# Projective variety

A projective variety is a **projective curve** if its dimension is one; it is a **projective surface** if its dimension is two; it is a **projective hypersurface** if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial.

If *X* is a projective variety defined by a homogeneous prime ideal *I*, then the quotient ring

is called the homogeneous coordinate ring of *X*. Basic invariants of *X* such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring.

Projective varieties arise in many ways. They are complete, which roughly can be expressed by saying that there are no points "missing". The converse is not true in general, but Chow's lemma describes the close relation of these two notions. Showing that a variety is projective is done by studying line bundles or divisors on *X*.

A particularly rich theory, reaching back to the classics, is available for complex projective varieties, i.e., when the polynomials defining *X* have complex coefficients. Broadly, the GAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. For example, the theory 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.

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

An equivalent but streamlined construction is given by the Proj construction, which is an analog of the spectrum of a ring, denoted "Spec", which defines an affine scheme.^{[5]} For example, if *A* is a ring, then

We can give a coordinate-free analog of the above. Namely, given a finite-dimensional vector space *V* over *k*, we let

Projective space over any scheme *S* can be defined as a fiber product of schemes

A scheme *X* → *S* is called **projective** over *S* if it factors as a closed immersion

By definition, a variety is complete, if it is proper over *k*. The valuative criterion of properness expresses the intuition that in a proper variety, there are no points "missing".

There is a close relation between complete and projective varieties: on the one hand, projective space and therefore any projective variety is complete. The converse is not true in general. However:

Some properties of a projective variety follow from completeness. For example,

for any projective variety *X* over *k*.^{[10]} This fact is an algebraic analogue of Liouville's theorem (any holomorphic function on a connected compact complex manifold is constant). In fact, the similarity between complex analytic geometry and algebraic geometry on complex projective varieties goes much further than this, as is explained below.

Quasi-projective varieties are, by definition, those which are open subvarieties of projective varieties. This class of varieties includes affine varieties. Affine varieties are almost never complete (or projective). In fact, a projective subvariety of an affine variety must have dimension zero. This is because only the constants are globally regular functions on a projective variety.

The product of two projective spaces is projective. In fact, there is the explicit immersion (called Segre embedding)

As the prime ideal *P* defining a projective variety *X* is homogeneous, the homogeneous coordinate ring

is a graded ring, i.e., can be expressed as the direct sum of its graded components:

If the homogeneous coordinate ring *R* is an integrally closed domain, then the projective variety *X* is said to be projectively normal. Note, unlike normality, projective normality depends on *R*, the embedding of *X* into a projective space. The normalization of a projective variety is projective; in fact, it's the Proj of the integral closure of some homogeneous coordinate ring of *X*.

where *d* is the dimension of *X* and *H*_{i}'s are hyperplanes in "general positions". This definition corresponds to an intuitive idea of a degree. Indeed, if *X* is a hypersurface, then the degree of *X* is the degree of the homogeneous polynomial defining *X*. The "general positions" can be made precise, for example, by intersection theory; one requires that the intersection is proper and that the multiplicities of irreducible components are all one.

The other definition, which is mentioned in the previous section, is that the degree of *X* is the leading coefficient of the Hilbert polynomial of *X* times (dim *X*)!. Geometrically, this definition means that the degree of *X* is the multiplicity of the vertex of the affine cone over *X*.^{[12]}

where *Z _{i}* are the irreducible components of the scheme-theoretic intersection of

*X*and

*H*with multiplicity (length of the local ring)

*m*.

_{i}A complex projective variety can be viewed as a compact complex manifold; the degree of the variety (relative to the embedding) is then the volume of the variety as a manifold with respect to the metric inherited from the ambient complex projective space. A complex projective variety can be characterized as a minimizer of the volume (in a sense).

Let *X* be a projective variety and *L* a line bundle on it. Then the graded ring

is called the canonical ring of *X*. If the canonical ring is finitely generated, then Proj of the ring is called the canonical model of *X*. The canonical ring or model can then be used to define the Kodaira dimension of *X*.

Projective schemes of dimension one are called *projective curves*. Much of the theory of projective curves is about smooth projective curves, since the singularities of curves can be resolved by normalization, which consists in taking locally the integral closure of the ring of regular functions. Smooth projective curves are isomorphic if and only if their function fields are isomorphic. The study of finite extensions of

This result is the projective analog of Noether's normalization lemma. (In fact, it yields a geometric proof of the normalization lemma.)

determined by the linear system *V*, where *B*, called the base locus, is the intersection of the divisors of zero of nonzero sections in *V* (see for the construction of the map).

Let *X* be a smooth projective variety where all of its irreducible components have dimension *n*. In this situation, the canonical sheaf ω_{X}, defined as the sheaf of Kähler differentials of top degree (i.e., algebraic *n*-forms), is a line bundle.

Serre duality is also a key ingredient in the proof of the Riemann–Roch theorem. Since *X* is smooth, there is an isomorphism of groups

which can be readily proved.^{[25]} A generalization of the Riemann-Roch theorem to higher dimension is the Hirzebruch-Riemann-Roch theorem, as well as the far-reaching Grothendieck-Riemann-Roch theorem.

Chow's theorem provides a striking way to go the other way, from analytic to algebraic geometry. It states that every analytic subvariety of a complex projective space is algebraic. The theorem may be interpreted to saying that a holomorphic function satisfying certain growth condition is necessarily algebraic: "projective" provides this growth condition. One can deduce from the theorem the following:

Chow's theorem can be shown via Serre's GAGA principle. Its main theorem states:

and are also referred to as complex tori. Here, *g* is the dimension of the torus and *L* is a lattice (also referred to as period lattice).

For higher dimensions, the notions of complex abelian varieties and complex tori differ: only polarized complex tori come from abelian varieties.