Bo BERNDTSSON, Chalmers University, Sweden
Title: Three Failed Projects
Abstract: I will describe three problems in the interface between convex and complex geometry, together with ideas of how to approach them. As the title indicates, these ideas have so far come to nothing, but hopefully they can have some interest nevertheless.
Zbigniew BŁOCKI, Jagiellonian University, Poland
Title: Some Estimates for the Real and Complex Monge-Ampère Operators
Abstract: We discuss some well-known estimates from the real and complex analysis, like Alexandrov-Bakelman-Pucci, Khovanskii-Teissier, and Alexandrov-Fenchel inequalities. In particular, we improve the first one applying the Bourgain-Milman inequality.
Sébastien BOUCKSOM, École Polytechnique, France
Title: Non-Archimedean Pluripotential Theory and K-Stability
Abstract: Non-Archimedean pluripotential theory studies plurisubharmonic (psh) functions and Monge--Ampère equations over spaces of valuations; in the trivially valued case, it provide a useful analytic tool to study K-stability. The goal of this talk is to introduce this circle of ideas, emphasizing the relations to convex geometry in the toric case.
Young-Jun CHOI, Pusan National University, Korea
Title: Curvature of Higher Direct Images
Abstract: Let f : X → S be a smooth family of compact K¨ahler manifolds and L is a hermitian line bundle on X. In 2009, Berndtsson proved that the direct image sheaf f∗(KX/S)(L) is Nakano (semi-)positive if E is (semi- )positive. He also computed the explicit curvature formula of f∗(KX/S)(L) in his sequent paper. Following his method, Mourougane and Takayama proved the positivity of Rqf∗(KX/S)(E) and Liu and Yang computed the curvature formula for f∗(KX/S)(E) where E is a hermitian vector bundle on X . Recently, Berndtsson, Paun and Wang computed the explicit curvature formula of Rn−pf∗(Ωp X/S)(L). On the other hand, in 2012 Schumacher proved the positivity of the relative canonical line bundle on X and computed the curvature formula for Rn−pf∗(Ωp X/S)(K⊗m X/S). After that, Naumann also computed the curvature formula for Rn−pf∗(Ωp X/S)(L). In this talk, we will discuss the curvature formula of Rqf∗Ω p X/S(E) and various special cases. This is a joint work with Georg Schumache.
Tamás DARVAS, University of Maryland, USA
Title: Existence of Twisted Kahler-Einstein Metrics in Big Classes
Abstract: We prove existence of twisted Kahler-Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when -K_X is big, we obtain a uniform Yau-Tian-Donaldson existence theorem for Kahler-Einstein metrics. To achieve this, we build up from scratch the theory of Fujita-Odaka type delta invariants in the transcendental big setting, using pluripotential theory. This is joint work with Kewei Zhang.
Jakob HULTGREN, Chalmers University, Sweden
Title: Collapsing Calabi-Yau Manifolds, Singular Affine Structures and Real Monge-Ampère Equations
Abstract: Recent developments in complex geometry have highlighted the importance of the real Monge-Ampère equation on singular affine manifolds. I will explain how this equation arises in topics concerning collapsing Calabi-Yau manifolds, in particular the Strominger-Yau-Zaslov conjecture about special Lagrangian torus fibrations and the Kontsevich-Soibelmann conjecture about Gromov-Hausdroff limits of degenerate families of Calabi-Yau manifolds. I will present joint work with Matttias Jonsson, Enrica Mazzon and Nick McCleerey showing that for symmetric data, the real Monge-Ampère equation on the unit simplex admits a unique Alexandrov solution and discuss applications.
Chenzi JIN, University of Maryland, USA
Title: Some Explicit Computations of Chebyshev Potentials
Abstract: The Chebyshev potential of a Kähler potential on a projective variety, introduced by Witt Nystrom, is a convex function defined on the Okounkov body. It is a generalization of the symplectic potential of a torus-invariant Kahler potential on a toric variety, introduced by Guillemin, that is a convex function on the Delzant polytope. A folklore conjecture asserts that a curve of Chebyshev potentials associated to a curve in the space of Kähler potentials is linear in the time variable if and only if the latter curve is a geodesic in the Mabuchi metric. This is classically true in the special toric setting, and in general Witt Nyström established the sufficiency. In joint work with Y. Rubinstein we disprove this conjecture. More generally, we characterize the Fubini--Study geodesics for which the conjecture is true in the special case of projective space. The proof involves explicitly solving the Monge-Ampere equation describing geodesics on the subspace of Fubini-Study metrics and computing their Chebyshev potentials.
Mattias JONSSON, University of Michigan, USA
Title: Flavors of Convexity and the Monge-Ampère Equation
Abstract: A focus of this conference is the strong interplay between convex and complex geometry. This interplay has a counterpart in the non-Archimedean world. I will survey some of this landscape, with a focus on the Monge--Ampère equation. The talk will in part be based on joint work with Boucksom and Favre, and with Hultgren, Mazzon, and McCleerey.
