Seminars from 2015, 2016, 2017, and 2018.

#### 2018

**2018-11-29: Håkan Hedenmalm** (KTH)*Off-spectral analysis of Bergman kernels*

**Abstract:** In the context of Kähler geometry in one complex variable, we have a weight e^{-mQ} where m is a parameter which tends to infinity. In the Kähler setting, the potential Q has positive Laplacian (curvature), but now we allow Q to have negative Laplacian somewhere. Potential analysis then gives us a contact set (spectrum) as where the maximal subharmonic minorant function reaches up to Q, and the off-spectral region is where it cannot. This split is analogous to the analytic continuation issue for functions of several complex variables: we would say that the part of the boundary where any holomorphic function extends across is off-spectral, the rest is spectral.

It is well-known how to obtain the Bergman kernel expansion in the bulk of the spectrum. If time permits, I will explain how a flow of domains around the expansion point may be introduced, which is intimately connected with the expansion and may be used to show that the expansion holds. The main result is however an expansion of the Bergman kernel based at an off-spectral point. The proof involves the construction of a flow which is analogous to the flow for bulk points. Morally speaking, this is like expanding the Bergman kernel of several variables at a boundary point where all functions extend holomorphically across.

This reports on joint work with A. Wennman. (pdf-file of the presentation)

**2018-11-22: Ragnar Sigurðsson** (University of Iceland)*Monge-Ampère measures of plurisubharmonic exhaustions associated to the Lie norm of holomorphic maps*

**Abstract:** This is a joint work with Auðunn Skúta Snæbjarnarson, arXiv:1810.01326v2. We look at psh functions of the form $\log|\Phi|_c$ where $\Phi$ is a holomorphic map on a complex manifold $X$ of dimension $n$ with values in ${\mathbb C}^{n+1}\setminus \{0\}$ and $|\cdot|_c$ is the Lie norm on ${\mathbb C}^{n+1}$, i.e., the largest complex norm which extends the euclidean norm from ${\mathbb R}^{n+1}$. Every such function is maximal on $\Phi^{-1}({\mathbb C}^{n+1}\setminus {\mathbb C}{\mathbb R}^{n+1})$, so the support of the complex Monge-Ampère measure $(dd^c(\log|\Phi|_c$))^n$ is contained in $\Phi^{-1}({\mathbb C}{\mathbb R}^{n+1})$. We derive an explicit formula for the Monge-Ampère measure, which hopefully is of interest for the study of $S$-parabolic Stein manifolds.

**2018-11-08: Takayuki Koike** (Osaka City University, Japan)*Gluing construction of non-projective K3 surfaces and holomorphic tubular neighbourhoods of elliptic curves*

**Abstract:** In this talk, we construct K3 surfaces by gluing two rational surfaces given by blowing-up the projective plane at "general" nine points. For such K3 surfaces, one can concretely calculate the period maps. By observing the result of this calculation, one can show that such K3 surfaces constitute a large family which includes non-projective K3 surfaces as general elements. This gluing construction is based on Arnold's theorem on the existence of nice neighbourhoods of an elliptic curve embedded in a surface whose normal bundle satisfies Diophantine-type condition in the Picard variety. This is a joint work in progress with T. Uehara.

**2018-10-22: Ya Deng** (Göteborg)*Hyperbolicity of moduli spaces of higher dimensional projective manifolds*

**Abstract:** In ICM 1962 Shafarevich conjectured that the base quasi-projective curve of any smooth, non-isotrivial family of projective curves with genus >1 is hyperbolic, which was proved by Parshin and Arakelov. The higher dimensional Shafarevich hyperbolicity conjecture (SHC) can be formulated as follows: let Y be the quasi-projective base of any smooth family of polarized minimal manifolds (i.e. the canonical bundles are semi-ample) with maximal variation (i.e. the Kodaira-Spencer map is generically injective). Then Y is of log general type (algebraic version) , and Y is pseudo Kobayashi hyperbolic (analytic version). The algebraic SHC was proved by Campana-Păun in 2015, combining previous work by Viehweg-Zuo. In this talk, I will present my recent work on the proof of analytic SHC. If time allows, I will briefly explain another work (jointly with Dan Abramovich) on the Kobayashi hyperbolicity of moduli spaces of minimal general type manifolds.

**2018-09-24: Dennis Eriksson** (Göteborg)*An invariant for Calabi-Yau manifolds through analytic torsion*

**Abstract:** String theorists have predicted the existence of an invariant constructed through analytic torsion for Calabi-Yau manifolds, which should, unlike the analytic torsions themselves, only depend on the complex structure. The construction should, by mirror symmetry principles, be a birational invariant and count genus 1 curves on a Calabi-Yau 'mirror'.

The construction of the invariant was done by Fang-Lu-Yoshikawa in dimension 3, and in this talk, we generalize the construction to higher dimensions. A key feature is that we can control, topologically, the asymptotic behaviour of the invariant along 1-parameter degenerations.

This is joint work with Gerard Freixas and Christophe Mourougane.

**2018-09-11: Lucas Kaufmann** (National University of Singapore)*Dynamics of correspondences on Riemann Surfaces*

**Abstract:** Let X be a compact Riemann surface. A correspondence f on X is a multi-valued map from X to itself. Each point of X has d images and d’ pre-images counting multiplicity.

As in the case of maps we can iterate f and study its dynamics. When d and d’ are different the global dynamics of f is well understood and f admits a canonical invariant measure with many good properties.

In this talk I intend to present some results concerning the case d = d’. Surprisingly, under a mild and necessary condition, f admits two canonical measures that share the role of the global description of the system. As an application, we can consider the action of a finitely generated subgroup of PSL(2,C) on P^1 and recover some classical results about random products of matrices.

