# Very stable bundles and properness of the Hitchin map

Research paper by Christian Pauly, Ana Peón-Nieto

Indexed on: 01 Mar '18Published on: 27 Feb '18Published in: Geometriae Dedicata

#### Abstract

Let X be a smooth complex projective curve of genus $$g\ge 2$$ and let K be its canonical bundle. In this note we show that a stable vector bundle E on X is very stable, i.e. E has no non-zero nilpotent Higgs field, if and only if the restriction of the Hitchin map to the vector space of Higgs fields $$H^0(X, \mathrm {End}(E) \otimes K)$$ is a proper map.