| 1. |
- Andersson, Mats, 1957, et al.
(author)
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Non-pluripolar energy and the complex Monge-Ampere operator
- 2022
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In: Journal Fur Die Reine Und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 0075-4102 .- 1435-5345. ; 2022:792, s. 145-188
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Journal article (peer-reviewed)abstract
- Given a domain Omega subset of C-n we introduce a class of plurisubharmonic (psh) functions G(Omega) and Monge-Ampere operators u -> [dd(c)u](p), p <= n, on G(Omega) that extend the Bedford-Taylor-Demailly Monge-Ampere operators. Here [dd(c)u](p) is a closed positive current of bidegree (p, p) that dominates the non-pluripolar Monge-Ampere current < dd(c)u >(p). We prove that [dd(c)u](p) is the limit of Monge-Ampere currents of certain natural regularizations of u. On a compact Kahler manifold (X, omega) we introduce a notion of non-pluripolar energy and a corresponding finite energy class G(X, omega) subset of PSH(X, omega) that is a global version of the class G(Omega). From the local construction we get global Monge-Ampere currents [dd(c)phi + omega](p) for phi is an element of G(X, omega) that only depend on the current dd(c)phi + omega. The limits of Monge-Ampere currents of certain natural regularizations of phi can be expressed in terms of [dd(c)phi + omega](j) for j <= p. We get a mass formula involving the currents [dd(c)phi + omega](p) that describes the loss of mass of the non-pluripolar Monge-Ampere measure < dd(c)phi + omega >(n). The class G(X, omega) includes omega-psh functions with analytic singularities and the class E(X, omega) of omega-psh functions of finite energy and certain other convex energy classes, although it is not convex itself.
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| 2. |
- Berman, Robert, 1976, et al.
(author)
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Convergence of Bergman measures of high powers of a line bundle
- 2008
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Other publication (other academic/artistic)abstract
- Let L be a holomorphic line bundle on a compact complex manifold X of dimension n, and let exp(-\phi) be a continuous metric on L. Fixing a measure dμ on X gives a sequence of Hilbert spaces consisting of holomorphic sections of tensor powers of L. We prove that the corresponding sequence of scaled Bergman measures converges, in the high tensor power limit, to the equilibrium measure of the pair (K,\phi), where K is the support of dμ, as long as dμ is stably Bernstein-Markov with respect to (K,\phi). Here the Bergman measure denotes dμ times the restriction to the diagonal of the pointwise norm of the corresponding orthogonal projection operator. In particular, an extension to higher dimensions is obtained of results concerning random matrices and classical orthogonal polynomials.
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| 3. |
- Berman, Robert, 1976, et al.
(author)
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Fekete points and convergence towards equilibrium measures on complex manifolds
- 2011
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In: Acta Mathematica. - : International Press of Boston. - 1871-2509 .- 0001-5962. ; 207:1, s. 1-27
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Journal article (other academic/artistic)abstract
- Building on the first two authors' previous results, we prove a general criterion for convergence of (possibly singular) Bergman measures towards equilibrium measures on complex manifolds. The criterion may be formulated in terms of growth properties of balls of holomorphic sections, or equivalently as an asymptotic minimization of generalized Donaldson L-functionals. Our result yields in particular the proof of a well-known conjecture in pluripotential theory concerning the equidistribution of Fekete points, and it also gives the convergence of Bergman measures towards equilibrium for Bernstein-Markov measures. Applications to interpolation of holomorphic sections are also discussed.
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| 4. |
- Hultgren, Jakob, 1986, et al.
(author)
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Coupled Kähler-Einstein Metrics
- 2019
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In: International Mathematics Research Notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; 2019:21, s. 6765-6796
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Journal article (peer-reviewed)abstract
- We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler–Einstein metrics, generalizing Kähler–Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler–Einstein Fano. In the Fano case, we also prove that existence of coupled Kähler–Einstein metrics imply a certain algebraic stability condition, generalizing K-polystability.
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| 5. |
- Ross, J., et al.
(author)
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Analytic test configurations and geodesic rays
- 2014
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In: The Journal of Symplectic Geometry. - : International Press of Boston, Inc.. - 1527-5256. ; 12:1, s. 125-169
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Journal article (peer-reviewed)abstract
- Starting with the data of a curve of singularity types, we use the Legendre transform to construct weak geodesic rays in the space of locally bounded metrics on an ample line bundle L over a compact manifold. Using this we associate weak geodesics to suitable filtrations of the algebra of sections of L. In particular this works for the natural filtration coming from an algebraic test configuration, and we show how this recovers the weak geodesic ray of Phong-Sturm.
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| 6. |
- Ross, J., et al.
(author)
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Differentiability of the Argmin Function and a Minimum Principle for Semiconcave Subsolutions
- 2020
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In: Journal of Convex Analysis. - 0944-6532. ; 27:3, s. 811-832
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Journal article (peer-reviewed)abstract
- Suppose f(x, y) + k/2 parallel to x parallel to(2 )- sigma/2 parallel to y parallel to(2) is convex where kappa >= 0 sigma > 0, and the argmin function gamma(x) = {gamma : inf(y) f(x, y) = f(x, gamma)} exists and is single valued. We will prove gamma is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.
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| 7. |
- Ross, Julius, et al.
(author)
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Interpolation, Prekopa and Brunn-Minkowski for F-subharmonicity
- 2024
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In: Advances in Mathematics. - 1090-2082 .- 0001-8708. ; 436
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Journal article (peer-reviewed)abstract
- We extend Prekopa's Theorem and the Brunn-Minkowski Theorem from convexity to F-subharmonicity. We apply this to the interpolation problem of convex functions and convex sets introducing a new notion of “harmonic interpolation” that we view as a generalization of Minkowski-addition.
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| 8. |
- Ross, J., et al.
(author)
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Minimum Principle for Convex Subequations
- 2022
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 32:1
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Journal article (peer-reviewed)abstract
- A subequation, in the sense of Harvey-Lawson, on an open subset X subset of R-n is a subset F of the space of 2-jets on X with certain properties. A smooth function is said to be F-subharmonic if all of its 2-jets lie in F, and using the viscosity technique one can extend the notion of F-subharmonicity to any upper-semicontinuous function. Let P denote the subequation consisting of those 2-jets whose Hessian part is semipositive. We introduce a notion of product subequation F#P on X x R-m and prove, under suitable hypotheses, that if F is convex and f (x, y) is F#P-subharmonic then the marginal function g(x) := inf y f (x, y) is F-subharmonic. This generalises the classical statement that the marginal function of a convex function is again convex. We also prove a complex version of this result that generalises the Kiselman minimum principle for the marginal function of a plurisubharmonic function.
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| 9. |
- Ross, J., et al.
(author)
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ON THE MAXIMAL RANK PROBLEM FOR THE COMPLEX HOMOGENEOUS MONGE-AMPERE EQUATION
- 2019
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In: Analysis & Pde. - : Mathematical Sciences Publishers. - 1948-206X .- 2157-5045. ; 12:2, s. 493-504
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Journal article (peer-reviewed)abstract
- We give examples of regular boundary data for the Dirichlet problem for the complex homogeneous Monge-Ampere equation over the unit disc, whose solution is completely degenerate on a nonempty open set and thus fails to have maximal rank.
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| 10. |
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