This is joint work with Tien Cuong Dinh and Hao Wu.

**2018-05-29: Richard Lärkäng** (Göteborg)*Residue currents and the Euler characteristic of a complex of vector bundles*

**Abstract:** I will describe a factorization of the fundamental cycle of a bounded generically exact complex of vector bundles in terms of certain differential forms and residue currents associated to this complex. This is a generalization of previous results in the case when the complex is a locally free resolution of the structure sheaf of an analytic space, which in turn is a generalization of the Poincaré-Lelong formula. This is joint work with Elizabeth Wulcan.

**2018-04-25: Xueyuan Wan** (Göteborg)*The second variation of analytic torsion on Teichmüller space and curvature asymptotics*

**Abstract:** Given a holomorphic family of Riemann surfaces, and a relative ample line bundle over the total space such that its curvature gives a metric with constant scalar curvature on each fiber. We will compute the curvature asymptotics of L^2 metric and Quillen metric on direct image bundles. By comparing, we obtain the second variation of analytic torsion on Teichmüller space.

**2018-03-28: David Witt Nyström** (Göteborg)*Deforming a Kähler manifold to the normal cone of a subvariety*

**Abstract:** A classical construction in algebraic/complex geometry, known as degeneration to the normal cone, allows you to degenerate a variety to the normal cone of a subvariety. In this talk I will discuss what happens when you add a Kähler structure to this picture.

**2018-03-08: Zakarias Sjöström Dyrefelt** (Göteborg)*Introduction to the Yau-Tian-Donaldson conjecture for Kähler manifolds*

**Abstract:** A central open problem in Kähler geometry is the Yau-Tian-Donaldson conjecture, which predicts that existence of constant scalar curvature Kähler (cscK) metrics on a given Kähler manifold is equivalent to a certain algebro-geometric obstruction, called K-stability. This obstruction was originally introduced to characterize existence of Kähler-Einstein metrics on Fano manifolds, and was later extended to study existence of cscK metrics on polarized manifolds. In this talk I will explain how to extend the notion of K-stability further, in order to give a coherent theory for arbitrary Kähler manifolds (not necessarily admitting any ample line bundles). More precisely, I will discuss K-stability and the YTD conjecture from the perspective of pluripotential theory, and explain how this allows to simultaneously simplify and generalize more classical approaches due to Donaldson, Mabuchi, Stoppa, Berman, Boucksom-Hisamoto-Jonsson, and others. If time permits I will explain that all cscK Kähler manifolds are (generalized) K-stable, thus proving one direction of the Yau-Tian-Donaldson conjecture for Kähler manifolds.

**2018-02-22: Xu Wang** (NTNU Trondheim)*A remark on the Alexandrov-Fenchel inequality*

**Abstract:** We shall first recall a few notions in convex geometry: Minkowski sum, Legendre transform, gradient map, mixed volume, etc. Our main result is a complex-geometric proof of the Alexandrov-Fenchel inequality without using toric compactifications. The idea is to use the Legendre transform and develop the Brascamp-Lieb proof of the Pr\'ekopa theorem. New ingredients in our proof include an integration of Timorin's mixed Hodge-Riemann bilinear relation and a mixed norm version of H\"ormander's $L^2$-estimate, which also implies a non-compact version of the Khovanski\u{i}-Teissier inequality.

**2018-02-15: Bo Berndtsson** (Göteborg)*Supercurrents and minimal manifolds *

**Abstract:** Originally positive closed supercurrents were constructed as objects associated to tropical varieties, much the same way as ordinary closed positive currents are associated to complex subvarieties. We will show how one can also associate a supercurrent to any smooth submanifold of R^n. The construction depends on the choice of a scalar product on R^n, and the current reflects the induced Riemannian structure on the submanifold. This permits us to build a calculus on the submanifold analogous to the Kahler formalism on complex manifolds. The supercurrents that we define are not closed (except when the submanifold is linear), but it turns out the currents associated to minimal submanifolds satisfy another condition which is almost as strong. From there one can deduce e g area estimates for minimal surfaces long the lines of the classical results of Lelong and others.

**2018-02-08: Magdalena Larfors** (Uppsala Universitet)*Geometry and moduli of heterotic G2 systems*

**Abstract:** A heterotic G2 system is a quadruple [(Y,φ),(V,A),(TY,Θ),H ] consisting of a 7-dimensional real manifold Y with G2 structure φ, a vector bundle V over Y with connection A, a connection Θ on the tangent bundle TY of Y, and a 3-form H. These objects form a heterotic system if φ is conformally co-closed, A and Θ are G2 instanton connection, and H satisfies the so-called anomaly cancellation condition. The simplest instance of a heterotic G2 system has H=0; in this case the system reduces to a G2 holonomy manifold with instanton connections.

In this talk, I will discuss the geometry and moduli of heterotic G2 systems, with a particular focus on the G2 holonomy case. Defining a bundle Q on Y which is topologically Q=T∗Y ⊕End(TY)⊕End(V), I will show that the geometry of a heterotic G2 system is equivalent to a differential D acting on Q-valued forms which satisfies a certain nilpotency condition. Moreover, the infinitesimal moduli of the heterotic structure correspond to classes in an associated cohomology. This mimics Atiyah’s discussion of the geometry and moduli of holomorphic bundles over complex manifolds. The analogy is particularly striking for the G2 holonomy case, where the bundle Q is an Atiyah extension bundle.

