| 1. |
- Berndtsson, Bo, 1950, et al.
(författare)
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ALGEBRAIC FIBER SPACES and CURVATURE of HIGHER DIRECT IMAGES
- 2022
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Ingår i: Journal of the Institute of Mathematics of Jussieu. - 1474-7480 .- 1475-3030. ; 21:3, s. 973-1028
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Tidskriftsartikel (refereegranskat)abstract
- Let p : X → Y be an algebraic fiber space, and let L be a line bundle on. In this article, we obtain a curvature formula for the higher direct images of ΩiX/Y ⊗ L restricted to a suitable Zariski open subset of X. Our results are particularly meaningful if L is semi-negatively curved on X and strictly negative or trivial on smooth fibers of p. Several applications are obtained, including a new proof of a result by Viehweg-Zuo in the context of a canonically polarized family of maximal variation and its version for Calabi-Yau families. The main feature of our approach is that the general curvature formulas we obtain allow us to bypass the use of ramified covers - and the complications that are induced by them.
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| 2. |
- Berndtsson, Bo, 1950
(författare)
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Bergman kernels for Paley-Wiener spaces and Nazarov’s proof of the Bourgain-Milman theorem
- 2022
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Ingår i: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 18:2, s. 395-409
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Tidskriftsartikel (refereegranskat)abstract
- We give a general inequality for Bergman kernels of Bergman spaces defined by certain convex weights in Cn. We also discuss how this can be used in Nazarov’s proof of the BourgainMilman theorem, as a substitute for Hörmander’s estimates for the ¯∂-equation.
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| 3. |
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| 4. |
- Berndtsson, Bo, 1950, et al.
(författare)
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Complex interpolation of R-norms, duality and foliations
- 2020
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Ingår i: Journal of the European Mathematical Society. - : European Mathematical Society - EMS - Publishing House GmbH. - 1435-9855. ; 22:2, s. 477-505
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Tidskriftsartikel (refereegranskat)abstract
- The complex method of interpolation, going back to Calderon and Coifman et al., on the one hand, and the Alexander-Wermer-Slodkowski theorem on polynomial hulls with convex fibers, on the other hand, are generalized to a method of interpolation of real (finite-dimensional) Banach spaces and of convex functions. The underlying duality in this method is given by the Legendre transform. Our results can also be interpreted as new properties of solutions of the homogeneous complex Monge-Ampere equation.
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| 5. |
- Berndtsson, Bo, 1950, et al.
(författare)
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Complex interpolation of R-norms, duality and foliations
- 2020
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Ingår i: Journal of the European Mathematical Society. - 1435-9863 .- 1435-9855. ; 22:2, s. 477-505
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Tidskriftsartikel (refereegranskat)abstract
- The complex method of interpolation, going back to Calderón and Coifman et al., on the one hand, and the Alexander–Wermer–Słodkowski theorem on polynomial hulls with convex fibers, on the other hand, are generalized to a method of interpolation of real (finite-dimensional) Banach spaces and of convex functions. The underlying duality in this method is given by the Legendre transform. Our results can also be interpreted as new properties of solutions of the homogeneous complex Monge–Ampère equation.
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| 6. |
- Berndtsson, Bo, 1950, et al.
(författare)
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COMPLEX LEGENDRE DUALITY
- 2020
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Ingår i: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 142:1, s. 323-339
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Tidskriftsartikel (refereegranskat)abstract
- We introduce complex generalizations of the classical Legendre transform, operating on Kahler metrics on a compact complex manifold. These Legendre transforms give explicit local isometric symmetries for the Mabuchi metric on the space of Kaler metrics around any real analytic Kahler metric, answering a question originating in Semmes' work.
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| 7. |
- Berndtsson, Bo, 1950, et al.
(författare)
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L P-POLARITY, MAHLER VOLUMES, AND THE ISOTROPIC CONSTANT
- 2024
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Ingår i: Analysis and PDE. - 2157-5045 .- 1948-206X. ; 17:6, s. 2179-2245
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Tidskriftsartikel (refereegranskat)abstract
- This article introduces L p versions of the support function of a convex body K and associates to these canonical L p-polar bodies K degrees ;p and Mahler volumes M p (K). Classical polarity is then seen as L 1-polarity. This one-parameter generalization of polarity leads to a generalization of the Mahler conjectures, with a subtle advantage over the original conjecture: conjectural uniqueness of extremizers for each p 2 (0, 1 ). We settle the upper bound by demonstrating the existence and uniqueness of an L p-Santal & oacute; point and an L p-Santal & oacute; inequality for symmetric convex bodies. The proof uses Ball's Brunn- Minkowski inequality for harmonic means, the classical Brunn-Minkowski inequality, symmetrization, and a systematic study of the M p functionals. Using our results on the L p-Santal & oacute; point and a new observation motivated by complex geometry, we show how Bourgain's slicing conjecture can be reduced to lower bounds on the L p-Mahler volume coupled with a certain conjectural convexity property of the logarithm of the Monge-Amp & egrave;re measure of the L p-support function. We derive a suboptimal version of this convexity using Kobayashi's theorem on the Ricci curvature of Bergman metrics to illustrate this approach to slicing. Finally, we explain how Nazarov's complex-analytic approach to the classical Mahler conjecture is instead precisely an approach to the L 1-Mahler conjecture.
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| 8. |
- Berndtsson, Bo, 1950
(författare)
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Lelong Numbers and Vector Bundles
- 2020
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 30:3, s. 2361-2376
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Tidskriftsartikel (refereegranskat)abstract
- We study Lelong numbers and integrability indices for (Formula presented.)-invariant singular metrics on vector bundles over the disk.
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| 9. |
- Berndtsson, Bo, 1950
(författare)
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Long Geodesics in the Space of Kähler Metrics
- 2022
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Ingår i: Analysis Mathematica. - : Springer Science and Business Media LLC. - 0133-3852 .- 1588-273X. ; 48:2, s. 377-392
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Tidskriftsartikel (refereegranskat)abstract
- We give some remarks on geodesics in the space of Kähler metrics that are defined for all time. Such curves are conjecturally induced by holomorphic vector fields, and we show that this is indeed so for regular geodesics, whereas the question for generalized geodesics is still open (as far as we know). We also give a result about the derivative of such geodesics which implies a variant of a theorem of Atiyah and Guillemin-Sternberg on convexity of the image of certain moment maps.
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| 10. |
- Berndtsson, Bo, 1950
(författare)
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Plurisubharmonic functions and real submanifolds of ℂ n
- 2023
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Ingår i: Indagationes Mathematicae. - : Elsevier BV. - 0019-3577. ; 34:2, s. 357-366
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Tidskriftsartikel (refereegranskat)abstract
- We give an estimate for the volume of an analytic variety (or more generally the mass of a positive closed current) close to a real submanifold M. Applications are given to the Hausdorff measure of the intersection of the variety with M and the exponential integrability of plurisubharmonic functions on M.
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