Seminars 2014:

December 4, 2014, 10:00

Gerard Freixas (Jussieu, Paris) : Reciprocity laws on Riemann surfaces, connections on Deligne pairings and holomorphic torsio.

Abstract:

In this talk I will recall classical reciprocity laws on Riemann surfaces and explain how they translate into the language of connections on Deligne pairings. I will give a couple of applications. One explains a construction of Hitchin on hyperkähler varieties of lambda-connections, the other explains a result of Fay on holomorphic extensions of analytic torsion on spaces of characters of the fundamental group of a Riemann surface. The contents of this talk are based on joint work with Richard Wentworth (University of Maryland).

December 3, 2014, 10:00 Eleonora Di Nezza (Imperial College London):Regularizing properties and uniqueness of

the Kaehler-Ricci flow Abstract: Let X be a compact Kaehler manifold. I will show that the Kaehler-Ricci flow,

as well as its twisted version, can be run from an arbitrary positive closed current,

and that it is immediately smooth in a Zariski open subset of X. Moreover, if the initial

data has positive Lelong number we indeed have propagation of singularities for short time.

Finally, I will prove a uniqueness result in the case of zero Lelong numbers. (This is a joint work with Chinh Lu)

November 26, 2014, 10:00

Chinh H. Lu (Göteborg): The completion of the space of Kähler metrics.

Abstract: We will talk about recent results of T. Darvas concerning the completion of the space of Kähler metrics

Chinh H. Lu (Göteborg): The completion of the space of Kähler metrics.

Abstract: We will talk about recent results of T. Darvas concerning the completion of the space of Kähler metrics

Ivan Cheltsov (Edinburgh): Cylinders in del Pezzo surfaces. Abstract: For an ample divisor H on a variety V, an H-polar cylinder in V is an open ruled affine subset whose complement is a support of an effective Q-divisor that is Q-rationally equivalent to H. In the case when V is a Fano variety and H is its anticanonical divisor, this notion links together affine, birational and Kahler geometries. I prove existence and non-existence of H-polar cylinders in smooth and mildly singular (with at most du Val singularities) del Pezzo surfaces for different ample divisors H. In particular, I will answer an old question of Zaidenberg and Flenner about additive group actions on the cubic Fermat affine threefold cone. This is a joint work with Park and Won.

May 21, 2014, 10:00

Bo Berndtsson: Yet another proof of the Ohsawa-Takegoshi theorem.

Abstract: A very simple proof of optimal lower bounds for the Bergman kernel was given recently by Blocki and Lempert. I will show how their method generalizes to at least one version of the Ohsawa-Takegoshi theorem.

Abstract: A very simple proof of optimal lower bounds for the Bergman kernel was given recently by Blocki and Lempert. I will show how their method generalizes to at least one version of the Ohsawa-Takegoshi theorem.

May 15, 2014, 10:00

Robert Berman (Göteborg) : A primer on toric manifolds – from the analytic point of view

Abstract: In this talk I will give an elementary and “hands-on” introduction to toric manifolds, from an analytic point of view. More precisely, I will explain how to build a polarized compact complex manifold X from a given polytope P (satisfying the Delzant conditions). Topologically, the manifold X is fibered over the polytope P in real tori, in much the same way as the Riemann sphere may be fibered over the unit-interval by embedding it in R^3 and projecting onto a coordinate axis.

April 30, 2014, 10:00

Chinh H. Lu (Göteborg): Complex Hessian equations on compact Kähler manifolds.

Abstract: The complex Hessian equation is an interpolation between the Laplace and complex Monge-Ampère equation. An analogous version of the Calabi-Yau equation has been recently solved by Dinew and Kolodziej.

In this talk we first survey some known results about existence and regularity of solutions in the non-degenerate case. We then discuss how to solve the degenerate equation by a variational approach due to Berman-Boucksom-Guedj-Zeriahi. The key (and new) point is a regularization result for m-subharmonic functions. This is joint work with Van-Dong Nguyen (Ho Chi Minh city University of Pedagogy).

Abstract: The complex Hessian equation is an interpolation between the Laplace and complex Monge-Ampère equation. An analogous version of the Calabi-Yau equation has been recently solved by Dinew and Kolodziej.

In this talk we first survey some known results about existence and regularity of solutions in the non-degenerate case. We then discuss how to solve the degenerate equation by a variational approach due to Berman-Boucksom-Guedj-Zeriahi. The key (and new) point is a regularization result for m-subharmonic functions. This is joint work with Van-Dong Nguyen (Ho Chi Minh city University of Pedagogy).

April 23, 2014, 10:00

**Yanir Rubinstein (**University of Maryland): Logzooet - beyond elephants

Abstract: This is the second talk in the series. It should be of interest to differential/algebraic/complex geometers

The compact four-manifolds that admit a Kähler metric with positive Ricci curvature have been classified in the 19th century: they come in 10 families. In analogy with conical Riemann surfaces (e.g., football, teardrop) and hyperbolic 3-folds with a cone singularity along a link appearing in Thurston's program, one may consider 4-folds with a Kähler metric having "edge singularities", namely admitting a 2-dimensional cone singularity transverse to an immersed minimal surface, a `complex edge'. What are all the pairs (4-fold, immersed surface) that admit a Kähler metric with positive Ricci curvature away from the edge? In joint work with I. Cheltsov (Edinburgh) we classify all such pairs under some assumptions. These now come in infinitely-many families and we then pose the "Calabi problem" for these pairs: when do they admit Kähler-Einstein edge metrics? This problem is far from being solved, even in this low dimension, but we report on some initial progress: some understanding of the non-existence part of the conjecture, as well as several existence results.

