# Seminars (KASS)

Here you find an overview of upcoming and past talks in the so-called complex analysis special seminar (shortly KASS) of the SCV research group. For all related inquiries, please feel free to contact the current organizer, Mingchen Xia. Current regular seminar time: Mondays, 13:15 in MV:L14 (LP1 2019/20).

#### Next talks

2019-10-10: Burglind Juhl-Jöricke (MSRI)
Fundamental groups, slalom curves and extremal length
Abstract: We define the extremal length of elements of the fundamental group  of the twice punctured complex plane and give effective estimates for  this invariant. The main motivation comes from $3$-braid  invariants and their application, for instance to effective finiteness theorems in the spirit of the Geometric Shafarevich Conjecture over Riemann surfaces  of second kind.

#### Past talks

Non-proper intersections and generalized cycles

Abstract: (Joint with Eriksson, Samuelsson-Kalm, Wulcan, Yger). Let $Z_1,Z_2$ be subvarieties of $P^n$.  If $Z_j$ intersect properly, that is, the set-theoretical intersection $V$ has dimension $\dim Z_1+\dim Z_2-n$, then there is a well-defined cycle $Z_1\cdot Z_2$ with support $V$ called the proper intersection.  In the non-proper case the intersection, in the sense of Fulton-MacPherson, is no longer a well-defined cycle but a Chow class on $V$.  Tworzewski has defined local intersection numbers $\epsilon_k(Z_1,Z_2,x)$ at each point $x\in V$, $0\le k\le dim V$, even in the non-proper case.  We introduce the group $B(V)$ of (certain quotient classes of what we call) generalized cycles on $V$ that contains the usual cycles as a subgroup. Contrary to Chow classes on $V$ each $B(V)$-class has well-defined multiplicities at each point $x\in X$. Formally generalized cycles are currents, but they can/should be thought of as geometric objects.  Our main result is the definition of a $B(V)$-class $Z_1\bullet Z_2$ such that the multiplicities (for the components of various dimensions) at each point coincide with the local intersection numbers $\epsilon_k(Z_1,Z_2,x)$. Moreover, $Z_1\bullet Z_2$  and $Z_1\cdot Z_2$ define the same cohomology class on $V$.

2019-09-24: Richard Schoen (UCI, USA)
Positive scalar curvature, the Dirac operator and minimal hypersurfaces
Abstract: We will discuss the approaches which have been successful in understanding manifolds with positive scalar curvature. Two basic and related questions one can ask are the topological classification of such manifolds and the positive mass theorem. The two approaches which have worked in all dimensions are the Dirac operator and the  minimal hypersurface approach. Both have many successes but also limitations which we will describe. In the case of three dimensions there are also the inverse mean curvature  flow and the Ricci flow which yield strong results.

2019-09-02: Ken-Ichi Yoshikawa (Kyoto University, Japan)
Enriques 2n-folds and analytic torsion
Abstract: In this talk, a compact connected Kähler manifold of even dimension is called simple Enriques if it is not simply connected and its universal covering is either Calabi-Yau or hyperkähler. These manifolds were introduced and studied independently by Boissière-Nieper-Weisskirchen-Sarti and Oguiso-Schröer. We introduce a holomorphic torsion invariant of simple Enriques 2n-folds and study the function on the moduli space of such manifolds obtained in this way. In the talk, we report its basic properties such as the strong plurisubharmonicity and the automorphy, as well as possible (conjectural) applications. If time allows, we will also report the explicit formula for the invariant as an automorphic function on the moduli space in some cases.

2019-06-13: Martin Sera (Göteborg)
Regularization of mixed Monge-Ampère products of qpsh functions with analytic singularities
Abstract: This reports on an ongoing work, joint with Elizabeth Wulcan and Richard Lärkäng. We consider mixed Monge-Ampère products of quasiplurisubharmonic functions with analytic singularities. We show that such products may be regularized by explicit one-parameter sequence of mixed Monge-Ampère products of smooth functions, generalizing results of M. Andersson, Z. Błocki and E. Wulcan in the case of non-mixed Monge-Ampère products. Connections to the theory of residue currents, going back to N. Coleff & M. Herrera, M. Passare and others, play an important role in the proofs.

2019-06-11: Reza Seyyedali (IPM, Iran)
Relative Chow stability of extremal Kahler metrics
Abstract: In 2001, Donaldson proved that any constant scalar curvature polarized manifold is asymptotically Chow stable provided that the group of hamiltonian athromorphisms is discrete. In this talk, we discuss some generalization of Donaldson's result to extremal metrics.

2019-05-28: Jakob Hultgren (University of Oslo)
Products of Vandermonde Determinants, Fekete points and canonical point processes
Abstract: I will talk about a new construction in complex geometry related to Fekete points and canonical point processes. I will give a conjectural picture of its asymptotic behaviours and a theorem supporting this picture. This is partly based on joint work with Yanir Rubinstein.

2019-05-21: Tristan Collins (MIT, USA)
Stability and Nonlinear PDE in mirror symmetry
Abstract: A longstanding problem in mirror symmetry has been to understand the relationship between the existence of solutions to certain geometric nonlinear PDES (the special Lagrangian equation, and the deformed Hermitian-Yang-Mills equation) and algebraic notions of stability, mainly in the sense of Bridgeland. I will discuss progress in this direction through ideas originating in infinite dimensional GIT. This is joint work with S.-T. Yau.

