Lefschetz theorems for tamely ramified coverings

AbstractWe consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere $\hat {\mathbb {C}}$ ℂ ̂ . Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

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Weierstrass multiple points on algebraic curves and ramified coverings

Israel Journal of Mathematics ◽

10.1007/bf02897070 ◽

1998 ◽

Vol 104 (1) ◽

pp. 335-348 ◽

Cited By ~ 1

Author(s):

Edoardo Ballico ◽

Changho Keem

Keyword(s):

Algebraic Curves ◽

Multiple Points ◽

Ramified Coverings

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The signature of ramified coverings

Gesammelte Abhandlungen/Collected Papers ◽

10.1007/978-3-642-61711-9_8 ◽

1987 ◽

pp. 122-134

Author(s):

F. Hirzebruch

Keyword(s):

Ramified Coverings

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Morse inequalities for covering manifolds

Nagoya Mathematical Journal ◽

10.1017/s0027763000007947 ◽

2001 ◽

Vol 163 ◽

pp. 145-165 ◽

Cited By ~ 11

Author(s):

Radu Todor ◽

Ionuţ Chiose ◽

George Marinescu

Keyword(s):

Line Bundles ◽

Invariant Line ◽

Complex Structures ◽

Von Neumann ◽

Holomorphic Sections ◽

Galois Coverings ◽

Von Neumann Dimension ◽

The Stability ◽

Bounded Below ◽

Lefschetz Theorems

We study the existence of L2 holomorphic sections of invariant line bundles over Galois coverings. We show that the von Neumann dimension of the space of L2 holomorphic sections is bounded below under weak curvature conditions. We also give criteria for a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. As applications we prove the stability of the previous Moishezon pseudoconcave manifolds under perturbation of complex structures as well as weak Lefschetz theorems.

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