## Journal of the Mathematical Society of Japan

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

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

Rajendra Vasant GURJAR and Bangere P. PURNAPRAJNA

#### 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**

Received: 2 November 2015

Revised: 2 December 2016

First available in Project Euclid: 12 June 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/jmsj/73567356

**Mathematical Reviews number (MathSciNet)**

MR3830793

**Zentralblatt MATH identifier**

06966968

**Subjects**

Primary: 14F35: Homotopy theory; fundamental groups [See also 14H30]

Secondary: 14J29: Surfaces of general type 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 14H30: Coverings, fundamental group [See also 14E20, 14F35] 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 14J50: Automorphisms of surfaces and higher-dimensional varieties

**Keywords**

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

#### 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