,v)$, with the key property that $d$ is complete and comparable to $I_1(u,v)=\int_X |u-v|(H_m(u)+H_m(v))$; and a supersolution method where solutions to approximate problems are glued by envelopes. Metric completeness supplies the lower bound for supersolutions, replacing the relative $L^\infty$ estimate that is not available in the Hessian setting.

What would settle it

Let $1\le m<n$ be fixed. Construct a sequence $u_j\in E^1$ that is Cauchy in $d$ but whose pointwise limit does not belong to $E^1$, for instance because its Hessian energy is infinite or its total Hessian mass is not $1$. Such a sequence would falsify the completeness theorem and remove the lower bound used in the existence proof; alternatively, an explicit model potential $\varphi$ and non-$m$-polar measure $\mu$ with $\mu(X)=\int_X H_m(\varphi)>0$ but no normalized $u\in E_\varphi$ solving $H_m(u)=\mu$ would directly disprove the main existence theorem.

Watch

Extended reading notes

Core claim

The central discovery is that the obstacle to solving Hessian equations with prescribed singularity is only the total mass of the Hessian measure, not the shape or concentration of the measure. For a model potential $\varphi$, meaning an $\omega$-$m$-subharmonic function with $P[\varphi]=\varphi$, and for any positive measure $\mu$ that vanishes on $m$-polar sets and satisfies $\mu(X)=\int_X H_m(\varphi)>0$, the equation $H_m(u)=\mu$ has a unique solution $u\in E_\varphi$ normalized by $\sup_X u=0$. This is proved by first establishing mass monotonicity with respect to singularity, then building envelopes, comparison and domination principles, and then constructing solutions as lower envelopes of approximate supersolutions. A byproduct is the inequality $\int_X H_m(u_1,\ldots,u_m)\ge \prod_{k=1}^m \left(\int_X H_m(u_k)\right)^{1/m}$ for any $\omega$-$m$-subharmonic functions $u_1,\ldots,u_m$.

Load-bearing premise

The proof depends on the completeness of the metric $d$ on the Hessian finite-energy class $E^1$ and on the two-sided estimate $d\approx I_1$, facts imported from the $m=n$ setting by analogy; if they fail for $m<n$, the subextension step and the existence proof collapse.

Editorial extensions

If this is right

  • For every model potential $\varphi$, the normalized solution map $\mu\mapsto u\in E_\varphi$ is a bijection from non-$m$-polar measures of mass $\int_X H_m(\varphi)$ onto $E_\varphi$.
  • The Hodge-index type inequality gives log-concavity-type control on mixed Hessian masses, so products of Hessian measures obey the predicted lower bound.
  • The Aubin–Yau type equation $H_m(u)=e^u\mu$ is solvable under the same assumptions on $\mu$ and $\varphi$.
  • The relative potential theory developed here makes the comparison and domination principles available in the Hessian category, so further equations with prescribed singularity can be treated by the same envelope machinery.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I infer that the same supersolution plus metric-completeness strategy should solve Hessian equations in big cohomology classes, provided the $d\approx I_1$ comparison can be re-derived without a Kähler reference form.
  • I infer that if metric completeness is the true bottleneck, a self-contained proof of completeness for the Hessian $E^1$ would remove the paper's main imported assumption; until then the existence theorem inherits that assumption.
  • The uniqueness proof via contact sets does not rely on geodesics, so I expect it to generalize to equations of Monge–Ampère type on non-Kähler manifolds where geodesic methods are unavailable.
  • The mass monotonicity might yield a full Brunn–Minkowski-type inequality for mixed Hessian measures, extending the Hodge-index bound.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies complex Hessian equations (ω + ddc u)^m ∧ ω^{n-m} = μ on a compact Kähler manifold, with a prescribed singularity type encoded by a model potential φ satisfying P[φ]=φ. The three main claims are: (i) Theorem 1.1, monotonicity of the total mass of the non-m-polar Hessian product under singularity ordering; (ii) Theorem 1.2, existence and uniqueness of a solution u ∈ E_φ for any positive measure μ vanishing on m-polar sets with μ(X)=∫ H_m(φ); (iii) Theorem 1.3, a Hodge-index type inequality for positive currents. The proof of Theorem 1.1 is based on a slope formula (Lemma 3.1) and energy monotonicity; Theorem 1.2 is approached via the supersolution method, with a metric d on the Hessian finite-energy class E^1 and a subextension theorem (Theorem 4.11); Theorem 1.3 is derived from Theorems 1.1 and 1.2 and the mixed Hessian inequality.

