Wall-Chamber Decompositions for Generalized Monge-Ampère Equations

Sohaib Khalid; Zakarias Sjöström Dyrefelt

doi:10.48550/arXiv.2412.20089

Wall-Chamber Decompositions for Generalized Monge-Ampère Equations

Sohaib Khalid, Zakarias Sjöström Dyrefelt

Aarhus Institute of Advanced Studies

Research output: Working paper/Preprint › Preprint

Abstract

Generalized Monge-Ampère equations form a large class of PDE including Donaldson's J-equation, inverse Hessian equations, some supercritical deformed Hermitian-Yang Mills equations, and some Z-critical equations. Solvability of these equations is characterized by numerical criteria involving intersection numbers over all subvarieties, and in this paper, we aim to characterize algebraically what happens when these nonlinear Nakai-Moishezon type criteria fail. As a main result, we show that under mild positivity assumptions, there is a finite number of subvarieties violating the Nakai type criterion, and such subvarieties are moreover rigid in a suitable sense. This gives first effective solvability criteria for these familes of PDE, thus improving on work of Gao Chen, Datar-Pingali, Song and Fang-Ma. As an application, we obtain first examples in higher dimension of polarized compact Kähler manifolds exhibiting a natural PDE analog of Bridgeland's locally finite wall-chamber decomposition.

Original language	English
DOIs	https://doi.org/10.48550/arXiv.2412.20089
Publication status	Published - 28 Dec 2024

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@misc{367684877c04466b9a675fa5ff0d44b9,

title = "Wall-Chamber Decompositions for Generalized Monge-Amp{\`e}re Equations",

abstract = "Generalized Monge-Amp{\`e}re equations form a large class of PDE including Donaldson's J-equation, inverse Hessian equations, some supercritical deformed Hermitian-Yang Mills equations, and some Z-critical equations. Solvability of these equations is characterized by numerical criteria involving intersection numbers over all subvarieties, and in this paper, we aim to characterize algebraically what happens when these nonlinear Nakai-Moishezon type criteria fail. As a main result, we show that under mild positivity assumptions, there is a finite number of subvarieties violating the Nakai type criterion, and such subvarieties are moreover rigid in a suitable sense. This gives first effective solvability criteria for these familes of PDE, thus improving on work of Gao Chen, Datar-Pingali, Song and Fang-Ma. As an application, we obtain first examples in higher dimension of polarized compact K{\"a}hler manifolds exhibiting a natural PDE analog of Bridgeland's locally finite wall-chamber decomposition.",

author = "Sohaib Khalid and {Sj{\"o}str{\"o}m Dyrefelt}, Zakarias",

year = "2024",

month = dec,

day = "28",

doi = "10.48550/arXiv.2412.20089",

language = "English",

type = "WorkingPaper",

}

TY - UNPB

T1 - Wall-Chamber Decompositions for Generalized Monge-Ampère Equations

AU - Khalid, Sohaib

AU - Sjöström Dyrefelt, Zakarias

PY - 2024/12/28

Y1 - 2024/12/28

N2 - Generalized Monge-Ampère equations form a large class of PDE including Donaldson's J-equation, inverse Hessian equations, some supercritical deformed Hermitian-Yang Mills equations, and some Z-critical equations. Solvability of these equations is characterized by numerical criteria involving intersection numbers over all subvarieties, and in this paper, we aim to characterize algebraically what happens when these nonlinear Nakai-Moishezon type criteria fail. As a main result, we show that under mild positivity assumptions, there is a finite number of subvarieties violating the Nakai type criterion, and such subvarieties are moreover rigid in a suitable sense. This gives first effective solvability criteria for these familes of PDE, thus improving on work of Gao Chen, Datar-Pingali, Song and Fang-Ma. As an application, we obtain first examples in higher dimension of polarized compact Kähler manifolds exhibiting a natural PDE analog of Bridgeland's locally finite wall-chamber decomposition.

AB - Generalized Monge-Ampère equations form a large class of PDE including Donaldson's J-equation, inverse Hessian equations, some supercritical deformed Hermitian-Yang Mills equations, and some Z-critical equations. Solvability of these equations is characterized by numerical criteria involving intersection numbers over all subvarieties, and in this paper, we aim to characterize algebraically what happens when these nonlinear Nakai-Moishezon type criteria fail. As a main result, we show that under mild positivity assumptions, there is a finite number of subvarieties violating the Nakai type criterion, and such subvarieties are moreover rigid in a suitable sense. This gives first effective solvability criteria for these familes of PDE, thus improving on work of Gao Chen, Datar-Pingali, Song and Fang-Ma. As an application, we obtain first examples in higher dimension of polarized compact Kähler manifolds exhibiting a natural PDE analog of Bridgeland's locally finite wall-chamber decomposition.

U2 - 10.48550/arXiv.2412.20089

DO - 10.48550/arXiv.2412.20089

M3 - Preprint

BT - Wall-Chamber Decompositions for Generalized Monge-Ampère Equations

ER -

Wall-Chamber Decompositions for Generalized Monge-Ampère Equations

Abstract

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