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July, 2018 On the fundamental group of a smooth projective surface with a finite group of automorphisms
Rajendra Vasant GURJAR, Bangere P. PURNAPRAJNA
J. Math. Soc. Japan 70(3): 953-974 (July, 2018). DOI: 10.2969/jmsj/73567356


In this article we prove new results on fundamental groups for some classes of fibered smooth projective algebraic surfaces with a finite group of automorphisms. The methods actually compute the fundamental groups of the surfaces under study upto finite index. The corollaries include an affirmative answer to Shafarevich conjecture on holomorphic convexity, Nori’s well-known question on fundamental groups and free abelianness of second homotopy groups for these surfaces. We also prove a theorem that bounds the multiplicity of the multiple fibers of a fibration for any algebraic surface with a finite group of automorphisms $G$ in terms of the multiplicities of the induced fibration on $X/G$. If $X/G$ is a $\mathbb{P}^1$-fibration, we show that the multiplicity actually divides $|G|$. This theorem on multiplicity, which is of independent interest, plays an important role in our theorems.

Funding Statement

The research was partially supported by NSF grant 1206434.


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Rajendra Vasant GURJAR. Bangere P. PURNAPRAJNA. "On the fundamental group of a smooth projective surface with a finite group of automorphisms." J. Math. Soc. Japan 70 (3) 953 - 974, July, 2018.


Received: 2 November 2015; Revised: 2 December 2016; Published: July, 2018
First available in Project Euclid: 12 June 2018

zbMATH: 06966968
MathSciNet: MR3830793
Digital Object Identifier: 10.2969/jmsj/73567356

Primary: 14F35
Secondary: 14D05 , 14H30 , 14J29 , 14J50 , 14L30

Keywords: base change , fibrations , finite group actions on varieties , fundamental groups of algebraic surfaces , holomorphic convexity , Shafarevich conjecture , surfaces of general type

Rights: Copyright © 2018 Mathematical Society of Japan


Vol.70 • No. 3 • July, 2018
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