This talk is based on joint work with Xenia de la Ossa and Eirik Svanes, see preprints 1709.06974, 1704.08717 and 1607.03473.

**2018-02-01: Antonio Trusiani** (Göteborg/Rome)*Multipoint Okounkov Bodies*

**Abstract:** Starting from the data of a big line bundle L on a projective manifold X with a choice of N different points on X, I'll give a new construction of N Okounkov bodies that encodes important geometric features of (L \to X; p_1,...,p_N) such as the volume of L, the (moving) multipoint Seshadri constant of L at p_1,...,p_N and the possibility to construct a Kähler packing centered at p_1,...,p_N. In the toric case the multipoint Okounkov bodies at N torus--fixed points can be obtained subdividing the polytope, while in the surface case it is possible to show their polyhedrality, and in the case X = P2 their shape is strongly connected to the renowned Nagata's Conjecture.

**2018-01-24: Martin Sera** (Göteborg)*Chern forms of Hermitian metrics with analytic singularities on vector bundles*

**Abstract:** We would like to study Griffiths positive singular Hermitian metrics on vector bundles in the sense of Bo Berndtsson and Mihai Paun. An example of Hossein Raufi shows, for these metrics, it is in general not possible to define the curvature as a current of order zero. Nevertheless, in a joint work with Richard Lärkäng, Hossein Raufi and Jean Ruppenthal, we present a definition of Chern and Segre currents up to a certain degree, which are closed and of order zero. In a joint work with Richard Lärkäng, Hossein Raufi and Elizabeth Wulcan, we study Griffiths positive metrics with analytic singularities. We present a definition of Chern and Segre currents of arbitrary degree as closed currents of order zero by using the Monge-Ampère operator for plurisubharmonic functions with analytic singularities given by Mats Andersson and Elizabeth Wulcan. These Chern and Segre currents represent the Chern and Segre classes of the vector bundle.

#### 2017

**2017-12-07: Jakob Hultgren** (Göteborg)*Coupled Kähler-Einstein Metrics on toric manifolds*

**Abstract:** Problems of canonical metrics is a classical topic in geometry. Modern branches of this include Kähler-Einstein metrics and cscK metrics. These have been studied extensively for their connections to stability notions in algebraic geometry. This seminar will be about a type of canonical metrics introduced by David Witt Nyström and myself which (probably) share these connections to algebraic geometry: Coupled Kähler-Einstein metrics. In particular, I will talk about a necessary and sufficient condition for existence of coupled Kähler-Einstein metrics on toric Fano manifolds. The condition leads to several questions, some of which can be formulated using only polytopes in Euclidean space. If time permits I will also talk about a "soliton" versions of coupled Kähler-Einstein metrics.

**2017-11-30: Andreas Andersson** (University of Oslo)*Hermitian Yang--Mills metrics in multivariable operator theory *

**Abstract:** The algebra of smooth functions on a smooth projective variety can be approximated by finite-dimensional algebras (``Berezin--Toeplitz quantization''). With a special kind of such approximation (introduced in arXiv: 1506.01454) one enters the realm of multivariable operator theory based on Hilbert spaces of analytic functions on the unit ball. In this talk we discuss how to quantize Hermitian metrics on vector bundles over the variety. One of the main goals of this ongoing work is to obtain a new characterization of those bundles which admit Hermitian Yang--Mills metrics.

**2017-11-23: Tuyen Truong** (University of Oslo)*Wedge intersection of some singular currents*

**Abstract:** Let f:X->Y be a pseudo-automorphism of compact Kahler 3-folds. Let T be a smooth closed (1,1) form satisfying the following cohomology condition: \int _X T\wedge C =0 for all curves C in the indeterminacy locus of f^{-1}. While in general f^*(T) may be very singular near the indeterminacy locus of f, I will show that there is a well-defined Monge-Ampère operator MA(f^*(T)) satisfying a Bedford - Taylor type convergence. Some extensions to the case where T is singular at a finite number of points will also be given. If time permits, I will also present an approach to define a Monge-Ampère operator for an arbitrary positive closed (1,1) current on any compact Kähler manifold, provided that we allow the resulting to be more general than a (signed) measure.

**2017-11-21: Ruadhai Dervan** (University of Cambridge)*Stable maps in higher dimensions*

**Abstract:** Kontsevich's version of Gromov-Witten theory rests on the notion of a "stable map" from a curve to a variety. I will discuss a notion of stability for maps between arbitrary varieties, which generalises Kont- sevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the tar- get is a point. I will discuss some examples, and also an analogue of the Yau-Tian-Donaldson conjec- ture in this setting, which relates stability to the existence of certain canonical Kähler metrics. This is joint work with Julius Ross.

**2017-11-16: Lucas Kaufmann** (Göteborg)*Density of currents (and a generalized Monge-Ampère operator?) *

**Abstract:** Given a collection of positive closed (1,1)-currents it is useful in many situations to give a meaning to their wedge product (or intersection). These products can be defined, for instance, when the singular set of these currents is not too big (Bedford-Taylor, Demailly, ...).

In this talk I will recall the notion of density of positive closed currents (Dinh-Sibony) and show how this can be used to recover the wedge product in known cases. Also, I will try to give a meaning to these products when the classical ones are not well-defined and relate them to the non-pluripolar product (BEGZ) and the Andersson-Wulcan product.

This is based on a work in progress with D.-V. Vu (KIAS) and Elizabeth Wulcan.

**2017-11-02: Xueyuan Wan** (Göteborg)*The existence of Finsler-Einstein metrics*

**Abstract:** In this talk, we will show that the existence of Finsler-Einstein metrics and Hermitian-Einstein metrics are equivalent by using Berndtsson’s curvature formula.

