Some examples of stable generalized complex 6-manifolds with either 0 or negative Euler characteristic

Abstract Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.

Download Full-text

Vortices over Riemann surfaces and dominated splittings

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.142 ◽

2021 ◽

pp. 1-26

Author(s):

THOMAS METTLER ◽

GABRIEL P. PATERNAIN

Keyword(s):

Euler Characteristic ◽

Tangent Bundle ◽

Riemann Surfaces ◽

Dominated Splitting ◽

Vortex Equations ◽

Special Cases ◽

Unit Tangent Bundle ◽

Anosov Flows

Abstract We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .

Download Full-text

Tropical geometry and the motivic nearby fiber

Compositio Mathematica ◽

10.1112/s0010437x11005446 ◽

2011 ◽

Vol 148 (1) ◽

pp. 269-294 ◽

Cited By ~ 6

Author(s):

Eric Katz ◽

Alan Stapledon

Keyword(s):

Euler Characteristic ◽

Hodge Structure ◽

Tropical Geometry ◽

Mixed Hodge Structure ◽

General Fiber ◽

Algebraic Torus ◽

Tropical Varieties

AbstractWe construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the ‘tropical motivic nearby fiber’. This invariant specializes in the schön case to the Hodge–Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge–Deligne polynomial in the cases of schön hypersurfaces and matroidal tropical varieties. We also deduce a formula for the Euler characteristic of a general fiber of the degeneration.

Download Full-text

Topology of Subvarieties of Complex Semi-abelian Varieties

International Mathematics Research Notices ◽

10.1093/imrn/rnaa242 ◽

2020 ◽

Author(s):

Yongqiang Liu ◽

Laurentiu Maxim ◽

Botong Wang

Keyword(s):

Euler Characteristic ◽

Morse Theory ◽

Characteristic Property ◽

Abelian Varieties ◽

Betti Numbers ◽

Local Systems ◽

Affine Manifolds ◽

Cyclic Covers ◽

Rank One ◽

Generic Vanishing

Abstract We use the non-proper Morse theory of Palais–Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh’s extension of Varchenko’s conjecture to very affine manifolds and provide a generalization of this result in the context of smooth closed sub-varieties of semi-abelian varieties.

Download Full-text

Kinematic formula and tube formula in space of constant curvature

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150002890x ◽

1990 ◽

Vol 33 (1) ◽

pp. 79-88

Author(s):

Sungyun Lee

Keyword(s):

Euclidean Space ◽

Euler Characteristic ◽

Constant Curvature ◽

Kinematic Formula ◽

Curvature Invariants ◽

Space Of Constant Curvature

The Euler characteristic of an even dimensional submanifold in a space of constant curvature is given in terms of Weyl's curvature invariants. A derivation of Chern's kinematic formula in non-Euclidean space is completed. As an application of above results Weyl's tube formula about an odd-dimensional submanifold in a space of constant curvature is obtained.

Download Full-text