# L^1 metric geometry of big cohomology classes

@article{Darvas2018L1MG, title={L^1 metric geometry of big cohomology classes}, author={Tam'as Darvas and Eleonora Di Nezza and Chinh H. Lu}, journal={arXiv: Differential Geometry}, year={2018} }

Suppose $(X,\omega)$ is a compact K\"ahler manifold of dimension $n$, and $\theta$ is closed $(1,1)$-form representing a big cohomology class. We introduce a metric $d_1$ on the finite energy space $\mathcal{E}^1(X,\theta)$, making it a complete geodesic metric space. This construction is potentially more rigid compared to its analog from the K\"ahler case, as it only relies on pluripotential theory, with no reference to infinite dimensional $L^1$ Finsler geometry. Lastly, by adapting the…

## 22 Citations

Lp Metric Geometry of Big and Nef Cohomology Classes

- MathematicsActa Mathematica Vietnamica
- 2019

Let $(X,\omega)$ be a compact Kahler manifold of dimension $n$, and $\theta$ 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…

Mabuchi geometry of big cohomology classes with prescribed singularities

- Mathematics
- 2019

Let $X$ be a compact Kahler manifold. Fix a big class $\alpha$ with smooth representative $\theta$ and a model potential $\phi$ with positive mass. We study the space $\mathcal{E}^p(X,\theta;[\phi])$…

The metric geometry of singularity types

- Mathematics
- 2019

Let $X$ be a compact Kahler manifold. Given a big cohomology class $\{\theta\}$, there is a natural equivalence relation on the space of $\theta$-psh functions giving rise to $\mathcal S(X,\theta)$,…

On sharp lower bounds for Calabi-type functionals and destabilizing properties of gradient flows

- MathematicsAnalysis & PDE
- 2021

Let $X$ be a compact K\"ahler manifold with a given ample line bundle $L$. In \cite{Don05}, Donaldson proved that the Calabi energy of a K\"ahler metric in $c_1(L)$ is bounded from below by the…

$L^{1}$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds

- Mathematics
- 2019

Given $(X,\omega)$ compact Kahler manifold and $\psi\in\mathcal{M}^{+}\subset PSH(X,\omega)$ a model type envelope with non-zero mass, i.e. a fixed potential determing some singularities such that…

The closures of test configurations and algebraic singularity types

- Mathematics
- 2020

Given a Kahler manifold $X$ with an ample line bundle $L$, we consider the metric space of $L^1$ geodesic rays associated to the first Chern class $c_1(L)$. We characterize rays that can be…

Complex Hessian equations with prescribed singularity on compact Kähler manifolds

- Mathematics
- 2019

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…

S ep 2 01 9 L 1 metric geometry of potentials with prescribed singularities on compact Kähler manifolds

- 2019

Given (X,ω) compact Kähler manifold and ψ ∈ M ⊂ PSH(X,ω) a model type envelope with non-zero mass, i.e. a fixed potential determing some singularities such that ∫ X (ω+ddψ) > 0, we prove that the…

Comparison of Monge-Amp\`ere capacities

- Mathematics
- 2020

Since Yau’s solution to Calabi’s conjecture [29] geometric pluripotential theory has found its important place in the development of differential geometry. An important tool in the theory is the…

A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations

- Mathematics
- 2021

For any polarized variety (X,L), we show that test configurations and, more generally, R-test configurations (defined as finitely generated filtrations of the section ring) can be analyzed in terms…

## References

SHOWING 1-10 OF 66 REFERENCES

The Mabuchi completion of the space of Kähler potentials

- Mathematics
- 2014

Suppose $(X,\omega)$ is a compact K\"ahler manifold. Following Mabuchi, the space of smooth K\"ahler potentials ${\cal H}$ can be endowed with a Riemannian structure, which induces an infinite…

Degenerate complex Hessian equations on compact K\"ahler manifolds

- Mathematics
- 2014

Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix $m\in \mathbb{N}$ such that $1\leq m \leq n$. We prove that any $(\omega,m)$-sh function can be approximated from above by…

Geodesic Rays and K\"ahler-Ricci Trajectories on Fano Manifolds

- Mathematics
- 2014

Suppose $(X,J,\omega)$ is a Fano manifold and $t \to r_t$ is a diverging Kahler-Ricci trajectory. We construct a bounded geodesic ray $t \to u_t$ weakly asymptotic to $t \to r_t$, along which Ding's…

On the constant scalar curvature Kähler metrics (II)—Existence results

- Mathematics
- 2018

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature K\"ahler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As…

Regularity of weak minimizers of the K-energy and applications to properness and K-stability

- Mathematics
- 2016

Let $(X,\omega)$ be a compact K\"ahler manifold and $\mathcal H$ the space of K\"ahler metrics cohomologous to $\omega$. If a cscK metric exists in $\mathcal H$, we show that all finite energy…

The metric completion of the Riemannian space of K\"{a}hler metrics

- Mathematics
- 2014

Let $X$ be a compact K\"ahler manifold and $\a \in H^{1,1}(X,\R)$ a K\"ahler class. We study the metric completion of the space $\HH_\a$ of K\"ahler metrics in $\a$, when endowed with the Mabuchi…

A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry

- Mathematics
- 2013

For $$\phi $$ϕ a metric on the anticanonical bundle, $$-K_X$$-KX, of a Fano manifold $$X$$X we consider the volume of $$X$$X$$\begin{aligned} \int _X e^{-\phi }. \end{aligned}$$∫Xe-ϕ.In earlier…

Geometry and topology of the space of Kähler metrics on singular varieties

- MathematicsCompositio Mathematica
- 2018

Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space…

Metric geometry of normal K\"ahler spaces, energy properness, and existence of canonical metrics

- Mathematics
- 2016

Let $(X,\omega)$ be a compact normal K\"ahler space, with Hodge metric $\omega$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical…

From Monge–Ampère equations to envelopes and geodesic rays in the zero temperature limit

- Mathematics, Physics
- 2013

Let $$(X,\theta )$$(X,θ) be a compact complex manifold X equipped with a smooth (but not necessarily positive) closed (1, 1)-form $$\theta .$$θ. By a well-known envelope construction this data…