A note on the characteristic $p$ nonabelian Hodge theory in the   geometric case
  We provide a construction of associating a de Rham subbundle to a Higgs subbundle in characteristic $p$ in the geometric case. As applications, we obtain a Higgs semistability result and a $W_2$-unliftable result. 
Mao Sheng, Xin He and Kang Zuo
math.AG