Significance. If the main theorems are correct, the paper would be a substantial contribution: it would establish a Hessian analogue of the Monge-Ampère theory of prescribed singularity, including solvability for the natural class of non-m-polar measures, and a Hodge-index inequality. The proof of Theorem 1.1 appears convincing and is a worthwhile new step, avoiding geodesic methods. The construction of the complete metric d on E^1 and the subextension theorem are valuable tools, and the envelope-based uniqueness proof is a genuine novelty. However, the current written proof of Theorem 1.2 contains a load-bearing gap in the reduction to dominated measures and a further gap in the construction of supersolutions; until these are fixed, the main existence theorem is established only conditionally.

major comments (3)
  1. [Section 5.2, Theorem 5.4] The proof begins with the claim "It suffices to treat the case when μ ≤ A H_m(ψ_0)", followed by "The general case will follow by a well-known projection argument due to Cegrell as shown in [39,21]". No proof of this projection in the Hessian setting is given, and both [39] and [21] concern the Monge-Ampère case m=n. This is not a routine modification: the paper's own introduction to Section 5 states that "In the general case of non-m-polar measures the approach in [20] using Cegrell's method [11] also breaks down in the Hessian setting." Since the rest of the proof, including the use of Theorem 4.11, requires the domination hypothesis H_m(v_j) ≤ A H_m(ψ_0) with ψ_0 bounded, the written proof establishes Theorem 1.2 only for μ of the dominated form, not for arbitrary non-m-polar μ.
  2. [Section 5.2, proof of the claim in Theorem 5.4] After defining v_k := P(b u_k - (b-1) max(φ,-k)), the paper states "Since 0 = P[u_k], it follows from Corollary 3.20 (with u,v ∈ E hence P[u]=P[v]=0) that v_k ∈ E." This application is not justified: v = max(φ,-k) is not necessarily an element of E, and P[max(φ,-k)] is not generally 0 for a model potential φ with P[φ]=φ. The conclusion v_k ∈ E is essential, as the subsequent use of Theorem 4.11 requires a sequence in E with sup_X = 0. The argument needs to be repaired, for instance by using a different reduction or by proving directly that v_k has full Hessian mass.
  3. [Section 4.3, Theorem 4.10] The completeness of the metric space (E^1, d) is asserted via references to Darvas [15,16] and [19], with phrases such as "The argument is due to Darvas" and "As in the proof of [16, Theorem 9.2]". While the outline is plausible, the proof depends on results that have not been fully verified in the Hessian setting m<n, namely the energy monotonicity and convergence properties summarized in Proposition 2.14 and the comparison between d and I_1 (Theorem 4.8). Since Theorem 4.11 and, through it, the lower bounds in Theorem 5.4 rely on this completeness, the authors should either provide a complete and self-contained proof or state the Hessian analogue as a precise theorem with an exact reference.
minor comments (5)
  1. [Section 2.2, Theorem 2.6 and Section 2.3, Propositions 2.13 and 2.14] Several foundational results are justified as "obvious modifications" of the Monge-Ampère case. Given that these results are used repeatedly in the proofs of Theorems 3.3, 4.8 and 4.10, a brief indication of what must be modified would improve verifiability.
  2. [Section 4.1] The sentence "The proof of [16, Theorem 3.6], applied to the Hessian setting, shows that P(u,v) ∈ E^1" states a non-trivial fact without giving the adaptation. This point is used in the definition of d and should be elaborated or explicitly referenced.
  3. [Section 5.4, Theorem 5.6] Theorem 5.6 (the Aubin-Yau type equation) is stated without proof, with the reader referred to [21,20]. Since the introduction suggests this is a direct consequence of Theorem 1.2, it would be clearer to label it as a corollary or to provide a short proof.
  4. [Theorem 1.2 and Theorem 5.4] The terminology could be made precise: the abstract and Theorem 1.2 speak of "non-m-polar" measures, while Theorem 5.4 speaks of measures "vanishing on m-polar sets". These are not identical notions, and the intended meaning should be stated consistently.
  5. [Proposition 3.2] There is a typo: "assume" is spelled "ssume" in the statement of Proposition 3.2.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: the derivation chain is largely self-contained, with one unproved imported projection in Theorem 5.4 that is a completeness gap rather than circularity.