**2017-10-26: Jincao Wu** (Göteborg)*Super-canonical metric and an analytic approach to Kawamata’s basepoint-free theorem*

**Abstract:** Our main goal in this note is to study the super-canonical metric on the pluricanonical bundles. And as an application, it can be used to indicate the relationship between the Kawamata’s basepoint-free theorem and the ﬁnitely generated canonical ring.

**2017-10-19: Dennis Eriksson** (Göteborg)*Singularities of metrics on Hodge bundles and their topological invariants *

**Abstract:** For a family of compact complex manifolds, the direct image of the canonical bundle (or Hodge bundle) admits a natural metric. Approaching singular fibers this metric degenerates, and we provide explicit expressions for the dominant terms in the Calabi-Yau case. These are described in terms of topological invariants coming from vanishing cycles or limit Hodgestructures. I will also discuss applications to BCOV-torsion or metrics, whose motivation originally came from mirror symmetry. This is joint work with Gerard Freixas and Christophe Mourougane.

**2017-10-05: Mingchen Xia **(ENS, Paris)*Axiomatization of Deligne Pairings and a Generalization of Deligne-Riemann-Roch Theorem*

**Abstract:** Let $f:X\rightarrow Y$ be a smooth projective morphism between arithmetic varieties. Let $E$ be a Hermitian vector bundle on $X$. Let $\lambda(E)$ be the determinant line bundle of $E$ along $f$ equipped with Quillen metric.

The famous arithmetic Riemann-Roch theorem calculates the arithmetic Chern chern class of $\lambda(E)$ in terms of the arithmetic characteristic classes of $E$ and $T_{X/Y}$. The arithmetic Chern class determines the determinant line bundle up to torsion. We are interested in reconstructing the determinant line bundle directly from the characteristic classes themselves.

The main tool was proposed by Deligne, which is known as Deligne pairing nowadays. For a projective flat morphism of varieties of pure relative dimension $n$, say $f:X\rightarrow Y$, given some line bundles $E_i$ on $X$ and a homogeneous polynomial $P$ of degree (n+1) in the Chern classes of $E_i$, the Deligne pairing $

_{X/Y}$ is a line bundle on $Y$ whose chern class is the fibre integration $\int_{X/Y} P$. Certain functorial properties of this construction are required. The Deligne pairings have been constructed by a number of people through different methods, among them are Elkik and Ducrot. We propose a number of axioms of Deligne pairings and prove a uniqueness theorem for them. Using this uniqueness theorem, we shall try to construct a functorial isomorphism

\lambda(E)\stackrel{\sim}{\rightarrow} \langle RR^{n+1}(E) \rangle_{X/Y},

where $E$ is a vector bundle on $X$, $RR^{n+1}$ denotes the degree $n+1$ part of the Riemann-Roch form. The $n=1$ case is proved by Deligne himself. We give a different proof in that case. When $n=2$, we have also proved the isomorphism, which seems to be unknown in the previous literatures. Roughly speaking, we resolve the above isomorphism formally to define a theory of Deligne pairing, using the uniqueness theorem, we can then obtain the desired isomorphism. Hopefully, when $n>2$, the same method may work, although we have not carried out the calculations.

There is a further point, when we are in the complex quasi-projective setting, and assume that every bundle under consideration is Hermitian, then both sides of the above equation have natural metrics. Using some results of Bismut-Lebeau and Gillet-Soul\'{e}, we can calculate their difference explicitly if the isomorphism of underlying line bundles is already established.

**2017-09-21: Julie Rowlett** (Göteborg)**& 09-14: Ksenia Fedosova** (Göteborg)*Variational formulas, zeta functions, and plurisubharmonicity*

**Abstract:** We will give a two-part talk on our joint work with Genkai Zhang. In the first part, we will introduce the Selberg zeta function, its many relatives, and the Selberg trace formula. The aforementioned trace formula gives an equality which on one side contains spectral information and on the other side contains geometric information. We will give some insights on the proof of the Selberg trace formula as well as some of its consequences and applications. In conclusion, we will discuss a result of Takhtadzhyan and Zograf in the spirit of the Selberg trace formula which will be a key ingredient in the proof of the results discussed in the second talk.

In the second talk, we will specify to the context of our recent joint work. Namely, we will discuss the Selberg zeta function (and its many relatives) associated to a Riemann surface of genus g \geq 2. We shall identify Riemann surfaces of genus g with points in Teichmüller space. In this way, we will demonstrate first and second variational formulas which show how the zeta functions vary in Teichmüller space. Taking integer points in the Selberg zeta function, Z, we will show that the second variation of log Z(m) is related to a certain curvature formula of Bo Berndtsson. Using that curvature formula, together with the asymptotics of the curvature also computed by Bo, we will show that -log Z(m) is a plurisubharmonic function on relatively compact open sets of Teichmüller space, for m sufficiently large. In so doing, we shall also compute the asymptotic behaviour of -log Z(m) in terms of m as m tends to infinity. These talks require minimal pre-requisite knowledge. We have planned the first talk specifically for the KASS audience, to introduce the possibly less familiar topics and concepts which will be discussed further in the second talk. Nonetheless, both talks are aimed at a broad mathematical audience, and only basic knowledge of differential and complex geometry is required!

