Valentino Tosatti, Ben Weinkove
We prove the existence of plurisubharmonic functions with prescribed logarithmic singularities on complex 3-folds equipped with a nef class of positive volume. We prove the same result for rational classes on Moishezon n-folds.
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Let $(X,ω)$ be a compact Kähler manifold of dimension $n$, and $θ$ be a closed smooth real $(1,1)$-form representing a big and nef cohomology class. We introduce a metric $d_p, p\geq 1$, on the finite energy space $\mathcal{E}^p(X,θ)$, making it a complete geodesic metric space.
The goal of this work is to prove the regularity of certain quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a compact complex manifold. We prove that the complex Hessian forms of these envelopes are locally bounded outside an analytic set of singularities. It is furthermore shown that a parametrized version of this result yields a priori inequalities for the solution of the Dirichlet problem for a degenerate Monge-Ampere operator; applications to geodesics in the space o…
The Kobayashi-Hitchin correspondence for nef and big classes
通过引入适应闭正(1,1)-流和适应Hermitian-Yang-Mills度量的概念，建立了nef和大上同调类的Kobayashi-Hitchin对应，并应用于Bogomolov-Gieseker不等式和Miyaoka-Yau不等式的等式情形。
A note on Teissier problem for nef classes
本文解决了紧致 Kähler 流形上仅假定为 nef 的 $(1,1)$-类 Khovanskii-Teissier 型不等式取等号的 Teissier 问题，并将结果推广到部分类非 nef 的情形，且通过例子证明结果最优。
We prove the qualitative part of Demailly's conjecture on transcendental Morse inequalities for differences of two nef classes satisfying a numerical relative positivity condition on an arbitrary compact Kähler (and even more general) manifold. The result improves on an earlier one by J. Xiao whose constant $4n$ featuring in the hypothesis is now replaced by the optimal and natural $n$. Our method follows arguments by Chiose as subsequently used by Xiao up to the point where we introduce a new way of handling the estimates in a certain Monge-Ampère equation. This result is needed to extend to …
Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their "dual" objects, the plurisubharmonic functions. These functions are defined classically by requiring that the restriction to each pseudo-holomorphic curve is subharmonic. In this paper subharmonic functions are defined by applying the viscosity approach to a version of the complex hessian which exists intrinsically on any almost complex manifold. Three theorems are proved. The first is a restriction theorem which establishes the equivalence of our definition with the "classical" definition. I…
Let $\{α\}$ and $\{β\}$ be nef cohomology classes of bidegree $(1,\,1)$ on a compact $n$-dimensional Kähler manifold $X$ such that the difference of intersection numbers $\{α\}^n - n\,\{α\}^{n-1}.\,\{β\}$ is positive. We solve in a number of special but rather inclusive cases the quantitative part of Demailly's Transcendental Morse Inequalities Conjecture for this context predicting the lower bound $\{α\}^n-n\,\{α\}^{n-1}.\,\{β\}$ for the volume of the difference class $\{α-β\}$. We completely solved the qualitative part in an earlier work. We also give general lower bounds for the volume of $…
On $L^\infty$ estimates for Monge-Ampère and Hessian equations on nef classes
本研究将前三位作者此前发展的Kähler流形上完全非线性方程$L^\infty$估计的PDE方法推广至nef类情形，给出了Monge-Ampère方程已有估计的新证明，并将估计拓展到Hessian方程。
Geodesic connectivity and rooftop envelopes in the Cegrell classes
本文在Cegrell类中证明复Monge-Ampère方程可比奇异性解的唯一性，解决Cegrell提出的未决问题，并借此推广测地连通性判据及屋顶包络的幂等性条件。
In this note we revisit and extend few classical and recent results on the definition and use of the Futaki invariant in connection with the existence problem for Kaehler constant scalar curvature metrics on polarized algebraic manifolds, especially in the case of resolution of singularities. The general inspiration behind this work is no doubt the beautiful 1992 paper by Ding and Tian which contains the germs of a huge amount of the successive developments in this fundamental problem, and it is a great pleasure to dedicate this to Professor G. Tian on the occasion of his birthday!
The canonical structures of the limit of the Yang-Mills flows for nef and big classes
本文延续此前关于nef且大的上同调类的Kobayashi-Hitchin对应研究，证明紧Kähler流形上全纯向量丛中对应这类上同调类的$T$-Yang-Mills流全局可解且在Uhlenbeck极限下收敛，并刻画了极限联络在丰沛轨迹上的复规范等价结构。
Rigid and Nonrigid Isotropic Nef Classes on K3 Surfaces
该论文在K3曲面上建立了闭正电流刚性与Ricci-flat度量塌缩之间的等价关系，并首次给出非刚性无理nef类例子，同时构造了超Kähler流形上多个新刚性类。
We pose a conjecture about Morse-type integrals in nef (1,1) classes on compact Hermitian manifolds, and we show that it holds for semipositive classes, or when the manifold admits certain special Hermitian metrics.