April 9, 2014, 10:00

**Yanir Rubinstein (**University of Maryland): Logzooet

Abstract: This is the first talk out of two and will require little prerequisites.

The compact four-manifolds that admit a Kähler metric with positive Ricci curvature have been classified in the 19th century: they come in 10 families. In analogy with conical Riemann surfaces (e.g., football, teardrop) and hyperbolic 3-folds with a cone singularity along a link appearing in Thurston's program, one may consider 4-folds with a Kähler metric having "edge singularities", namely admitting a 2-dimensional cone singularity transverse to an immersed minimal surface, a `complex edge'. What are all the pairs (4-fold, immersed surface) that admit a Kähler metric with positive Ricci curvature away from the edge? In joint work with I. Cheltsov (Edinburgh) we classify all such pairs under some assumptions. These now come in infinitely-many families and we then pose the "Calabi problem" for these pairs: when do they admit Kähler-Einstein edge metrics? This problem is far from being solved, even in this low dimension, but we report on some initial progress: some understanding of the non-existence part of the conjecture, as well as several existence results.

April 2, 2014, 10:00

**Choi Young-Jun (**KIAS): Variations of Kähler-Einstein metrics on strongly pseudoconvex domainsAbstract: By a celebrated theorem of Cheng and Yau, every bounded strongly pseudo-convex domain with smooth boundary admits a unique complete Kähler-Einstein metric. In this talk, we discuss the plurisubharmonicity of variations of the Kähler- Einstein metrics of strongly pseudoconvex domains.

March 19, 2014, 10.00

**Nikolay Shcherbina**(Wuppertal University): Bounded plurisubharmonic functions, cores and Liouville type property.

Abstract. For a domain G in a complex manifold M of dimension n let F be the family of all bounded above plurisubharmonic functions. Define a notion of the core c(G) of G as:

c(G) = {z \in G: rank L(z,f) < n for all functions f \in F},

where L(z,f) is the Levi form of f at z. The main purpose of the talk is to discuss a Liouville type property of c(G), namely, to investigate if it is true or not that every function g from F has to be a constant on each connected component of c(G).

March 12, 2014, 10.00

**Robert Berman**(Göteborg): The convexity of the K-energy on the space of all Kähler metrics and applications

Abstract: The K-energy is a functional on the space H of all Kähler metrics in a given cohomology class and was introduced by Mabuchi in the 80's. Its critical points are Kähler metrics with constant scalar curvature. As shown by Mabuchi the K-energy is convex along smooth geodesics in the space H. This was result was later put into a the framework of geometric invariant theory in infinite dimensions, by Donaldson. However, when studying the geometry of H one is, in general, forced to work with a weaker notion of geodesics introduced by Chen (due to lack of a higher order regularity theory for the PDE describing the corresponding geodesic equation). In this talk, which is based on a joint work with Bo Berndtsson, it will be shown that the K-energy remains convex along weak geodesics, thus confirming a conjecture by Chen. Some applications in Kähler geometry will also be briefly discussed.

February 26, 2014 9.30

**Bo Berndtsson**(Göteborg): Complex interpolation of real norms

Abstract: The method of complex interpolation is a way to, given a family of complex Banach norms, find intermediate or averaged norms. I will describe an extension of this to real norms and also mention the relation to the boundary value problem for geodesics in the space of metrics on a line bundle. This is joint work with Bo'az Klartag, Dario Cordero and Yanir Rubinstein.

February 12, 2014, 10.00

**Xu Wang**(Tongji University, Shanghai): Variation of the Bergman kernels of pseudoconvex domains.

Abstract: We shall give a variational formula of the full Bergman kernels associated to smoothly bounded strongly pseudoconvex domains. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.

January 27, 2014, 10.00

**Håkan Samuelsson Kalm**(Göteborg): On analytic structure in maximal ideal spacesAbstract: John Wermer's classical maximality theorem says the following: Let $f$ be a continuous function on the unit circle $b\Delta$, where $\Delta \subset \mathbb{C}$ is the unit disk. Then, either $f$ is the boundary values of a holomorphic function on $\Delta$ or the uniform algebra generated by $z$ and $f$ on the unit circle equals the algebra of all continuous functions on the unit circle. I will discuss to what extent Wermer's maximality theorem extends to the setting of several complex variables. In particular, we will answer a question posed by Lee Stout concerning the presence of analytic structure for a uniform algebra whose maximal ideal space is a manifold. This is joint work with A. Izzo (Bowling Green) and E. F. Wold (Oslo).

**Seminars 2007, 2008, 2009, 2010, 2011, 2012 2013:**See attached pdf below.

**Links to lists of previous seminars:**

- Seminars 2005/2006
- Seminars 2004/2005
- Seminars 2003/2004
- Seminars 2002/2003
- Seminars 2001/2002
- Seminars 2000/2001
- Seminars 1999/2000
- Seminars 1998/1999
- Seminars 1997/1998
- Seminars 1996/1997
- Seminars 1995/1996
- Seminars 1994/1995
- Seminars 1993/1994
- Seminars 1992/1993
- Seminars 1991/1992
- Seminars 1990/1991