of holomorphic vector bundles (more generally coherent analytic sheaves) on X coincide with that of algebraic vector bundles. Chow's theorem says that a subset of projective space is the zero-locus of a family of holomorphic functions if and only if it is the zero-locus of homogeneous polynomials. The combination of analytic and algebraic methods for complex projective varieties lead to areas such as Hodge theory. Variety and scheme structure Variety structure Let k be an algebraically closed field. The basis of the definition of projective varieties is projective space , which can be defined in different, but equivalent ways: as the set of all lines through the origin in (i.e., all one-dimensional vector subspaces of ) as the set of tuples , with not all zero, modulo the equivalence relation for any . The equivalence class of such a tuple is denoted by This equivalence class is the general point of projective space. The numbers are referred to as the homogeneous coordinates of the point. A projective variety is, by definition, a closed subvariety of , where closed refers to the Zariski topology. In general, closed subsets of the Zariski topology are defined to be the common zero-locus of a finite collection of homogeneous polynomial functions. Given a polynomial , the condition does not make sense for arbitrary polynomials, but only if f is homogeneous, i.e., the degrees of all the monomials (whose sum is f) are the same. In this case, the vanishing of is independent of the choice of . Therefore, projective varieties arise from homogeneous prime ideals I of , and setting Moreover, the projective variety X is an algebraic variety, meaning that it is covered by open affine subvarieties and satisfies the separation axiom. Thus, the local study of X (e.g., singularity) reduces to that of an affine variety. The explicit structure is as follows. The projective space is covered by the standard open affine charts which themselves are affine n-spaces with the coordinate ring Say i = 0 for the notational simplicity and drop the superscript (0). Then is a closed subvariety of defined by the ideal of generated by for all f in I. Thus, X is an algebraic variety covered by (n+1) open affine charts . Note that X is the closure of the affine variety in . Conversely, starting from some closed (affine) variety , the closure of V in is the