2019-05-07: Sławomir Kołodziej (Jagiellonian University, Krakow)
On volumes of nef classes on compact Hermitian manifolds
Abstract: pdf-file
(joint work with V. Tosatti)

2019-04-18: Stefano Trapani (Tor Vergata, Rome)
A survey on recent results about curvature and canonical bundle
Abstract: In this talk I will describe some recent results by various people relating canonical bundle and curvature on compact Kahler manifolds, time permitting I will relate this to a conjecture by Serge Lang.

2019-03-12: Xueyuan Wan (Göteborg)
Plurisuperharmonicity of reciprocal energy function on Teichmüller space and Weil-Petersson metrics
Abstract: We consider harmonic maps u(z) in a fixed homotopy class from Riemann surfaces varying in the Teichmüller space to a Riemannian manifold N with non-positive Hermitian sectional curvature. The energy function E(z)=E(u(z)) can be viewed as a smooth function on Teichmüller space,\mathcal{T} and we study its first and the second variations. We prove that the reciprocal energy function 1/E(z) is plurisuperharmonic on Teichmüller space. We also obtain the (strict) plurisubharmonicity of \log E(z) and E(z). As an application, we obtain that the the second variation of logarithmic energy function is exactly the Weil-Petersson metric if the harmonic map is holomorphic or anti-holomorphic and totally geodesic. This work is joint with Professors Inkang Kim (KIAS) and Genkai Zhang (Chalmers & GU).

2019-03-05: Mingchen Xia (Göteborg)
Destabilizing properties of gradient flows in Kähler geometry
Abstract: In Kähler geometry, a central problem is to understand the relation between the existence of cscK metrics on a polarized Kähler manifold and various GIT stability conditions. Last year, after the break through of Chen—Cheng, it has been understood that the existence of cscK metric is equivalent to certain geodesic stability. In this talk, I will discuss what happens when cscK metric does not exist. I will construct a geodesic ray using the weak Calabi flow and show that in the unstable case, this geodesic ray is the unique most unstable geodesic ray in the sense that it maximizes the analogue of the (minus) Donaldson—Futaki invariant in metric geometry, answering the metric version of a conjecture of Donaldson. In the Fano setting, the result is also obtained independently by T.Hisamoto.

2019-02-22: Yanir Rubinstein (University of Maryland, USA)
Differential, algebraic, and convex geometry arising from asymptotic positivity
Abstract: A general theme in geometry is the classification of algebraic/differential geometric structures which satisfy a positivity property. I will describe an "asymptotic" version of this theme based on joint work with Cheltsov, Martinez-Garcia, and Zhang. On the algebraic side, we introduce the class of asymptotically log Fano varieties and state a classification theorem in dimension 2, generalizing the classical efforts of the 19th century Italian school. The novelty here is the use of a convex optimization theorem that reduce the asymptotic positivity to determining intersection properties of high-dimensional convex bodies. On the differential side, I will give a conjectural picture for existence of singular Kähler-Einstein metrics and explain progress towards this conjecture making use of symmetry, log canonical thresholds, test configurations, and Fujita-Odaka's basis type invariant. Time permitting, I will also touch on relations to singular Kahler-Ricci solitons, mention some conjectures and results about the `small angle limit' when the angle tends to zero, and tie this picture to non-compact Calabi-Yau fibrations, steady Ricci solitons, and recent work of Liu on wall-crossings in moduli space.

A pointwise norm on a non-reduced space
Abstract: Let $X$ be a non-reduced analytic space of pure dimension $n$. I will describe how one can define an, essentially intrinsic, pointwise norm of functions (and differential forms) on $X$. The construction relies on Noetherian differential operators on $X$. I will also indicate why the structure sheaf $\mathcal O_X$ of holomorphic functions on $X$ is complete with respect to the natural topology induced by this norm.

2019-02-05: Bo Berndtsson (Göteborg)
Weierstrass points and Bergman kernels
Abstract: This is basically a survey lecture in the sense that no essentially new results will be shown. I will start by discussing a higher order version of Bergman kernels that I learned about in a recent paper by Hedenmalm and Wennman. These kernels make sense also for a Riemann surface equipped with a positive line bundle and then define metrics on associated adjoint bundles. Then I will show a formula relating the Bergman kernels to Wronskian determinants, whose zeros are the Weierstrass points for the line bundle. (This is implicitly contained in a paper by J Lewittes from -69.) The classical case is when the line bundle is the canonical bundle of a surface of genus at least 2. This means that the asymptotics of the Bergman kernels for higher and higher powers of the line bundle should be related to the distribution of Weierstrass points, as studied by Mumford and Neeman. I will speculate on this and time permitting I'll continue to speculate on possible higher dimensional versions and their relations to Okounkov bodies.

2019-01-29: Xu Wang (NTNU Trondheim)
Geometry of Poisson Kahler fibrations.
Abstract: This is a joint work with Xueyuan Wan. A relative Kahler fibration is said to be Poisson if the top power of the relative Kahler form vanishes (here top power means one plus the dimension of the fibers). We will discuss basic properties of a Poisson Kahler fibration and its relation with stability of vector bundles, Kahler metric geodesics, Levi flatness of Donaldson Fujiki moment map and positivity of relative canonical bundle. The main result is the following: if the Kodaira Spencer map of a Poisson Kahler fibration is injective then there is a Kahler metric on the base manifold with strictly negative holomorphic sectional curvature and semi-negative holomorphic bisectional curvature. The idea is to consider an average of a family of Lu's Hodge metrics and use the standard Higgs bundle package. Our main result is also obtained independently by Bo Berndtsson (using a different method) recently.

You can find a list of former talks here (2015–2018) and here (2014 and before) (see also here).​​

Published: Fri 11 Oct 2019.