SCV Seminar

Monday, Jan. 12, 09	Lina Lee (University of Michigan) Kobayashi, Carath\'eodory and Sibony metric : We study the metrics near a pseudoconcave boundary point and show that the Sibony metric is strictly greater than the Carath\'eodory metric and strictly less than the Kobayashi metric. It is known that the Kobayashi metric blows up at a rate of $1/d^{3/4}$, $d$ being the distance to the boudnary, in the normal direction. Carath\'eodory metric does not blow up near a pseudoconcave point. We estimate the Sibony metric and show that it blows up at a rate of $1/d^{1/2}$ in the normal direction. This is a joint work with J. E. Fornaess.
Monday, Jan. 26, 09	Al Taylor (University of Michigan) Extremal plurisubharmonic functions for linear growth (Part I): We will discuss properties of the extremal function $\Lambda_E(z)$, associated to plurisubharmonic functions of linear growth, i.e. the upper envelope of all psh functions $u$ that are bounded above by zero on the set $E$ and satisfy $u(z) \le \|z\| + o(\|z\|)$. Questions about this function arise naturally in trying to classify the algebraic varieties with sufficiently many real points that they satisfy the strong radial Phragmen-Lindelof condition. This extremal function is also an analogue of the Siciak-Zaharuta extremal psh function $L_E$ of logarithmic growth. However, we will show that it fails to have most of the properties that make $L_E$ such a useful function in pluripotential theory. In Monday's seminar talk, we will give an overview of these results, but very few proofs. In the working seminar on Wednesday, we will present enough of the proofs to explain all the main techniques used in the work. All of the material is taken from the 1998 Ph.D. thesis of David Bainbridge.
Monday, Feb. 2, 09	Berit Stensones (University of Michigan) Finite D'Angelo type in C^3: We shall study bounded pseodoconvex domains of finite type in C^3. The goal of the talk is to describe an algorithm that identifies the complex curves that has an exceptional high order of contact with the boundary of such a domain.This enables us to show that there is only "finitely" many such domains.
Monday, Feb. 9, 09	Araceli Bonifant (University of Rhode Island) Cubic Polynomial Maps with Periodic Critical Orbit: The parameter space ${\mathcal S}_p$ for cubic polynomial maps with a marked critical point of period $p$ is a complicated algebraic curve whose genus increases rapidly with $p$. Each ${\mathcal S}_p$ consists of a compact connectedness locus together with finitely many escape regions, each biholomorphic to a punctured disk. The parameter rays in the various escape regions provide a tool for studying the dynamics.
Tuesday, Feb. 17, 09 3-4 pm (3088 EH)	Aron Lagerberg (Chalmers University of Technology) An analyticity theorem for a generalized Lelong number: Lelong numbers, introduced by Pierre Lelong in the 50's, provide us with a way of measuring singularities of plurisubharmonic functions. I will discuss a generalization of Lelong numbers, introduced by Bo Berndtsson, and prove that they satisfy an analyticity property. More precisely, we prove that, for $c>0$, the upper level set $$\{ x \in \Omega : \nu_{x,\psi}( f ) \geq c \}$$ defines an analytic set in $\Omega$, where $\nu_{x, \psi}(f)$ denotes Berndtsson's generalized Lelong number of $f$, with respect to a sufficiently nice function $\psi$, at the point $x$. For $\psi=(n-1) \log\|z\|$, this number is just the classical Lelong number, and we obtain the classical semi-continuity theorem as proved by Siu in 1974.
Wednesday, Feb. 18, 09 4-6 pm (3088 EH)	David Witt Nystroem (Gothenburg University) - joint with Algebraic Geometry seminar Chebyshev functions on Okounkov bodies
Monday, Mar. 9, 09	Jasmin Raissy (Universita de Pisa) Simultaneous linearization in presence of resonances: In this talk I shall discuss the linearization problem of germs of biholomorphisms of C^n fixing the origin in presence of resonances. I shall describe certain arithmetic conditions on the eigenvalues of df_O and some restrictions on the resonances implying that such a germ is holomorphically linearizable if and only if there exists an invariant complex manifold with a specific additional structure. Most of the classical linearization results can be obtained as corollaries of this result. Then I shall describe a way to use that result in the problem of simultaneous linearization of germs of biholomorphisms of C^n fixing the origin in presence of resonances.
Monday, Mar. 30, 09	Kyounghee Kim (Florida State University / Indiana University) Linearization on a large set: We will discuss a discrete family of automorphisms. These maps have a curve of fixed points. There are two cases. In the first case, the curve is attracting/repelling, and its basin has full volume. In the other case, the curve is the "\emph{center}" of a rotation domain, which is "\emph{large}" because it contains both a curve of fixed points, as well as isolated fixed points. We will discuss how to linearize these mappings.
Monday, Apr. 6, 09	Tatyana Foth (University of Western Ontario) Complex submanifolds, connections and asymptotics: Let L be a positive line bundle on a compact complex manifold X. Suppose Y,S are compact submanifolds of X and Y -> S is a holomorphic submersion. I shall talk about two natural connections in certain line bundles on S.
Monday, Apr. 13, 09
Monday, Apr. 20, 09