| 1. |
- Berman, Robert, 1976, et al.
(författare)
-
Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics
- 2017
-
Ingår i: Journal of the American Mathematical Society. - : American Mathematical Society (AMS). - 0894-0347 .- 1088-6834. ; 30:4, s. 1165-1196
-
Tidskriftsartikel (refereegranskat)abstract
- We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kähler potentials on a compact Kähler manifold, thus confirming a conjecture of Chen, and give some applications in Kähler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogeneous Monge-Ampère equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.
|
|
| 2. |
- Berndtsson, Bo, 1950
(författare)
-
A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry
- 2015
-
Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 200:1, s. 149-200
-
Tidskriftsartikel (refereegranskat)abstract
- For ϕ a metric on the anticanonical bundle, −KX , of a Fano manifold X we consider the volume of X ∫Xe−ϕ. In earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on −KX . Our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on X , even with very low regularity assumptions on the geodesic. As a consequence we get a simplified proof of the Bando–Mabuchi uniqueness theorem for Kähler–Einstein metrics. A generalization of this theorem to ‘twisted’ Kähler–Einstein metrics and some classes of manifolds that satisfy weaker hypotheses than being Fano is also given. We moreover discuss a generalization of the main result to other bundles than −KX , and finally use the same method to give a new proof of the theorem of Tian and Zhu on uniqueness of Kähler–Ricci solitons.
|
|
| 3. |
- Berndtsson, Bo, 1950
(författare)
-
A comparison principle for bergman kernels
- 2017
-
Ingår i: Trends in Mathematics. - Cham : Springer. - 2297-0215 .- 2297-024X. - 9783319524719 - 9783319524696 ; , s. 121-126
-
Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- © 2017 Springer International Publishing. We give a version of the comparison principle from pluripotential theory where the Monge–Ampère measure is replaced by the Bergman kernel and use it to derive a maximum principle.
|
|
| 4. |
- Berndtsson, Bo, 1950, et al.
(författare)
-
A proof of the Ohsawa-Takegoshi theorem with sharp estimates
- 2016
-
Ingår i: Journal of the Mathematical Society of Japan. - : Mathematical Society of Japan (Project Euclid). - 0025-5645 .- 1881-1167. ; 68:4, s. 1461-1472
-
Tidskriftsartikel (refereegranskat)abstract
- We give a proof of the Ohsawa-Takegoshi extension theorem with sharp estimates. The proof is based on ideas of Blocki to use variations of domains to simplify his proof of the Suita conjecture, and also uses positivity properties of direct image bundles.
|
|
| 5. |
- Berndtsson, Bo, 1950
(författare)
-
Complex Brunn-Minkowski theory and positivity of vector bundles
- 2018
-
Ingår i: Proceedings of the International Congress of Mathematicians, ICM 2018. ; 2, s. 877-902
-
Konferensbidrag (refereegranskat)abstract
- This is a survey of results on positivity of vector bundles, inspired by the BrunnMinkowski and Preoókopa theorems. Applications to complex analysis, Kähler geometry and algebraic geometry are also discussed.
|
|
| 6. |
- Berndtsson, Bo, 1950
(författare)
-
Probability measures associated to geodesics in the space of kähler metrics
- 2018
-
Ingår i: Springer Proceedings in Mathematics and Statistics. - Cham : Springer International Publishing. - 2194-1017 .- 2194-1009. ; 269, s. 395-419
-
Konferensbidrag (refereegranskat)abstract
- We associate certain probability measures on R to geodesics in the space HL of positively curved metrics on a line bundle L, and to geodesics in the finite dimensional symmetric space of hermitian norms on H0(X, kL). We prove that the measures associated to the finite dimensional spaces converge weakly to the measures related to geodesics in HL as k goes to infinity. The convergence of second order moments implies a recent result of Chen and Sun on geodesic distances in the respective spaces, while the convergence of first order moments gives convergence of Donaldson’s Z-functional to the Aubin–Yau energy. We also include a result on approximation of infinite dimensional geodesics by Bergman kernels which generalizes work of Phong and Sturm.
|
|
| 7. |
- Berndtsson, Bo, 1950
(författare)
-
Superforms, supercurrents, minimal manifolds and Riemannian geometry
- 2019
-
Ingår i: Arnold Mathematical Journal. - : Springer Science and Business Media LLC. - 2199-6792 .- 2199-6806. ; 5:4, s. 501-532
-
Tidskriftsartikel (refereegranskat)abstract
- Supercurrents, as introduced by Lagerberg, were mainly motivated as a way to study tropical varieties. Here we will associate a supercurrent to any smooth submanifold of Rn. Positive supercurrents resemble positive currents in complex analysis, but depend on a choice of scalar product on Rn and reflect the induced Riemannian structure on the submanifold. In this way we can use techniques from complex analysis to study real submanifolds. We illustrate the idea by giving area estimates of minimal manifolds and a monotonicity property of the mean curvature flow. We also use the formalism to give a relatively short proof of Weyl’s tube formula.
|
|
| 8. |
|
|
| 9. |
- Berndtsson, Bo, 1950, et al.
(författare)
-
The volume of Kahler-Einstein varieties and convex bodies
- 2017
-
Ingår i: Journal für die Reine und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 0075-4102 .- 1435-5345. ; :723, s. 127-152
-
Tidskriftsartikel (refereegranskat)abstract
- We show that the complex projective space Pn has maximal degree (volume) among all n-dimensional Kähler–Einstein Fano manifolds admitting a non-trivial holomorphic C∗-action with a finite number of fixed points. The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart’s volume conjecture for a large class of rational polytopes, including duals of lattice polytopes. The case of spherical varieties/multiplicity free symplectic manifolds is also discussed. The proof uses Moser–Trudinger type inequalities for Stein domains and also leads to criticality results for mean field type equations in Cn of independent interest. The paper supersedes our previous preprint [5] concerning the case of toric Fano manifolds.
|
|
