and L is a lattice (also referred to as period lattice). According to the uniformization theorem already mentioned above, any torus of dimension 1 arises from an abelian variety of dimension 1, i.e., from an elliptic curve. In fact, the Weierstrass's elliptic function attached to L satisfies a certain differential equation and as a consequence it defines a closed immersion: There is a p-adic analog, the p-adic uniformization theorem. For higher dimensions, the notions of complex abelian varieties and complex tori differ: only polarized complex tori come from abelian varieties. Kodaira vanishing The fundamental Kodaira vanishing theorem states that for an ample line bundle on a smooth projective variety X over a field of characteristic zero, for i > 0, or, equivalently by Serre duality for i < n. The first proof of this theorem used analytic methods of Kähler geometry, but a purely algebraic proof was found later. The Kodaira vanishing in general fails for a smooth projective variety in positive characteristic. Kodaira's theorem is one of various vanishing theorems, which give criteria for higher sheaf cohomologies to vanish. Since the Euler characteristic of a sheaf (see above) is often more manageable than individual cohomology groups, this often has important consequences about the geometry of projective varieties. Related notions Multi-projective variety Weighted projective variety, a closed subvariety of a weighted projective space See also Algebraic geometry of projective spaces Adequate equivalence relation Hilbert scheme Lefschetz hyperplane theorem Minimal model program Notes References R. Vakil, Foundations Of Algebraic Geometry External links The Hilbert Scheme by Charles Siegel - a blog post varieties Ch. 1 Algebraic geometry Algebraic varieties Projective geometry