full rationale

I walked the claimed derivation chain and found no equation or theorem that reduces by construction to its own input. Theorem 1.1 is proved from the slope formula (Lemma 3.1) and energy monotonicity, both derived in the paper from definitions and prior independent results; Theorem 1.2 is proved by the supersolution method using the existence result from [52] for truncated equations, the metric completeness of E^1 (Theorem 4.10, credited to Darvas), and the subextension theorem (Theorem 4.11), whose proof is actually given in the paper rather than merely cited. The self-citations to [19], [20], [21], [49], and [52] are frequent, but the cited results are published with independent proofs and are not shown to assume the target theorem. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work to forbid alternatives. One issue deserves explicit flagging as a correctness risk rather than circularity: in the proof of Theorem 5.4, the paper reduces the general non-m-polar measure case to the dominated case μ ≤ A H_m(ψ_0) by saying 'The general case will follow by a well-known projection argument due to Cegrell as shown in [39,21]', while earlier the paper concedes that 'In the general case of non-m-polar measures the approach in [20] using Cegrell's method [11] also breaks down in the Hessian setting.' This is an omitted proof and an apparent tension, and the written proof of Theorem 5.4 as it stands covers only the dominated case. However, this is a gap or unsupported import, not a circular reduction: the projection argument, if supplied, would be an external input rather than an equivalent reformulation of the conclusion. The central mathematical content of the paper remains independent of its own conclusions. Overall circularity score: 2, reflecting only the heavy but transparent reliance on the authors' prior work and the unproved imported step, neither of which makes the derivation circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The central claim rests on established pluripotential theory for m-subharmonic functions, on the Darvas-type complete metric imported from [19], and on the unproved Cegrell projection reduction for arbitrary non-m-polar measures. The metric completeness and the projection reduction are the externally supplied tools that do the heaviest lifting.

assumptions (6)
  • domain assumption (X,ω) is a compact Kähler manifold of dimension n and ω is normalized so that ∫_X ω^n=1.
    Assumed throughout the paper, stated in the introduction. The normalization simplifies mass comparisons and is harmless.
  • standard math The Bedford-Taylor-type complex Hessian operator H_m is well defined for bounded ω-m-subharmonic functions and extends by truncation to general ω-m-subharmonic functions.
    Background established in [52,49,36] and cited in Section 2.2. The extension is used to define H_m(u_1,...,u_m) for unbounded functions.
  • standard math The mixed Hessian inequality (Lemma 2.12) and the domination principle (Theorem 3.15) hold for ω-m-subharmonic functions.
    The mixed Hessian inequality is proven by reducing to [32]; the domination principle is proven in the paper using Lemma 3.14. These give comparison and uniqueness for solutions.
  • standard math The space (E^1,d) with d(u,v)=E(u)+E(v)-2E(P(u,v)) is complete and d is comparable with the I_1 distance.
    Adapted from Darvas [15,16] and [19] in Section 4.3. The proof is largely imported from the Monge-Ampère case and is load-bearing for the subextension theorem.
  • standard math There exists a solution in the full-mass class E to H_m(u)=µ for every non-m-polar positive measure with µ(X)=1.
    Used in Theorem 5.4 to construct approximate supersolutions; cited to [52, Theorem 1.3].
  • standard math Every non-m-polar positive measure can be reduced, via Cegrell's projection argument, to a measure bounded by A H_m(ψ_0) for some bounded ω-m-sh ψ_0.
    Invoked in Section 5.2 with the words 'the general case will follow by a well-known projection argument due to Cegrell as shown in [39,21]'. This step is not written out for the Hessian setting and is a genuine gap in presentation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Complex Hessian equations with prescribed singularity on compact K\"ahler manifolds." pith.science (2026). https://pith.science/paper/VV2LBVYP

@misc{pith2026190902469,
  author       = {Pith},
  title        = {Pith review of: Complex Hessian equations with prescribed singularity on compact K\"ahler manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VV2LBVYP}},
  note         = {Machine review of arXiv:1909.02469}
}
abstract

Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix $1\leq m\leq n$. We prove that the total mass of the complex Hessian measure of $\omega$-$m$-subharmonic functions is non-decreasing with respect to the singularity type. We then solve complex Hessian equations with prescribed singularity, and prove a Hodge index type inequality for positive currents.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

66 extracted references · 64 canonical work pages

  1. [20]

    Lu, Log-concavity of volume and complex Monge-Amp` ere equations with prescribed singular ity, arXiv:072018 (2018)

