# 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… Expand

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#### 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… Expand

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… Expand

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… Expand

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… Expand

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… Expand

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… Expand

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… Expand

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

- Mathematics
- Compositio 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… Expand

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… Expand

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… Expand