Weighted complete intersections

In the talk we observe what is known about complete intersections in weighted projective spaces. We mainly focus on the case of smooth Fano varieties. We discuss which standard theorems for usual complete intersections hold in the weighted case and present boundness results in the smooth Fano case. We study Hodge diamonds of weighted complete intersections, extremal, from this point of view, cases, and their connections with semiorthogonal decompositions of derived categories. If time permits we discuss Mirror Symmetry for weighted complete intersections and connections of invariants of their Landau–Ginzburg models with invariants of the initial varieties.