## Journal of the Mathematical Society of Japan

### On the fundamental group of a smooth projective surface with a finite group of automorphisms

#### Abstract

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.

#### Note

The research was partially supported by NSF grant 1206434.

#### Article information

Source
J. Math. Soc. Japan, Volume 70, Number 3 (2018), 953-974.

Dates
Revised: 2 December 2016
First available in Project Euclid: 12 June 2018

https://projecteuclid.org/euclid.jmsj/1528790543

Digital Object Identifier
doi:10.2969/jmsj/73567356

Mathematical Reviews number (MathSciNet)
MR3830793

Zentralblatt MATH identifier
06966968

#### Citation

GURJAR, Rajendra Vasant; PURNAPRAJNA, Bangere P. On the fundamental group of a smooth projective surface with a finite group of automorphisms. J. Math. Soc. Japan 70 (2018), no. 3, 953--974. doi:10.2969/jmsj/73567356. https://projecteuclid.org/euclid.jmsj/1528790543

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