Singular Kähler-Einstein metrics

By Philippe Eyssidieux, Vincent Guedj, Ahmed Zeriahi. 2006

428 citations recorded by the source corpus.

We study degenerate complex Monge-Ampere equations of the form $(\omega+dd^c\f)^n = e^{t \f}\mu$ where $\omega$ is a big semi-Kahler form on a compact Kahler manifold $X$ of dimension $n$, $t \in \R^+$, and $\mu=f\omega^n$ is a positive measure with density $f\in L^p(X,\omega^n)$, $p>1$. We prove the existence and unicity of continuous $\o$-plurisubharmonic solutions. In case $X$ is projective and $\omega=\psi^*\omega'$, where $\psi:X\to V$ is a proper birational morphism to a normal projective variety, $[\omega']\in NS_{\R} (V)$ is an ample class and $\mu$ has only algebraic singularities, we prove that the solution is smooth in the regular locus of the equation. We use these results to construct singular Kahler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e6fb04bdc04dc6e7521be73826fce19710b4aaed

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Smooth Convergence Away from Singular Sets (2012, background): “[6], Huang [11], Ruan and Zhong [16], Sesum [17], Tian and Tosatti [20, 21] particularly in the setting of Kahler–Einstein manifolds.”
Compact Moduli Spaces of Del Pezzo Surfaces and K\"ahler-Einstein metrics (2012, methodology): “Finally we remark that the points in the boundary Mh\M h should correspond to Q-Fano varieties, admitting weak Kähler-Einstein metrics in the sense of pluripotential theory [21].”
Kähler currents and null loci (2013, background): “In this case there is a birational contraction map π : X → Y to a singular Calabi–Yau variety, which is an isomorphism away fromNull(α), and such thatω0 is the pullback of a singular Kähler–Einsteinmetric onY [25].”
Limits of Calabi–Yau metrics when the Kähler class degenerates (2007, background): “;0 that depends on the vanishing orders of the gi. Then (4.2) Z D F1+εωn= Z D FεΩ <∞. DEGENERATIONS 13 The compactness of Xgives F∈ L1+ε(ωn), and so we can apply Theorem 2.1 and Proposition 3.1 of [EGZ] (which rely on the seminal work of Ko lodziej [Ko l]) to get a unique bounded ϕ∈ PSH(X,ω) such”