    Tam´ as Darvas, Eleonora Di Nezza, and Chinh H. Lu, Log-concavity of volume and complex Monge-Amp` ere equations with prescribed singular ity, arXiv:072018 (2018)

    work page 2018

  2. [39]

    26, European Mathematical Society (EMS), Z¨ urich,

    Vincent Guedj and Ahmed Zeriahi, Degenerate complex Monge-Amp` ere equations , EMS Tracts in Mathematics, vol. 26, European Mathematical Society (EMS), Z¨ urich,

    work page

  3. [21]

    Lu, Monotonicity of nonpluripo- lar products and complex Monge-Amp` ere equations with pres cribed singularity, Anal

    Tam´ as Darvas, Eleonora Di Nezza, and Chinh H. Lu, Monotonicity of nonpluripo- lar products and complex Monge-Amp` ere equations with pres cribed singularity, Anal. PDE 11 (2018), no. 8, 2049–2087. MR 3812864

    work page 2018

  4. [11]

    180 (1998), no

    Urban Cegrell, Pluricomplex energy , Acta Math. 180 (1998), no. 2, 187–217. MR 1638768

    work page 1998

  5. [19]

    Lu, L1 metric geometry of big cohomology classes , Ann

    Tam´ as Darvas, Eleonora Di Nezza, and Chinh H. Lu, L1 metric geometry of big cohomology classes , Ann. Inst. Fourier (Grenoble) 68 (2018), no. 7, 3053–3086. MR 3959105

    work page 2018

  6. [1]

    Alesker and M

    S. Alesker and M. Verbitsky, Quaternionic Monge-Amp` ere equation and Calabi prob- lem for HKT-manifolds , Israel J. Math. 176 (2010), 109–138. MR 2653188

    work page 2010

  7. [2]

    Thierry Aubin, ´Equations du type Monge-Amp` ere sur les vari´ et´ es k¨ ahl´ eriennes com- pactes, Bull. Sci. Math. (2) 102 (1978), no. 1, 63–95. MR 494932

    work page 1978

  8. [3]

    Eric Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Amp` ere equation, Invent. Math. 37 (1976), no. 1, 1–44. MR 0445006

    work page 1976

Show all 66 references
  1. [4]

    Eric Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions , Acta Math. 149 (1982), no. 1-2, 1–40. MR 674165

    1982

  2. [5]

    Eric Bedford and B. A. Taylor, Fine topology, ˇSilov boundary, and (ddc)n, J. Funct. Anal. 72 (1987), no. 2, 225–251. MR 886812

    1987

  3. [6]

    Berman, From Monge-Amp` ere equations to envelopes and geodesic ray s in the zero temperature limit , Math

    Robert J. Berman, From Monge-Amp` ere equations to envelopes and geodesic ray s in the zero temperature limit , Math. Z. 291 (2019), no. 1-2, 365–394. MR 3936074

    2019

  4. [7]

    Berman, S´ ebastien Boucksom, Vincent Guedj, a nd Ahmed Zeriahi, A variational approach to complex Monge-Amp` ere equations , Publ

    Robert J. Berman, S´ ebastien Boucksom, Vincent Guedj, a nd Ahmed Zeriahi, A variational approach to complex Monge-Amp` ere equations , Publ. Math. Inst. Hautes ´Etudes Sci. 117 (2013), 179–245. MR 3090260

    2013

  5. [8]

    Zbigniew B/suppress l ocki,Weak solutions to the complex Hessian equation , Ann. Inst. Fourier (Grenoble) 55 (2005), no. 5, 1735–1756. MR 2172278

    2005

  6. [9]

    205 (2010), no

    S´ ebastien Boucksom, Philippe Eyssidieux, Vincent Gue dj, and Ahmed Zeriahi, Monge-Amp` ere equations in big cohomology classes , Acta Math. 205 (2010), no. 2, 199–262. MR 2746347

    2010

  7. [10]

    Caffarelli, L

    L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second- order elliptic equations. III. Functions of the eigenvalue s of the Hessian , Acta Math. 155 (1985), no. 3-4, 261–301. MR 806416

    1985

  8. [12]

    Mohamad Charabati, Modulus of continuity of solutions to complex Hessian equa- tions, Internat. J. Math. 27 (2016), no. 1, 1650003, 24. MR 3454681 COMPLEX HESSIAN EQUATIONS WITH PRESCRIBED SINGULARITY 33

