Following J. Stelzig and A. Milivojević’s introduction of the notion of Strong Formality for complex manifolds (a bigraded extension of the already well-known concept of formality for manifolds), it has been natural to ask which classes of complex manifolds are strongly formal. This question has become even more relevant after G. Placini, J. Stelzig, and L. Zoller have recently shown the existence of compact Kähler manifolds that are not strongly formal, thus marking a notable difference from the classical case (indeed, in 1975 P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan proved that all compact Kähler manifolds are formal).

In this talk, starting from an appropriate introduction to the the notion of formality, I will discuss the Strong Formality of a particular class of complex manifolds, namely compact Hyperkähler manifolds, aiming to highlight the role analogies between Hyperkähler manifolds/strong formality and Kähler manifolds/classical formality.