For a hyperkähler manifold of dimension , Huybrechts showed that there are constants such that

 

 

for any line bundle $L$ on , where is the Beauville--Bogomolov--Fujiki quadratic form of . Here the polynomial is called the Riemann--Roch polynomial of .

 

In this talk, I will discuss recent progress on the positivity of coefficients of the Riemann--Roch polynomial and also positivity of Todd classes. Such positivity results follows from a Lefschetz-type decomposition of the root of Todd genus via the Rozansky—Witten theory, following the ideas of Hitchin, Sawon, and Nieper-Wißkirchen.