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European Mathematical Society Publishing House
2024-03-29 16:57:48
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=CMH&vol=85&iss=2&update_since=2024-03-29
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
85
2010
2
Geometric approach to the Weil–Petersson symplectic form
Reynir
Axelsson
University of Iceland, REYKJAVIK, ICELAND
Georg
Schumacher
Philipps-Universität, MARBURG, GERMANY
Weil–Petersson form, Fenchel–Nielsen coordinates, symplectic geometry Publishers Note. Due to an error the submission date as printed in the article is not correct. The article was received by the editors on February 27, 2008.
In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel–Nielsen and Weil–Petersson symplectic forms on the Teichmüller spaces of compact Riemann surfaces in a purely geometric way. The method can also be applied to situations like moduli spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available.
Several complex variables and analytic spaces
Differential geometry
General
243
257
10.4171/CMH/194
http://www.ems-ph.org/doi/10.4171/CMH/194
A characterization of round spheres in terms of blocking light
Benjamin
Schmidt
Michigan State University, EAST LANSING, UNITED STATES
Juan
Souto
Université de Rennes 1, RENNES CEDEX, FRANCE
Riemannian manifold, cross blocking, sphere blocking, blocking number
A closed Riemannian manifold M is said to have cross (compact rank one symmetric space) blocking if whenever p ≠ q are less than the diameter apart, all light rays from p can be shaded away from q with at most two point shades. Similarly, a closed Riemannian manifold is said to have sphere blocking if for each p ∈ M all the light rays from p are shaded away from p by a single point shade. We prove that Riemannian manifolds with cross and sphere blocking are isometric to round spheres.
Differential geometry
Dynamical systems and ergodic theory
General
259
271
10.4171/CMH/195
http://www.ems-ph.org/doi/10.4171/CMH/195
Conformal arc-length as ½-dimensional length of the set of osculating circles
Rémi
Langevin
Université de Bourgogne, DIJON CEDEX, FRANCE
Jun
O'Hara
Tokyo Metropolitan University, TOKYO, JAPAN
Conformal arc-length, osculating circles, pseudo-Riemannian manifolds
The set of osculating circles of a given curve in S3 forms a lightlike curve in the set of oriented circles in S3. We show that its “½-dimensional measure” with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.
Differential geometry
General
273
312
10.4171/CMH/196
http://www.ems-ph.org/doi/10.4171/CMH/196
Finiteness results for flat surfaces: large cusps and short geodesics
John
Smillie
Cornell University, ITHACA, UNITED STATES
Barak
Weiss
Tel Aviv University, TEL AVIV, ISRAEL
Flat surface, Veech group, Fuchsian group, Markov partition
For fixed g and T we show the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a cusp of hyperbolic co-area less than T. We obtain new restrictions on Veech groups: any non-elementary Fuchsian group can appear only finitely many times in a fixed stratum; any non-elementary Veech group is of finite index in its normalizer; and the quotient of ℍ by a non-lattice Veech group admits arbitrarily large embedded disks. A key ingredient of the proof is the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a hyperbolic element with eigenvalue less than T.
Manifolds and cell complexes
Dynamical systems and ergodic theory
General
313
336
10.4171/CMH/197
http://www.ems-ph.org/doi/10.4171/CMH/197
Bounding the symbol length in the Galois cohomology of function fields of p-adic curves
Venapally
Suresh
Emory University, ATLANTA, UNITED STATES
Galois cohomology, central simple algebras, cyclic algebras, symbols
Let K be a function field of a p-adic curve and l a prime not equal to p. Assume that K contains a primitive l th root of unity. We show that every element in the l-torsion subgroup of the Brauer group of K is a tensor product of two cyclic algebras over K.
Number theory
General
337
346
10.4171/CMH/198
http://www.ems-ph.org/doi/10.4171/CMH/198
Automatic transversality and orbifolds of punctured holomorphic curves in dimension four
Chris
Wendl
Humboldt-Universität zu Berlin, BERLIN, GERMANY
Pseudoholomorphic curves, transversality, symplectic cobordisms, symplectic 4-manifolds, contact 3-manifolds
We derive a numerical criterion for J-holomorphic curves in 4-dimensional symplectic cobordisms to achieve transversality without any genericity assumption. This generalizes results of Hofer–Lizan–Sikorav [HLS97] and Ivashkovich–Shevchishin [IS99] to allow punctured curves with boundary that generally need not be somewhere injective or immersed. As an application, we combine this with the intersection theory of punctured holomorphic curves to prove that certain geometrically natural moduli spaces are globally smooth orbifolds, consisting generically of embedded curves, plus unbranched multiple covers that form isolated orbifold singularities.
Several complex variables and analytic spaces
Manifolds and cell complexes
General
347
407
10.4171/CMH/199
http://www.ems-ph.org/doi/10.4171/CMH/199
The K(π,1) conjecture for a class of Artin groups
Graham
Ellis
National University of Ireland, GALWAY, IRELAND
Emil
Sköldberg
National University of Ireland, GALWAY, IRELAND
Artin group, Eilenberg–Mac Lane space, cohomology groups
Salvetti constructed a cellular space BD for any Artin group AD defined by a Coxeter graph D. We show that BD is an Eilenberg–Mac Lane space if BD' is an Eilenberg–Mac Lane space for every subgraph D' of D involving no ∞-edges.
Algebraic topology
Group theory and generalizations
General
409
415
10.4171/CMH/200
http://www.ems-ph.org/doi/10.4171/CMH/200
Local signature defect of fibered complex surfaces via monodromy and stable reduction
Tadashi
Ashikaga
Tohoku Gakuin University, Tagajo, JAPAN
Signature, fibered surface, monodromy, eta invariant, stable reduction, signature defect
We consider the localization problem of signature of a fibered complex surface. In this paper, we analyze the contribution of topological monodromy to the local signature of a fiber germ. In order to realize it, we describe explicitly the behavior of Atiyah–Patodi–Singer’s eta invariant under the stable reduction of a fiber germ in terms of Nielsen’s data.
Algebraic geometry
Several complex variables and analytic spaces
Manifolds and cell complexes
General
417
461
10.4171/CMH/201
http://www.ems-ph.org/doi/10.4171/CMH/201
A converse theorem for Dirichlet L-functions
Jerzy
Kaczorowski
Adam Mickiewicz University, POZNAN, POLAND
Giuseppe
Molteni
Università di Milano, MILANO, ITALY
Alberto
Perelli
Università di Genova, GENOVA, ITALY
Dirichlet L-functions, converse theorems, Hamburger theorem, Selberg class
It is known that the space of solutions (in a suitable class of Dirichlet series with continuation over ℂ) of the functional equation of a Dirichlet L-function L(s,χ) has dimension ≥ 2 as soon as the conductor q of χ is at least 4. Hence the Dirichlet L-functions are not characterized by their functional equation for q ≥ 4. Here we characterize the conductors q such that for every primitive character χ (mod q), L(s,χ) is the only solution with an Euler product in the above space. It turns out that such conductors are of the form q = 2a3bm with any square-free m coprime to 6 and finitely many a and b.
Number theory
General
463
483
10.4171/CMH/202
http://www.ems-ph.org/doi/10.4171/CMH/202