arXiv subjects

David Witt Nyström

Publications and source records attributed to David Witt Nyström.

21 records · Page 2Linked to original sources

arXiv2015

Okounkov bodies and the Kähler geometry of projective manifolds

Given a projective manifold $X$ equipped with an ample line bundle $L$, we show how to embed certain torus-invariant domains $D \subseteq\mathbb{C}^n$ into $X$ so that the Euclidean Kähler form on $D$ extends to a Kähler form on X lying in the first Chern class of $L$. This is done using Okounkov bodies $Δ(L)$, and the image of $D$ under the standard moment map will approximate $Δ(L)$. This means that the volume of $D$ can be made to approximate the Kähler volume of $X$ arbitrarily well. As a special case we can let $D$ be an ellipsoid. We also have similar results when $L$ is just big.

math.AG↗

arXiv2014

Harmonic Discs of Solutions to the Complex Homogeneous Monge-Ampère Equation

We study regularity properties of solutions to the Dirichlet problem for the complex Homogeneous Monge-Ampère equation. We show that for certain boundary data on $\mathbb P^1$ the solution $Φ$ to this Dirichlet problem is connected via a Legendre transform to an associated flow in the complex plane called the Hele-Shaw flow. Using this we determine precisely the harmonic discs associated to $Φ$. We then give examples for which these discs are not dense in the product, and also prove that this situation persists after small perturbations of the boundary data.

math.CV↗

arXiv2013

Transforming metrics on a line bundle to the Okounkov body

Let $L$ be a big holomorphic line bundle on a complex projective manifold $X.$ We show how to associate a convex function on the Okounkov body of $L$ to any continuous metric $ψ$ on $L.$ We will call this the Chebyshev transform of $ψ,$ denoted by $c[ψ].$ Our main theorem states that the difference of metric volume of $L$ with respect to two metrics, a notion introduced by Berman-Boucksom, is equal to the integral over the Okounkov body of the difference of the Chebyshev transforms of the metrics. When the metrics have positive curvature the metric volume coincides with the Monge-Ampère energy, which is a well-known functional in Kähler-Einstein geometry and Arakelov geometry. We show that this can be seen as a generalization of classical results on Chebyshev constants and the Legendre transform of invariant metrics on toric manifolds. As an application we prove the differentiability of the metric volume in the cone of big metrized $\mathbb{R}$-divisors. This generalizes the result of Boucksom-Favre-Jonsson on the differentiability of the ordinary volume of big $\mathbb{R}$-divisors and the result of Berman-Boucksom on the differentiability of the metric volume when the underlying line bundle is fixed.

math.CV↗