**2017-06-22: Yanir Rubinstein** (University of Maryland, USA)*Lagrangian potential theory*

**Abstract:** Special Lagrangian are a curious and mostly mysterious object of interest in symplectic geometry, complex geometry, and elsewhere. Their study goes back to Harvey--Lawson (1982). In some aspects they resemble minimal surfaces, in others they resemble canonical metrics. Slodkowski (1988) and Harvey--Lawson (2009) observed that a ``potential theory" is more or less canonically attached to many nonlinear equations of second order. In this talk I will give motivation to developing the potential theory associated to the special Lagrangian equation. There are beautiful analogies with convex analysis and pluripotential theory, some dramatic differences, and, mostly, many open questions.

**2017-06-22: Tomoyuki Hisamoto** (Nagoya University, Japan)*On the weak Yau-Tian-Donaldson conjecture*

**Abstract:** We study the reduced norm of a torus-equivariant test configuration, in attempt to formulate a weak version of Yau-Tian-Donaldson conjecture for a general polarized manifold. The weak YTD is actually true in the toric case.

**2017-05-23: Loredana Lanzani **(Syracuse University, USA)*Holomorphic Cauchy integrals: a survey of the Lp-theory and two examples*

**Abstract:** I will begin with an overview of recent joint work with E. M. Stein on Lp-regularity of Cauchy-type integrals (Henkin-Ramirez; Leray) for domains with appropriate convexity and minimal boundary regularity. I will then discuss two examples showing that our assumptions (regularity; convexity) on the domain that supports the Leray integral are optimal.

(Abstract as pdf with references)

**2017-05-18: László Lempert** (Purdue University, USA)*Local isometries of the space of Kähler metrics*

**2017-05-10: Audunn Skuta Snaebjarnarson **(University of Iceland)*Rapid polynomial approximation on Stein manifolds*

**Abstract:** Let f be a holomorphic function of N complex variables defined in a neighbourhood of some compact set K. A theorem of Siciak (1962) describes the equivalence between possible holomorphic continuation of the function f and the decay of the sequence (d_n) of the best uniform approximation of f on K by polynomials of degree less than or equal to n. We start by discussing Siciak’s theorem and then we talk about ways to generalize this theorem to more general Stein manifolds.

**2017-04-19: Elizabeth Wulcan** (Göteborg)*Generalized Monge-Ampère operators *

**Abstract:** In a previous joint work with Andersson we introduced Monge-Ampère operators for plurisubharmonic functions with analytic singularities. Our construction extends Bedford-Taylor-Demailly's classical Monge-Ampère operators for plurisubharmonic functions with small unbounded locus, and it has an intersection-theoretical interpretation in terms of so-called Segre numbers.

I will discuss a joint work with Andersson and Błocki, in which we show that these Monge-Ampère operators are continuous for certain decreasing sequences of plurisubharmonic functions. We also prove and discuss a mass formula for the (full) Monge-Ampère operator of an ω-plurisubharmonic function on a compact Kähler manifold.

**2017-03-29: David Witt Nyström** (Göteborg)*Monotonicity of non-pluripolar Monge-Ampère masses*

**Abstract:** (from arXiv:1703.01950) We prove that on a compact Kahler manifold, the non-pluripolar Monge- Ampère mass of a θ-psh function decreases as the singularities increase. This was conjectured by Boucksom-Eyssidieux-Guedj-Zeriahi who proved it under the additional assumption of the functions having small unbounded locus. As a corollary we get a comparison principle for θ-psh functions, analogous to the comparison principle for psh functions due to Bedford-Taylor.

**2017-03-20: Robert Berman** (Göteborg)*A complex geometric approach to random matrices and the two-dimensional Coulomb gas*

**Abstract:** In this talk I will show how to apply complex geometry to the theory of random matrices (the complex Ginibre ensemble). According to Ginibre's classical “circular law” the eigenvalues of a random complex matrix of rank N, with normally distributed entries, become uniformly distributed on a disk, as N tends to infinity. In this talk, I will explain how to use complex geometry to get a quantitative version of this result. The key result is a sharp Gaussian deviation inequality, which is shown using a combination of Kähler geometry and potential theory. The inequality applies, more generally, to the two-dimensional Coulomb gas, where the role of complex eigenvalues is played by charged particles in electrostatic equilibrium. In the latter setting the inequality provides a rigorous version of some heuristics in Quantum Field Theory, which compares the fluctuations of the Coulomb gas with the Gaussian free field.

**2017-03-06: Zbigniew Blocki** (Jagiellonian University, Polen)*On the dimension of the Bergman space for pseudoconvex domains in C^n *

**Abstract:** We discuss a conjecture of Wiegerinck stating that a Bergman space of a pseudoconvex domain in C^n is either trivial or infinitely dimensional. Some partial results will be presented. This is a joint work with Wlodzimierz Zwonek.

**2017-02-27: Tristan Collins** (Harvard, USA)*Interior regularity for the Monge-Ampère equation in critical Sobolev spaces *

**2017-02-13: Martin Sera** (Göteborg)*A smoothness criterion for complex spaces in terms of differential forms*

**Abstract:** In an ongoing joint project with Håkan Samuelsson-Kalm, we made the observation: If X is a singular complex space and its sheaf of weakly holomorphic 1-forms is locally free, then X is smooth. After a short introduction in holomorphic differential forms on complex spaces, we would like to present a proof of this observation.

This is quite interesting in the context of the Zariski-Lipman conjecture which claims a complex space with locally free tangent sheaf is smooth. Even though the conjecture is proven for many cases, it is still unknown for surfaces. We notice a surface with rational singularities and locally free tangent sheaf is smooth.

