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SOME RESULTS ON THE TOPOLOGY OF ALGEBRAIC SURFACES

Part of: Surfaces and higher-dimensional varieties (Co)homology theory Holomorphic convexity

Published online by Cambridge University Press:  17 February 2025

RAJENDRA V. GURJAR Open the ORCID record for RAJENDRA V. GURJAR [Opens in a new window] ,
SUDARSHAN R. GURJAR Open the ORCID record for SUDARSHAN R. GURJAR [Opens in a new window]  and
SOUMYADIP THANDAR Open the ORCID record for SOUMYADIP THANDAR [Opens in a new window]
RAJENDRA V. GURJAR*
Affiliation:
Department of Mathematics Indian Institute of Technology Bombay Powai, Mumbai, 400 076 India
SUDARSHAN R. GURJAR
Affiliation:
Department of Mathematics Indian Institute of Technology Bombay Powai, Mumbai, 400 076 India sgurjar@math.iitb.ac.in
SOUMYADIP THANDAR
Affiliation:
School of Mathematics Tata Institute of Fundamental Research Mumbai Mumbai, 400 005 India stsoumyadip@gmail.com
*
rv.gurjar50@gmail.com
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Abstract

In this article, we discuss the topology of varieties over $\mathbb {C}$, viz., their homology and homotopy groups. We show that the fundamental group of a quasi-projective variety has negative deficiency under a certain hypothesis on its second homology and therefore a large class of groups cannot arise as fundamental groups of varieties. For a smooth projective surface admitting a fibration over a curve, we give a detailed analysis of the homology and homotopy groups of their universal cover via a case-by-case analysis, depending on the nature of the singular fibers. For smooth, projective surfaces whose universal cover is holomorphically convex (conjecturally always true), we show that the second and third homotopy groups are free abelian, often of infinite rank.

Keywords

quasi-projective varietyhomotopy groupsholomorphic convexity

MSC classification

Primary: 14F35: Homotopy theory; fundamental groups
Secondary: 14F45: Topological properties 14J10: Families, moduli, classification 32E05: Holomorphically convex complex spaces, reduction theory

Information

Type
Article
Information
Nagoya Mathematical Journal , Volume 258 , June 2025 , pp. 376 - 397
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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