标题： 墙腔分解用于广义蒙日-安培方程 显示英文标题

标题： Wall-chamber decompositions for generalized Monge-Ampère equations

摘要： 广义Monge-Ampère方程形成一个包含Donaldson的J方程、逆Hessian方程、一些超临界变形的Hermitian-Yang Mills方程和一些Z-临界方程的大型PDE类。这些方程的可解性由涉及所有子簇上的交数的数值准则来表征，在本文中，我们旨在代数上表征当这些非线性Nakai-Moishezon型准则失效时发生的情况。作为主要结果，我们在适度的正性假设下，证明了违反Nakai型准则的子簇数量是有限的，并且这些子簇在某种适当的意义上是刚性的。这为这些PDE族提供了第一个有效的可解性准则，从而改进了Gao Chen、Datar-Pingali、Song和Fang-Ma的工作。作为应用，我们得到了更高维的极化紧致Kähler流形的第一个例子，这些流形表现出Bridgeland的局部有限墙-房间分解的自然PDE类似物。 显示英文摘要

摘要： Generalized Monge-Amp\`ere 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\"ahler manifolds exhibiting a natural PDE analog of Bridgeland's locally finite wall-chamber decomposition.