**2017-01-23: Tristan Collins** (Harvard, USA)*The complexified Kahler cone*

**Abstract:** Mirror symmetry predicts that the moduli space of complex structures on one Calabi-Yau is dual to the moduli space of complexified Kahler forms on the mirror Calabi-Yau. However, the precise definition of a complexified Kahler form has remained mysterious. I will discuss the complexified Kahler cone in the setting of Strominger-Yau-Zaslow mirror symmetry, and a possible interpretation in terms of solvability of a certain fully-nonlinear PDE.

**2017-01-17: Bo Berndtsson** (Göteborg)*Curvature of higher direct images *

#### 2016

**2016-12-15 & 12-19: Xu Wang** (Göteborg)*Curvature restrictions on a manifold with a flat Higgs bundle & Flat Higgs bundle structure on the complex moduli space and the Kähler moduli space.*

**Abstract:** I will talk about a recent result of mine on a generalization of Griffiths-Schmid-Deligne-Lu's result (on the curvature property of the base manifold of a variation of Hodge structure) to the Higgs bundle case. I use the above generalization in another paper of mine, where I claim I solved a conjecture of Wilson on the negative curvature property of the K\"ahler cone.

[In the second part of my talk,] I will give a quick proof the negativity of the Hodge metric and then explain the Higgs bundle structure on the complex moduli space and the Kähler moduli space.

**2016-12-08: David Witt Nyström** (Göteborg)*On the Maximal Rank Problem for the Complex Homogeneous Monge-Ampère Equation*

**Abstract:** We give examples of regular boundary data for the Dirichlet problem for the Complex Homogeneous Monge-Ampère Equation over the unit disc, whose solution is completely degenerate on a non-empty open set and thus fails to have maximal rank.

**2016-11-17: Lucas Kaufmann** (Göteborg)*Intersection of positive closed currents and application to dynamics*

**Abstract:** A basic question in complex dynamics is to count the number of (isolated) periodic points of a given holomorphic or even meromorphic self-map f:X -> X of a projective manifold.

When f has a positive dimensional varieties of fixed points, the classical Lefschetz fixed point formula cannot be applied. However, since periodic points correspond to the intersection of the graph of f^n and the diagonal, this problem can be approached using the theory of intersection of positive closed currents.

The aim of this talk will be to present some elements of the theory of densities of Dinh and Sibony and a result of Dinh-Nguyen-Truong that allows us to bound the number of periodic points in a very general setting.

**2016-11-10: Richard Lärkäng** (Göteborg)*Chern classes of singular metrics on vector bundles*

**Abstract:** For holomorphic line bundles, it has turned out to be useful to not just consider smooth metrics, but also singular metrics which are not necessarily smooth, and which can degenerate. Mainly, such singular metrics are studied under some positivity condition of the curvature of the metric. More recently, singular metrics on holomorphic vector bundles have been studied, and one can give meaning to such a metric being positively curved in the sense of Griffiths. For a vector bundle with a Griffiths positive singular metric, there is a naturally defined first Chern form which is a closed positive (1,1)-current, but there are examples where the full curvature matrix is not of order 0. I will discuss joint work with Hossein Raufi, Jean Ruppenthal and Martin Sera, where we show that one can give a natural meaning to the k:th Chern form of a singular Griffiths positive metric as a closed (k,k)-current of order 0, as long as the set where the metric degenerates is small enough.

**2016-11-03: Eric Carlen** (Rutgers, USA)*Functional Inequalities and Gradient Flow for Quantum Markov Semigroups*

**Abstract:** In recent years, building on work of Felix Otto, much progress has been made in the study of a wide class of evolution equations for probability densities by viewing them as gradient flow for certain entropy functions with respect to mass-transportation metrics. The simplest example is the classical Fokker-Planck equation, which was shown by Jordan, Kinderleher and Otto to be gradient flow in the 2-Wasserstein metric for the relative entropy with respect to the steady-state Gaussian density. The Fokker-Planck equation has several natural quantum analogs, in particular one for fermions. This has the form of a Lindblad evolution equation for a time-dependent density matrix. There is a natural differential structure that allows this equation to be written as a "non-commutative partial differential equation", and also to define a natural analog of the 2-Wasserstein distance as a Riemannian distance on the manifold of density matrices such that the equation is, as in the classical case, gradient flow in this metric for the relative entropy with respect to the ground state. Recent joint work with Jan Maas has extended this to a wide class of quantum evolutions equations, linear and non-linear. As in the classical case, a wide range of functional inequalities governing the evolution can be seen as consequences of convexity with respect to the underlying transport metric. These lectures will provide an introduction form the beginning to quantum evolution equations and gradient flow. The close parallels with the classical theory will be emphasized, and a number of open problems will be pointed out.

**2016-10-06: Lucas Kaufmann** (Göteborg)*Foliation cycles on complex manifolds*

**Abstract:** A laminated compact set on a complex manifold is a compact subset given locally by a disjoint union of holomorphic graphs. Extremal examples are given by complex submanifolds and non-singular holomorphic foliations. Given a lamination, there is a natural notion of invariant measure, which in turn gives what we call a foliation cycle. Such an object can be thought as a diffuse version of a compact leaf.

In this talk I will present a result concerning the self-intersection of foliation cycles, whose proof uses a recent theory of intersection of positive closed currents introduced by T.-C. Dinh and N. Sibony. One of the its consequences is the non-existence of foliation cycles of low codimension on complex projective spaces, which generalizes results by Fornaess-Sibony and Dujardin in dimension 2.

