**Authors**: David Alfaya and Tomás L. Gómez (ICMAT-CSIC)

**Source**: *Advances in Mathematics* vol. 393

**Date of publication**: 24 December 2021

**Review**:

Let *X* be a complex smooth projective surface (equivalently, a compact Riemann surface), and let *D* be a finite set of points of *X*. A parabolic bundle is a vector bundle *E *on *X* together with a weighted filtration of the fibre of *E* over each point *x* in *D*. By a weighted filtration we mean a filtration by subspaces such that each subspace has associated certain real number called weight. Parabolic bundles were introduced by Seshadri (1977), and they are related to representations of the fundamental group of the punctured Riemann surface with fixed holonomy around the points in *D*.

We associate to the pair (*X*,*D*) a moduli space of parabolic vector bundles of fixed rank *r*, and the present paper studies certain geometric aspects of this moduli space. We first prove a Torelli type theorem for this moduli space. This means that, if *M* and *M’* are the moduli spaces associated to (*X*, *D*) and (*X’*, *D’*), and if *M* and *M’* are isomorphic, then there is an isomorphism between *X* and *X’* which sends *D* to *D’*. In other words, we can recover the curve *X* and the points *D* from the isomorphism class of the moduli space as an algebraic variety.

This was known by del Baño, Balaji and Biswas (2001) only for rank *r *= 2, degree 1 and small parabolic weights, but here it is proved in general.

Then the authors calculate the automorphism group of the moduli space *M*. The case where there is no parabolic structure was settled by Kouvidakis and Pantev (1995). In the situation considered, the parabolic structure produces new maps (defined using Hecke transforms). These maps might have singularities, so instead of morphisms they define 3-rational maps (this means a map which fails to be defined on a subset of codimension at least 3). This is due to the fact that the Hecke transform might not preserve the condition of stability (which depends on the parabolic weights) and it is a phenomenon which also happens for other types of transformations of parabolic bundles. Motivated by this, the researchers also calculate the group of 3-birational transformations.