Jacobian variety
Lefschetz's original proof worked on projective surfaces and used normal functions, which were introduced by Poincaré. Suppose that "C""t" is a pencil of curves on "X". Each of these curves has a Jacobian variety "JC""t" (if a curve is singular, there is an appropriate generalized Jacobian variety). These can be assembled into a family formula_1, the Jacobian of the pencil, which comes with a projection map π to the base "T" of the pencil. A normal function is a (holomorphic) section of π.