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European Mathematical Society Publishing House
2024-03-28 11:08:21
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=CMH&vol=84&iss=3&update_since=2024-03-28
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
84
2009
3
Simplicial volume of Hilbert modular varieties
Clara
Löh
Universität Regensburg, REGENSBURG, GERMANY
Roman
Sauer
Karlsruher Institut für Technologie, KARLSRUHE, GERMANY
Simplicial volume, Hilbert modular varieties
The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we extend this positivity result to certain non-compact locally symmetric spaces of finite volume, including Hilbert modular varieties. The key idea is to reduce the problem to the compact case by first relating the simplicial volume of these manifolds to the Lipschitz simplicial volume and then taking advantage of a proportionality principle for the Lipschitz simplicial volume. Moreover, using computations of Bucher–Karlsson for the simplicial volume of products of closed surfaces, we obtain the exact value of the simplicial volume of Hilbert modular surfaces.
Differential geometry
General
457
470
10.4171/CMH/169
http://www.ems-ph.org/doi/10.4171/CMH/169
Relations between tautological cycles on Jacobians
Ben
Moonen
University of Amsterdam, AMSTERDAM, NETHERLANDS
Jacobian varieties, Chow ring, tautological cycles
We study tautological cycle classes on the Jacobian of a curve. We prove a new result about the ring of tautological classes on a general curve that allows, among other things, easy dimension calculations and leads to some general results about the structure of this ring. Further we lift a result of Herbaut and van der Geer–Kouvidakis to the Chow ring (as opposed to its quotient modulo algebraic equivalence) and we give a method to obtain further explicit cycle relations. As an ingredient for this we prove a theorem about how Polishchuk’s operator $\mathcal{D}$ lifts to the tautological subalgebra of CH(J).
Algebraic geometry
General
471
502
10.4171/CMH/170
http://www.ems-ph.org/doi/10.4171/CMH/170
Relatively hyperbolic groups: geometry and quasi-isometric invariance
Cornelia
Druţu
Oxford University, OXFORD, UNITED KINGDOM
Relative hyperbolicity, rigidity, quasi-isometry
In this paper it is proved that relative hyperbolicity is a quasi-isometry invariant. As byproducts of the arguments, simplified definitions of relative hyperbolicity are provided. In particular we obtain a new definition very similar to the one of hyperbolicity, relying on the existence of a central left coset of a peripheral subgroup for every quasi-geodesic triangle.
Group theory and generalizations
Manifolds and cell complexes
General
503
546
10.4171/CMH/171
http://www.ems-ph.org/doi/10.4171/CMH/171
The geometry of genus-one helicoids
David
Hoffman
Stanford University, STANFORD, UNITED STATES
Brian
White
Stanford University, STANFORD, UNITED STATES
Complete embedded minimal surface, helicoid, variational methods
We prove: a properly embedded, genus-one, minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into two connected components that lie on either side of the helicoid. We prove an analogous result for periodic helicoid-like surfaces. We also give a simple condition guaranteeing that an immersed minimal surface with finite genus and bounded curvature is asymptotic to a helicoid at infinity.
Differential geometry
Calculus of variations and optimal control; optimization
Global analysis, analysis on manifolds
General
547
569
10.4171/CMH/172
http://www.ems-ph.org/doi/10.4171/CMH/172
Hamiltonian pseudo-representations
Vincent
Humilière
Université Pierre et Marie Curie, PARIS, FRANCE
Symplectic topology, Poisson brackets, Hamiltonian representation, Hamiltonian groups, Hofer's distance
The question studied here is the behavior of the Poisson bracket under C0-perturbations. For this purpose we introduce the notion of pseudo-representation and prove that the limit of a converging pseudo-representation of any normed Lie algebra is a representation. An unexpected consequence of this result is that for many non-closed symplectic manifolds (including cotangent bundles), the group of Hamiltonian diffeomorphisms (with no assumptions on supports) has no C−1 bi-invariant metric. Our methods also provide a new proof of the Gromov
Dynamical systems and ergodic theory
Differential geometry
General
571
585
10.4171/CMH/173
http://www.ems-ph.org/doi/10.4171/CMH/173
Chern numbers and the geometry of partial flag manifolds
Dieter
Kotschick
Universität München, MÜNCHEN, GERMANY
S.
Terzić
University of Montenegro, Podgorica, MONTENEGRO
Flag manifold, invariant complex structure, Chern class
We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds Fn = SU(n + 2)/S(U(n) × U(1) × U(1)). For all n > 1 there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles of ℂPn + 1. The projectivization of the cotangent bundle is the twistor space of a Grassmannian considered as a quaternionic Kähler manifold. There is also an invariant nearly Kähler structure, because Fn is a 3-symmetric space. We explain the relations between the different structures and their Chern classes, and we prove that Fn is not geometrically formal.
Differential geometry
Algebraic geometry
Manifolds and cell complexes
General
587
616
10.4171/CMH/174
http://www.ems-ph.org/doi/10.4171/CMH/174
New constructions of slice links
Tim
Cochran
Rice University, HOUSTON, UNITED STATES
Stefan
Friedl
Universität Regensburg, REGENSBURG, GERMANY
Peter
Teichner
University of California, BERKELEY, UNITED STATES
Slice link, slice knot, concordance, Milnor's invariants, infection, satellite link
We use techniques of Freedman and Teichner [FT] to prove that under certain circumstances the multi-infection of a slice link is again slice (not necessarily smoothly slice). We provide a general context for proving links are slice that includes many of the previously known results.
Manifolds and cell complexes
General
617
638
10.4171/CMH/175
http://www.ems-ph.org/doi/10.4171/CMH/175
Zariski k-plets via dessins d’enfants
Alex
Degtyarev
Bilkent University, ANKARA, TURKEY
Zariski pair, trigonal curve, dessin d’enfants, braid monodromy
We construct exponentially large collections of pairwise distinct equisingular deformation families of irreducible plane curves sharing the same sets of singularities. The fundamental groups of all curves constructed are abelian.
Algebraic geometry
General
639
671
10.4171/CMH/176
http://www.ems-ph.org/doi/10.4171/CMH/176
Totally umbilic surfaces in homogeneous 3-manifolds
Rabah
Souam
Institut Mathématiques de Jussieu, PARIS CEDEX 13, FRANCE
Eric
Toubiana
Institut Mathématiques de Jussieu, PARIS CEDEX 13, FRANCE
Totally umbilic, totally geodesic, homogeneous 3-manifolds
We discuss existence and classification of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We classify such surfaces in ℍ2 × ℝ, \mathbb{S}2 × ℝ and the Sol group. We prove nonexistence in the Berger spheres and in the remaining model geometries other than the space forms.
Differential geometry
General
673
704
10.4171/CMH/177
http://www.ems-ph.org/doi/10.4171/CMH/177