    2016

  9. [13]

    Pure Appl

    Kai-Seng Chou and Xu-Jia Wang, A variational theory of the Hessian equation , Comm. Pure Appl. Math. 54 (2001), no. 9, 1029–1064. MR 1835381

    2001

  10. [14]

    Collins and Sebastien Picard, The Dirichlet problem for the k-Hessian equation on a complex manifold , arXiv:1909.00447 (2019)

    Tristan C. Collins and Sebastien Picard, The Dirichlet problem for the k-Hessian equation on a complex manifold , arXiv:1909.00447 (2019)

    2019 arXiv

  11. [15]

    Tam´ as Darvas, The Mabuchi geometry of finite energy classes , Adv. Math. 285 (2015), 182–219. MR 3406499

    2015

  12. [16]

    Tam´ as Darvas,The Mabuchi completion of the space of K¨ ahler potentials , Amer. J. Math. 139 (2017), no. 5, 1275–1313. MR 3702499

    2017

  13. [17]

    Tam´ as Darvas,Metric geometry of normal K¨ ahler spaces, energy propernes s, and existence of canonical metrics , Int. Math. Res. Not. IMRN (2017), no. 22, 6752–6777. MR 3737320

    2017

  14. [18]

    Math., vol

    Tam´ as Darvas,Geometric pluripotential theory on K¨ ahler manifolds , Advances in complex geometry, Contemp. Math., vol. 735, Amer. Math. Soc ., Providence, RI, 2019, pp. 1–104. MR 3996485

    2019

  15. [22]

    Lu, On the singularity type of full mass currents in big cohomology classes , Compos

    Tam´ as Darvas, Eleonora Di Nezza, and Chinh H. Lu, On the singularity type of full mass currents in big cohomology classes , Compos. Math. 154 (2018), no. 2, 380–409. MR 3738831

    2018

  16. [23]

    Lu,The metric geometry of singu- larity types, arXiv:1909.00839 (2019)

    Tam´ as Darvas, Eleonora Di Nezza, and Chinh H. Lu,The metric geometry of singu- larity types, arXiv:1909.00839 (2019)

    2019 arXiv

  17. [24]

    Lu, Relative Pluripotential theory: a survey , Preprint (2020)

    Tam´ as Darvas, Eleonora Di Nezza, and Chinh H. Lu, Relative Pluripotential theory: a survey , Preprint (2020)

    2020

  18. [25]

    Rubinstein,Kiselman ’s principle, the Dirichlet problem for the Monge-Amp` ere equation, and rooftop obstacle probl ems, J

    Tam´ as Darvas and Yanir A. Rubinstein,Kiselman ’s principle, the Dirichlet problem for the Monge-Amp` ere equation, and rooftop obstacle probl ems, J. Math. Soc. Japan 68 (2016), no. 2, 773–796. MR 3488145

    2016

  19. [26]

    Lu, Complex Monge-Amp` ere equations on quasi- projective varieties, J

    Eleonora Di Nezza and Chinh H. Lu, Complex Monge-Amp` ere equations on quasi- projective varieties, J. Reine Angew. Math. 727 (2017), 145–167. MR 3652249

    2017

  20. [27]

    Elliptic Equ

    Nguyen Quang Dieu and Nguyen Thac Dung, Radial symmetric solution of complex Hessian equation in the unit ball , Complex Var. Elliptic Equ. 58 (2013), no. 9, 1261–

    2013

  21. [28]

    S/suppress lawomir Dinew,Uniqueness in E(X, ω ), J. Funct. Anal. 256 (2009), no. 7, 2113–

    2009

  22. [29]

    PDE 7 (2014), no

    S/suppress lawomir Dinew and S/suppress lawomir Ko/suppress lodziej,A priori estimates for complex Hessian equations, Anal. PDE 7 (2014), no. 1, 227–244. MR 3219505

    2014

  23. [30]

    S/suppress lawomir Dinew and S/suppress lawomir Ko/suppress lodziej,Liouville and Calabi-Yau type theorems for complex Hessian equations , Amer. J. Math. 139 (2017), no. 2, 403–415. MR 3636634

    2017

  24. [31]

    Notes Approx

    S/suppress lawomir Dinew and S/suppress lawomir Ko/suppress lodziej,Non standard properties of m-subharmonic functions, Dolomites Res. Notes Approx. 11 (2018), no. Special Issue Norm Leven- berg, 35–50. MR 3895934