Bo'az KLARTAG, Weizmann Institute of Science, Israel
Title: Convexity in High Dimension
Abstract: We will survey results and open problems regarding the distribution of volume in high-dimensional convex sets, discussing in particular Bourgain's slicing problem and the Kannan-Lovasz-Simonovits isoperimetric conjecture. We will try to make the presentation accessible to complex geometers, especially in view of the recent application of complex methods to convexity by Berndtsson, yielding a simpler proof of the best known bound for the symmetric Mahler conjecture due to Kuperberg.
Alexander KOLESNIKOV, HSE University, Russia
Title: Hessian Metrics in Convex Analysis
We discuss some recent developments in convex analysis connecting real Hessian metrics, optimal transportation and Minkowski-type inequalities.
Laszlo LEMPERT, Purdue University, USA
Title: Mapping Spaces and Holomorphic Functions
Abstract: Given a compact Hausdorff space, the space of its continuous maps to a fixed complex manifold has the natural structure of an infinite dimensional complex manifold. We will discuss two theorems on holorphic functions on such mapping spaces, one reminiscent of the Monodromy theorem, the other of Liouville's theorem
Vlassis MASTRANTONIS, University of Maryland, USA
Title: The Bergman kernel approach to the Bourgain—Milman inequality and L^p Mahler functionals
Abstract: We start by discussing Nazarov’s complex analytic proof of the Bourgain—Milman inequality and its adaptation to the non-symmetric setting. We continue with the observation that the Bergman kernels of tube domains for convex bodies resemble an L^1 version of the Mahler volume, which leads to a definition of L^p Mahler volumes, and we explore various properties of these volumes. This is joint work with Y. Rubinstein.
Mihai PĂUN, Universitat Bayreuth, , Germany
Title: Extension of Pluricanonical Forms
Abstract: We will present a revised and augmented version of a joint work with J. Cao. The main results concern infinitesimal extension of pluricanonical sections as well as some applications.
Rémi REBOULET, Chalmers University, Sweden
Title: Ding Stability for Manifolds with Big Anticanonical Class
Abstract: This is joint work with Ruadhai Dervan. I will present a notion of test configuration and (uniform) Dig stability for complex manifolds with big anticanonical class, i.e. when the dimension of the space of sections of powers -K_X has maximal growth. I will explain how this notion of stability is related to the existence of weak Kahler-Einstein metrics.
Julius ROSS, University of Illinois, USA
Title: General Brunn-Minkowski and Prekopa Theorems
Abstract: Will discuss a generalization of the Brunn-Minkowski Theorem and Prekopa's Theorem from convexity to the framework of $F$-subharmonicity. This is joint work with David Witt Nystrom.
Dror VAROLIN, Stony Brook University, USA
Title: BLS Fields
Abstract: Bo Berndtsson proved that for a proper holomorphic submersion $p:X \to B$ and a (semi-) positive line bundle $L \to X$ the direct image $p_*K_{X/B}\otimes L) \to B$ is (semi-) positive in the sense of Nakano. Berndtsson proved his theorem by a direct computation of the curvature. We recently gave another proof by embedding this vector bundle in an ambient object that is not a holomorphic vector bundle, but is similar in many respects. This category of geometric objects is called a BLS field. Computation of the curvature using BLS fields gives a different geometric perspective on Berndtsson’s curvature formula, and an infinitesimal version of the theory extends the computation of the curvature formula to the setting in which the line bundle $L \to X$ is not necessarily semi-positive. I will summarize this theory, the curvature computation, and some related work and questions.
Xu WANG, Norwegian University of Science and Technology, Norway
Title: A Hilbert Bundle Approach to Complex Brunn-Minkowski Theory
Abstract: This is a joint work with Tai Nguyen. Based on a recent regularity theorem of Berndtsson, we shall introduce the Hilbert bundle approach to the complex Brunn-Minkowski theory, which avoids the use of the Hamilton-Kohn regularity theory. We shall also discuss the applications to Ohsawa-Takegoshi theory and singularity theory of plurisubharmonic functions based on a generalized Berndtsson-Lempert method.
David WITT NYSTRÖM, Chalmers University, Sweden
Title: Harmonic Interpolation of Convex Sets
Abstract: Using Minkowski addition there is a natural way to interpolate between to two convex sets A_0 and A_1 in R^n, namely by letting A_t:=(1-t)A_0 + tA_1. The fundamental Brunn-Minkowski thorem then says that vol(A_t)^{1/n} is concave. Now let A_x be a family of convex sets parametrized by the boundary of a domain U in R^m. I will describe a natural generalization of the interpolation described above, giving a family of convex sets parametrized by U. We call this the harmonic interpolation as it uses Minkowski integrals with respect to harmonic measures. I will show how one can prove a Brunn-Minkowski type theorem, namely that vol(A_x)^{1/n} is superharmonic on U. I will also explain how this relates to one of Berndtsson's early complex Brunn-Minkowski results. This is based on joint work with Julius Ross.
Mingchen XIA, Chalmers University, Sweden
Title: Partial Okounkov Bodies of Hermitian Pseudo-Effective Line Bundles
Abstract: Given a big line bundle L on a projective manifold, Lazarsfeld–Mustată and Kaveh–Khovanskii introduced method of constructing convex bodies associated with L. These convex bodies are known as Okounkov bodies. When L is endowed with a singular positive Hermitian metric h, we will explain how to construct smaller convex bodies from the datum (L,h).