arXiv subjects
Publications and source records attributed to Quang-Tuan Dang.
In this paper, we establish a stability estimate and a uniform estimate for the modulus of continuity of solutions to the degenerate complex Monge-Ampère equation in big cohomology classes. Consequently, we prove the uniqueness of solutions to complex Monge-Ampère mean field equations for a sufficiently small parameter.
We prove uniform a priori estimates for solutions to degenerate complex Monge--Ampère equations in big cohomology classes, using both auxiliary-function technique developed by Guo, Phong and Tong [On $L^\infty$-estimates for complex Monge-Ampère equations, Ann. of Math. (2) 198 (2023), no.1, 393-418], and quasi-psh envelope approach developed by Guedj and Lu [Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, J. Eur. Math. Soc. (JEMS) 27 (2025), no. 3, 1185-1208.]. As an application, we apply our method to prove the Moser-Trudinger and Brezis-Merle-type inequalities for complex Monge-Ampère equations.
We establish a transcendental generalization of Nakamaye's theorem to compact complex manifolds when the form is not assumed to be closed. We apply the recent analytic technique developed by Collins and Tosatti to show that the non-Hermitian locus of a nef and big $(1,1)$-form, which is not necessarily closed, on a compact complex manifold equals the union of all positive-dimensional analytic subvarieties where the restriction of the form is not big (null locus). As an application, we can give an alternative proof of the Nakai--Moishezon criterion of Buchdahl and Lamari for complex surfaces and generalize this result in higher dimensions Finally, we investigate finite time non-collapsing singularities of the Chern--Ricci flow, partially answering a question raised by Tosatti and Weinkove.
In this paper, we prove a uniform estimate for the modulus of continuity of solutions to degenerate complex Monge--Ampère equation in big cohomology classes. This improves the previous results of Di Nezza--Lu and of the first author.
We study the stability and Hölder continuity of solutions to degenerate complex Monge--Ampère equations associated with a (non-closed) big form on compact Hermitian manifolds. We also show that the solution is globally continuous when the reference form is the pullback of a Hermitian metric. As a consequence, we establish a uniform diameter bound for the twisted Chern--Ricci flow.
We extend Tsuji's iterative construction of complete Kähler--Einstein metrics with negative scalar curvature to noncompact Kähler manifolds with bounded geometry, using Berndtsson's method from the compact setting. Consequently, given a holomorphic surjective map $p:X\to Y$, where $X$ is a weakly pseudoconvex Kähler manifold and $Y$ is a complex manifold, and where the smooth fibers admit Kähler--Einstein metrics with negative scalar curvature and bounded geometry, we show that the fiberwise Kähler--Einstein metrics induce a semipositively curved metric on the relative canonical bundle $K_{X/Y}$. Moreover, our approach also applies to the plurisubharmonic variation of cusp Kähler--Einstein metrics.
Let $X$ be a compact Kähler manifold and $D$ be a simple normal crossing divisor on $X$ such that $K_X+D$ is big and nef. We first prove that the singular Kähler--Einstein metric constructed by Berman--Guenancia is almost-complete on $X \backslash D$ in the sense of Tian--Yau. In our second main result, we establish the weak convergence of conic Kähler--Einstein metrics of negative curvature to the above-mentioned metric when $K_X+D$ is merely big, answering partly a recent question posed by Biquard--Guenancia. Potentials of low energy play an important role in our approach.
We study the nature of finite-time singularities for the Chern-Ricci flow, partially answering a question of Tosatti-Weinkove. We show that a solution of degenerate parabolic complex Monge-Ampère equations starting from arbitrarily positive (1,1)-currents are smooth outside some analytic subset, generalizing works by Di Nezza-Lu. We extend Guedj-Lu's recent approach to establish uniform a priori estimates for degenerate complex Monge-Ampère equations on compact Hermitian manifolds. We apply it to studying the Chern-Ricci flows on complex log terminal varieties starting from an arbitrary current.
The aim of this paper is to compare singularities of closed positive currents whose non-pluripolar complex Monge--Ampère masses equal. We also provide a short alternative proof for the monotonicity of non-pluripolar complex Monge--Ampère masses, generalizing results of Witt-Nyström, Darvas--Di Nezza--Lu, Lu--Nguyên and Vu.
We study the continuity of solutions to complex Monge-Ampere equations with prescribed singularities. This generalizes the previous results of DiNezza-Lu and the author. As an application, we can run the Monge-Ampere flow starting at a current with prescribed singularities.
We study pluripotential complex Monge-Ampère flows in big cohomology classes on compact K{ä}hler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential K{ä}hler-Ricci flows on compact K{ä}hler manifolds of general type as well as on stable varieties.
Extending a recent theory developed on compact Kähler manifolds by Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions to degenerate parabolic complex Monge-Ampère equations on compact Hermitian manifolds. Under natural assumptions on the Cauchy boundary data, we show that the pluripotential solution is semi-concave in time and continuous in space and that such a solution is unique. We also establish a partial regularity of such solutions under some extra assumptions of the densities and apply it to prove the existence and uniqueness of the weak Chern-Ricci flow on complex compact varieties with log terminal singularities.
Extending DiNezza-Lu's approach to the setting of big cohomology classes, we prove that solutions of degenerate complex Monge-Amp{è}re equations on compact K{ä}hler manifolds are continuous on a Zariski open set. This allows us to show that singular K{ä}hler-Einstein metrics on log canonical varieties of general type have continuous potentials on the ample locus outside of the non-klt part.