**2016-09-22 & 09-29: Martin Sera** (Göteborg)*Modifications of Coherent Analytic Sheaves & A generalization of Takegoshi's relative vanishing theorem*

**Abstract:** In the first talk, we introduce the notion of linear (fibre) spaces. Linear spaces can be seen as a generalization of vector bundles and help to comprehend coherent analytic sheaves in a different way. We proceed with the construction of a monoidal transformation with respect to a coherent analytic sheaf (originally just defined for ideal sheaves), i.e., to each coherent analytic sheaf on a complex space $X$, we associate a locally free sheaf on a modification of $X$. Motivated by this, we study the images of preimage sheaves and present some results from a joint work with Jean Ruppenthal.

In the second talk, we present an application of these results. We get a generalization of Takegoshi's relative vanishing result. Furthermore, we discuss the necessity of an additional assumption made in the generalization. This includes a deeper study of modifications of coherent analytic sheaves and their associated linear spaces.

**2016-09-15: Valentino Tosatti** (Northwestern University, USA)*On and around Nakamaye's Theorem*

**Abstract:** Nakamaye's Theorem from 2000 says that the augmented base locus of a nef and big line bundle on a smooth projective variety equals its null locus (the locus of subvarieties where the restriction of the line bundle is not big). I will discuss a transcendental generalization of this theorem to all compact complex manifolds (with line bundles replaced by (1,1) cohomology classes) and some of its applications. I will also discuss a conjectural generalization to big (1,1) classes which are not nef. Joint work with T. Collins.

**2016-05-24: Erwan Rousseau** (Université d'Aix-Marseille)*ABC inequalities*

**2016-05-13: Jakob Hultgren** (Göteborg)*Coupled Kähler-Einstein Metrics*

**Abstract:** A central theme in complex geometry has been to study various types of canonical metrics, for example Kähler-Einstein metrics and cscK metrics. In this talk we will introduce the notion of coupled Kähler-Einstein (cKE) metrics which are k-tuples of Kähler metrics that satisfy certain coupled Kähler-Einstein equations. We will discuss existence and uniqueness properties and elaborate on related algebraic stability conditions. (Joint work with David Witt Nyström)

(Combined Algebraic Geometry and Number Theory & KASS Seminar)

**2016-04-26: Dano Kim** (Seoul National University)*L2 extension theorems in the general codimension case*

**Abstract:** We will discuss several (old and recent) theorems of L2 extension of Ohsawa-Takegoshi type, for extending holomorphic sections of a line bundle, with some comparison among them. For this purpose, we will recall some basic notions of log-canonical center and subadjunction in algebraic geometry in the latter part of the talk.

**2016-04-21: Elizabeth Wulcan** (Göteborg)*Direct images of semi-meromorphic currents, joint work with Mats Andersson*

**2016-02-29: David Witt Nyström** (Göteborg)*Duality between the pseudoeffective and the movable cone on a projective manifold*

**Abstract:** The structure of projective algebraic manifolds is to a large extent governed by the geometry of its cones of divisors or curves. In the case of divisors, two cones are of primary importance: the cone of ample divisors and the cone of effective divisors (and the closure of these cones as well). These cones have natural transcendental analogues, namely the cone of Kähler classes (called the Kähler cone) and the cone of pseudoeffective (1,1)-classes (called the pseudoeffective cone).

I will discuss my recent proof of a conjecture of Boucksom-Demailly-Paun-Peternell which says that on a projective manifold the pseudoeffective cone is dual to the cone of movable classes.

**2016-02-01: Mattias Jonsson** (University of Michigan)*On the complex dynamics of birational surface maps defined over number fields*

**Abstract:** I will talk about joint work with Paul Reschke, where we prove that any birational self-map of a complex projective surface that has dynamical degree greater than one and is defined over a number field automatically satisfies the Bedford-Diller energy condition after a suitable birational conjugacy. As a consequence, the complex dynamics of the map is well-behaved.

**2016-01-13: Bo Berndtsson** (Göteborg)*Singular metrics on vector bundles over a disk *

**Abstract:** I will discuss a notion of Lelong number and integrability index for singular metrics on a vector bundle and show how they are interrelated. The talk treats a very particular case of bundles over a disk (or half plane) and metrics that have an extra property of S^1-invariance. I will also try to show how such bundles and metrics actually occur in 'nature' and that the Lelong numbers have natural interpretations in these cases.

#### 2015

**2015-12-10: Håkan Samuelsson Kalm** (Göteborg)*Differential forms on analytic varieties II*

**Abstract:** I will explain how to construct (local) solution formulas for the §\bar{\partial}§-equation for §(p,q)§-forms on reduced analytic varieties. This is the background for the results I presented last time. I will also indicate how these formulas can be used to get a concrete realization of Serre duality for complex spaces.

**2015-12-07: Mats Andersson** (Göteborg)*Local (non-)solvability of the dbar-equation on a singular space*

**Abstract:** Let $X$ be a reduced analytic space of pure dimension. I will recall how one can define smooth forms and currents on $X$ and discuss some classical and more recent results about local solvability and non-solvability of the $\dbar$-equation. In particular I will give an example of an "exotic" solution to $\dbar u=0$ when $u$ is a $(0,0)$-current.

**2015-11-18: Håkan Samuelsson Kalm** (Göteborg)*Differential forms on analytic varieties I*

**Abstract:** On a singular analytic variety there are several in general different notions of differential forms. These serve different purposes and each notion has its particular advantage. Relations and interplay between these notions has got increasing attention in recent years. I'll give a brief introduction to this area and present some new results. In a forthcoming talk I'll try to present the underlying theory in a more rigorous way.

**2015-10-23: David Witt Nyström** (Göteborg)*Okounkov bodies and embeddings of torus-invariant Kahler balls*

**Abstract:** In the 90's Okounkov found a way to associate a convex body to any ample line bundle. These convex bodies are now known as Okounkov bodies, and they generalize the moment polytopes from toric geometry.

