Jeehoon Park (Postech)

Period Integrals of Smooth Projective Hypersurfaces and $L_{\infty}$-Homotopy Theory

We provide a new point of view of understanding period integrals of differential forms on a smooth projective hypersurface in terms of $L_{\infty}$-homotopy theory and representation theory of Lie algebra. In order for that, we define a Lie algebra representation attached to a projective smooth hypersurface and a cochain complex associated to this representation. Then we show that the Griffiths' period integral can be understood as both a \textit{period integral} of this Lie algebra representation and a cochain map of certain cochain complexes whose interpretation leads us to $L_{\infty}$-homotopy theory. Consequently we verify that period integrals of smooth hypersurfaces are invariants of homotopy types of $L_{\infty}$-morphisms. In fact, the theory is more general; we define a notion of \textit{period integrals} of a Lie algebra representation and propose a general strategy of studying them via $L_{\infty}$-homotopy theory, when the representation space has an associative and commutative algebra structure. Then period integrals of a smooth projective hypersurface can be viewed as an explicit example of our general theory. This is a joint work with Jae-Suk Park.