    2018

  25. [32]

    Lu,Mixed Hessian inequalities and uniqueness in the class E(X, ω, m ), Math

    S/suppress lawomir Dinew and Chinh H. Lu,Mixed Hessian inequalities and uniqueness in the class E(X, ω, m ), Math. Z. 279 (2015), no. 3-4, 753–766. MR 3318249

    2015

  26. [33]

    Ayoub El-Gasmi, The Dirichlet problem for the complex Hessian operator in th e class Nm(H), arXiv:1712.06911

    arXiv

  27. [34]

    Pure Appl

    Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi, Viscosity solutions to degen- erate complex Monge-Amp` ere equations, Comm. Pure Appl. Math. 64 (2011), no. 8, 1059–1094. MR 2839271 34 CHINH H. LU AND V AN-DONG NGUYEN

    2011

  28. [35]

    Lars G ˚ arding,An inequality for hyperbolic polynomials , J. Math. Mech. 8 (1959), 957–965. MR 0113978

    1959

  29. [36]

    Dongwei Gu and Ngoc Cuong Nguyen, The Dirichlet problem for a complex Hessian equation on compact Hermitian manifolds with boundary , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 4, 1189–1248. MR 3829745

    2018

  30. [37]

    Lu, and Ahmed Zeriahi, Plurisubharmonic envelopes and supersolutions, arXiv:1703.05254, to appear in J

    Vincent Guedj, Chinh H. Lu, and Ahmed Zeriahi, Plurisubharmonic envelopes and supersolutions, arXiv:1703.05254, to appear in J. Differential Geom. (2017 )

    2017 arXiv

  31. [38]

    Vincent Guedj and Ahmed Zeriahi, The weighted Monge-Amp` ere energy of quasi- plurisubharmonic functions, J. Funct. Anal. 250 (2007), no. 2, 442–482. MR 2352488

    2007

  32. [40]

    Reese Harvey, H

    F. Reese Harvey, H. Blaine Lawson, Jr., and Szymon Pli´ s, Smooth approximation of plurisubharmonic functions on almost complex manifolds , Math. Ann. 366 (2016), no. 3-4, 929–940. MR 3563228

    2016

  33. [41]

    Zuoliang Hou, Complex Hessian equation on K¨ ahler manifold , Int. Math. Res. Not. IMRN (2009), no. 16, 3098–3111. MR 2533797

    2009

  34. [42]

    Zuoliang Hou, Xi-Nan Ma, and Damin Wu, A second order estimate for complex Hessian equations on a compact K¨ ahler manifold , Math. Res. Lett. 17 (2010), no. 3, 547–561. MR 2653687

    2010

  35. [43]

    Asma Jbilou, Complex Hessian equations on some compact K¨ ahler manifold s, Int. J. Math. Math. Sci. (2012), Art. ID 350183, 48. MR 3009566

    2012

  36. [44]

    V. N. Kokarev, Mixed volume forms and a complex equation of Monge-Amp` ere t ype on K¨ ahler manifolds of positive curvature, Izv. Ross. Akad. Nauk Ser. Mat. 74 (2010), no. 3, 65–78. MR 2682372

    2010

  37. [45]

    180 (1998), no

    S/suppress lawomir Ko/suppress lodziej,The complex Monge-Amp` ere equation , Acta Math. 180 (1998), no. 1, 69–117. MR 1618325

    1998

  38. [46]

    S/suppress lawomir Ko/suppress lodziej and Ngoc Cuong Nguyen,Weak solutions of complex Hessian equations on compact Hermitian manifolds , Compos. Math. 152 (2016), no. 11, 2221–

    2016

  39. [47]

    Mau-Hai Le and Van-Dung Trieu, Subextension of m-subharmonic functions, Vietnam Journal of Mathematics (2019)

    2019

  40. [48]

    Song-Ying Li, On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian , Asian J. Math. 8 (2004), no. 1, 87–106. MR 2128299

    2004

  41. [49]

    Lu, Solutions to degenerate complex Hessian equations , J

    Chinh H. Lu, Solutions to degenerate complex Hessian equations , J. Math. Pures Appl. (9) 100 (2013), no. 6, 785–805. MR 3125268

    2013

  42. [50]

    Lu, Viscosity solutions to complex Hessian equations , J

    Chinh H. Lu, Viscosity solutions to complex Hessian equations , J. Funct. Anal. 264 (2013), no. 6, 1355–1379. MR 3017267