A toric moment polytope can be thought of as the image of a moment map, corresponding to a torus-invariant Kahler form in the first Chern class of the line bundle. I will describe how this is approximately true for a general Okounkov body. Namely, I will show how to embed a torus-invariant Kahler ball into X so that the Kahler form on the ball extends to a Kahler form on X lying in the first Chern class of the line bundle, and so that the image of the moment map of the ball approximates the Okounkov body. This is inspired by recent work of Kaveh proving the symplectic version of this result.

**2015-10-21: Jean-Pierre Demailly** (Institut Fourier, Grenoble)*An extension theorem for non reduced subvarieties*

**2015-10-08: Jakob Hultgren** (Göteborg)*Real Monge-Ampère Equations and Permanental Point Processes *

**Abstract:** There is a large community of people working on topics in the overlap of probability theory and complex analysis. An example of such a topic is a recent work of Berman where he connects determinantal point processes with Kähler-Einstein metrics. I will present a real analog of this result where the role of Kähler-Einstein metrics is played by solutions to certain real Monge-Ampère equations and the determinantal point processes are replaced by so called permanental point processes. If time permits, I will also explain one of the tools from thermodynamics used in both Berman’s result and mine.

**2015-09-24: Xu Wang** (Göteborg)*Variation of the Green energy and positivity of the direct image*

**Abstract:** By using Berndtsson's method, we shall give a unified proof of a theorem of Yamaguchi (on variation of the Green function) and a theorem of Berndtsson (on variation of Bergman projections).

**2015-09-18: Mattias Jonsson** (University of Michigan)*Degenerations of Calabi-Yau manifolds and Berkovich spaces*

**Abstract:** Various considerations, from mirror symmetry and elsewhere, have lead people to consider 1-parameter degenerating families of Calabi-Yau manifolds, parameterized by the punctured unit disc. A conjecture by Kontsevich-Soibelman and Gross-Wilson describe what the limiting metric space should be, under suitable hypotheses. I will present joint work with Sebastien Boucksom, in which we show a measure theoretic version of this conjecture.

**2015-05-05: David Witt Nyström** (University of Cambridge)*A new invariant of ample (or big) line bundles *

**Abstract:** I will discuss a new construction which associates to any ample (or big) line bundle on a projective manifold a growth condition on the tangent space of any given point. The growth condition can be seen to encode such "classical" invariants as the volume and the Seshadri constant. It is inspired by toric geometry and the theory of Okounkov bodies, and I think of it as an (better?) alternative to the various Okounkov bodies. I will explain what it says about the Kähler geometry of the manifold.

**2015-05-15: Anthony G. O'Farrell** (National University of Ireland)*Boundary Smoothness Properties of Holomorphic Functions*

**Abstract:** The talk will cover some old and some recent results about the possibility of a differentiable extension to special boundary points of analytic functions that belong to some function space on an open set in the plane. We will also point out some related problems about analytic and harmonic functions.

The methods used will involve ideas about distributions, integral kernels, and potential theory, as well as duality methods from functional analysis. The new results are about holomorphic functions that satisfy a Lipschitz or Hölder condition, and involve extending classical operations on measures to elements of the dual of little-lip-alpha.

(Kombinerat analys- och KASS-seminarium)

**2015-03-23: Xiangyu Zhou** (Beijing)*Sharp L^2 extension problem and Demailly's strong openness conjecture*

**Abstract:** In this talk, we'll present our recent solutions of an sharp L^2 extension and strong openness conjecture, as well as the motivations of the problems, applications of the solutions, relations to some well-known results, and the main ideas of the proofs.

**2015-03-18: Bo Berndtsson** (Göteborg)*Generalized Legendre transforms and symmetries in the space of Kahler metrics*

**Abstract:** The classical Legendre transform is a symmetry of the space of convex functions which fixes precisely the function x^2. We show that given a real analytic plurisubharmonic function there is a symmetry that is defined near that function and fixes it. The construction works equally well (or even better) for metrics on line bundles and the symmetry is then an isometry for the Mabuchi metric. (This is joint work with Dario Cordero-E, Bo'az Klartag and Yanir Rubinstein.)

**2015-02-18 & 03-06: Magnus Önnheim** (Göteborg)*Propagation of chaos, Wasserstein gradient flows and toric Kähler-Einstein metrics*

**Abstract:** Motivated by the probabilistic approach to construction of canonical Kähler-Einstein metrics by Berman, we consider corresponding dynamical versions on R^n, analogous to the Calabi flow on a Kähler manifold. Convergence of many-particle dynamics to a limiting measure-valued dynamic can then be formulated as propagation of chaos. This is established by viewing the dynamics as gradient flows in the space of probability measures and use of a so called minimizing movement scheme.

As a first seminar in a series detailing the eponymous paper arXiv:1501.07820, this talk will be about metric theory for gradient flows and optimal transport, in particular as applied to the Wasserstein space of probability measures on R^n.

In the second seminar, I will let the number of particles N go to infinity, in particular I will explain the notion of chaos. I will also prove a main lemma, which is a discrete version of propagation of chaos.

**2015-02-04 & 02-09: Jakob Hultgren** (Göteborg)*Introduction to theta-functions *

**Abstract:** This is a series of two talks. The talk is aimed at people who, like the speaker, has not encountered theta-functions before but is familiar with a little bit of complex geometry. Part 1: We define theta functions and elliptic functions (in one complex variable) and explain how the former can be seen as holomorphic sections of certain line bundles over Riemann surfaces of genus 1. Part 2: We sketch an application in terms of elliptic integrals.