Let $X$ be a compact Kähler unibranch complex analytic space of pure dimension. Fix a big class $α$ with smooth representative $θ$ and a model potential $ϕ$ with positive mass. We define and the study non-pluripolar products of quasi-plurisubharmonic functions on $X$. We study the spaces $\mathcal{E}^p(X,θ;[ϕ])$ of finite energy Kähler potentials with prescribed singularities for each $p\geq 1$. We define a metric $d_p$ and show that $(\mathcal{E}^p(X,θ;[ϕ]),d_p)$ is a complete metric space. This construction generalizes the usual $d_p$-metric defined for an ample class.
Uniform estimates for complex Monge-Ampère equations: big cohomology classes
该研究针对大上同调类中的退化复Monge-Ampère方程，结合两种技术证明了一致先验估计，并将其应用于推导两类相关不等式，丰富了复几何中偏微分方程的理论结果。
We consider three fundamental classes of compact almost homogeneous manifolds and show that the complements of singular complex orbits in such manifolds are endowed with plurisubharmonic exhaustions satisfying complex homogeneous Monge-Ampère equations. This extends to a new family of mixed type examples various classical results on parabolic spaces and complexifications of symmetric spaces. Rigidity results on complex spaces modeled on such new examples are given.
We prove a geodesic distance formula for quasi-psh functions with finite entropy, extending results by Chen and Darvas. We work with big and nef cohomology classes: a key result we establish is the convexity of the $K$-energy in this general setting. We then study Monge-Ampère measures on contact sets, generalizing a recent result by the first author and Trapani.
Transcendental b-divisors II -- Monotonicity theorem
本文发展了 nef b-除子的一般交理论，扩展了可移动交理论，定义了 b-除子的限制体积，并证明了交积的单调性定量版本，从而得出新的当前体和上同调类体积不等式。
Degenerate quarternionic Monge-Ampère equations in weighted energy classes
本文研究了退化四元数Monge-Ampère方程在加权能量类中的性质，证明了算子的良定义性、Cegrell型类的精细性质及质量集中定理，并刻画了算子的有限能量范围。
Let $X$ be a compact Kähler manifold and $\om$ a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal E}_χ(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $χ$ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class ${\mathcal E}_χ(X,\om)$. This is done by establishing a priori estimates on the capacity of su…
In this paper we extend the notion of Futaki invariant to big and nef classes in such a way that it defines a continuous function on the \K\ cone up to the boundary. We apply this concept to prove that reduced normal crossing singularities are sufficient to check $K$-semistability. A similar improvement on Donaldson's lower bound for Calabi energy is given.
Weak limits of the $J$-flow and the deformed Hermitian-Yang-Mills flow on Kähler surfaces: boundary cases
研究证明了在Kähler类$J$- nef时，$J$-流在Kähler表面收敛于Monge-Ampère方程的弱解，同时建立了变形辛-Yang-Mills流的相同收敛行为，方法基于粘性子解极限的性质。
We prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local $C^{1,1}$ regularity of geodesic rays in the space of Kähler metrics associated to a test configuration, as well as the local $C^{1,1}$ regularity of quasi-psh envelopes in nef and big classes away from the non-Kähler locus.
We introduce two notions of hyperbolicity for not necessarily Kähler $n$-dimensional compact complex manifolds $X$. The first, called {\it balanced hyperbolicity}, generalises Gromov's Kähler hyperbolicity by means of Gauduchon's balanced metrics. The second, called {\it divisorial hyperbolicity}, generalises the Brody hyperbolicity by ruling out the existence of non-degenerate holomorphic maps from $\C^{n-1}$ to $X$ that have what we term a subexponential growth. Our main result in the first part of the paper asserts that every balanced hyperbolic $X$ is also divisorially hyperbolic. We provi…
Complex Monge-Ampère equation for positive $(p,p)$-forms on compact Kähler manifolds
该研究在紧Kähler流形上引入微分$(p,p)$-形式的复Monge-Ampère方程，证明了$1 \leq p < n$时光滑解的存在唯一性（可加常数），并定义了保持上同调类的高阶形式几何流，推广了Kähler-Ricci流且证明其收敛性。
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