    2013

  43. [51]

    Lu, A variational approach to complex Hessian equations in Cn, J

    Chinh H. Lu, A variational approach to complex Hessian equations in Cn, J. Math. Anal. Appl. 431 (2015), no. 1, 228–259. MR 3357584

    2015

  44. [52]

    Lu and Van-Dong Nguyen, Degenerate complex Hessian equations on compact K¨ ahler manifolds, Indiana Univ

    Chinh H. Lu and Van-Dong Nguyen, Degenerate complex Hessian equations on compact K¨ ahler manifolds, Indiana Univ. Math. J. 64 (2015), no. 6, 1721–1745. MR 3436233

    2015

  45. [53]

    Ngoc Cuong Nguyen, Subsolution theorem for the complex Hessian equation , Univ. Iagel. Acta Math. (2013), no. 50, [2012 on articles], 69–88. MR 3235004

    2013

  46. [54]

    41 (2014), no

    Ngoc Cuong Nguyen, H¨ older continuous solutions to complex Hessian equations, Po- tential Anal. 41 (2014), no. 3, 887–902. MR 3264825

    2014

  47. [55]

    Phong, Sebastien Picard, and Xiangwen Zhang, The Fu-Yau equation with negative slope parameter, Invent

    Duong H. Phong, Sebastien Picard, and Xiangwen Zhang, The Fu-Yau equation with negative slope parameter, Invent. Math. 209 (2017), no. 2, 541–576. MR 3674222

    2017

  48. [56]

    Phong, Sebastien Picard, and Xiangwen Zhang, Fu-Yau Hessian equations, arXiv:1801.09842 (2018)

    Duong H. Phong, Sebastien Picard, and Xiangwen Zhang, Fu-Yau Hessian equations, arXiv:1801.09842 (2018). COMPLEX HESSIAN EQUATIONS WITH PRESCRIBED SINGULARITY 35

    2018 arXiv

  49. [57]

    Phong, Sebastien Picard, and Xiangwen Zhang, On estimates for the Fu– Yau generalization of a Strominger system , J

    Duong H. Phong, Sebastien Picard, and Xiangwen Zhang, On estimates for the Fu– Yau generalization of a Strominger system , J. Reine Angew. Math. 751 (2019), 243–

    2019

  50. [58]

    Szymon Pli´ s,The smoothing of m-subharmonic functions , arXiv:1312.1906 (2013)

    2013 arXiv

  51. [59]

    Symplectic Geom

    Julius Ross and David Witt Nystr¨ om,Analytic test configurations and geodesic rays , J. Symplectic Geom. 12 (2014), no. 1, 125–169. MR 3194078

    2014

  52. [60]

    Sadullaev and B

    A. Sadullaev and B. Abdullaev, Potential theory in the class of m-subharmonic func- tions, Tr. Mat. Inst. Steklova 279 (2012), no. Analiticheskie i Geometricheskie Vo- prosy Kompleksnogo Analiza, 166–192. MR 3086763

    2012

  53. [61]

    Differential Geom

    G´ abor Sz´ ekelyhidi,Fully non-linear elliptic equations on compact Hermitian m ani- folds, J. Differential Geom. 109 (2018), no. 2, 337–378. MR 3807322

    2018

  54. [62]

    Trudinger, On the Dirichlet problem for Hessian equations , Acta Math

    Neil S. Trudinger, On the Dirichlet problem for Hessian equations , Acta Math. 175 (1995), no. 2, 151–164. MR 1368245

    1995

  55. [63]

    2, 579–591

    David Witt Nystr¨ om,Monotonicity of non-pluripolar Monge-Amp` ere masses, Indiana University Mathematics Journal 68 (2019), no. 2, 579–591

    2019

  56. [64]

    Mingchen Xia, Integration by parts formula for non-pluripolar product , arXiv:1907.06359 (2019)

    2019 arXiv

  57. [65]

    Jian Xiao, Hodge-index type inequalities, hyperbolic polynomials an d complex Hessian equations, arXiv:1810.04662 (2018)

    2018 arXiv

  58. [66]

    Shing Tung Yau, On the Ricci curvature of a compact K¨ ahler manifold and the c om- plex Monge-Amp` ere equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. MR 480350 Universit´e P aris Sud E-mail address : hoang-chinh.lu@u-psud.fr Ho Chi Minh city University of Educ...

    1978

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.