# Colloquium abstracts

**Anoop Singh**

TIFR, Mumbai

November 5, 2020

**Relative holomorphic connections and moduli space of logarithmic connections singular over a finite**: In this talk we discuss two problems. It is firstly about the relative holomorphic connections and we give a sufficient condition for the existence of relative holomorphic connections in a vector bundle over a complex analytic family of compact connected complex manifolds. we show that the relative Chern classes of a holomorphic vector bundle over a family of compact and K"ahler manifolds vanish if the bundle admits a relative holomorphic connection. Secondly, we give a description of certain invariants of the moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface with fixed residues. This moduli space is known to be quasi-projective variety. We compute the Picard group of the moduli space and show that the moduli space does not admit any non-constant algebraic functions, although it admits non-constant holomorphic functions.