to understand curves, first study the abel map. and then the torelli map. [perhaps I should expand this rather succinct answer.]
8320 Spring 2010, day one Introduction to Riemann Surfaces
We will describe how Riemann used topology and complex analysis to study algebraic
curves over the complex numbers. [The main tools and results have analogs in
arithmetic, which I hope are more easily understood after seeing the original versions.]
The idea is that an algebraic curve C, say in the plane, is the image by a holomorphic
map, of an abstract complex manifold, the Riemann surface X of the curve, where X has
an intrinsic complex structure independent of its representation in the plane. Then Riemann's approach to classifying all complex curves, is to classify such Riemann surfaces, and then for each such surface to classify all maps from it to projective space. Briefly, the Torelli maps classifies complex surfaces, and the abel maps classify all projective models of a given complex surface.
More precisely, we will construct two fundamental functors of an algebraic curve:
i) the Riemann surface X, and
ii) the Jacobian variety J(X), and
natural transformations X^(d)--->J(X), the Abel maps, from the “symmetric powers” X^(d) of X, to J(X).
The Riemann surface X
The first construction is the Riemann surface of a plane curve:
{irreducible plane curves C: f(x,y)=0} ---> {compact Riemann surfaces X}.
The first step is to compactify the affine curve C: f(x,y) =0 in A^2, the affine complex
plane, by taking its closure in the complex projective plane P^2. Then one separates
intersection points of C to obtain a smooth compact surface X. X inherits a complex
structure from the coordinate functions of the plane. If f is an irreducible polynomial, X
will be connected. Then X will have a topological genus g, and a complex structure, and
will be equipped with a holomorphic map ƒ:X--->C of degree one, i.e. ƒ will be an
isomorphism except over points where the curve C is not smooth, e.g. where C crosses
itself or has a pinch.
This analytic version X of the curve C retains algebraic information about C, e.g. the field
M(X) of meromorphic functions on X is isomorphic to the field Rat(C) of rational
functions on C, the quotient field k[x,y]/(f), where k = complex number field. It turns out
that two curves have isomorphic Riemann surfaces if and only if their fields of rational
functions are isomorphic, if and only if the curves are equivalent under maps defined by
mutually inverse pairs of rational functions.
Since the map X--->C is determined by the functions (x,y) on X, which generate the field Rat(C), classifying algebraic curves up to “birational equivalence” becomes the question of classifying these function fields, and classifying pairs of generators for each field, but Riemann’s approach to this algebraic problem will be topological/analytic.
We already can deduce that two curves cannot be birationally equivalent unless their Riemann surfaces have the same genus. This solves the problem that interested the Bernoullis as to why most integrals of form dx/sqrt(cubic in x) cannot be “rationalized” by rational substitutions. I.e. only curves of genus zero can be so rationalized and y^2 = (cubic in x) usually has positive genus.
The symmetric powers X^(d)
To recover C, we seek to encode the map ƒ:X--->C, i.e. ƒ:X--->P^2, by intrinsic
geometric data on X. If the polynomial f defining C has degree d, then each line L in the
plane P^2 meets C in d points, counted properly. Thus we get an unordered d tuple of
points L.C, possibly with repetitions, on C, hence when pulled back via ƒ, we get such a d
tuple called a positive “divisor” D = ƒ^(-1)(L) of degree d on X. (D = n1p1+...nk pk,
where nj are positive integers, n1+...nk = d.)
Since lines L in the plane move in a linear space dual to the plane, and (if d ≥ 2) each line is spanned by the points where it meets C, we get an injection P^2*--->{unordered d tuples of points of X}, taking L to ƒ^(-1)(L).
If X^d is the d - fold Cartesian product of X, and Sym(d) is the symmetric group of
permutations of d objects, and we define X^(d) = X^d/Sym(d) = the “symmetric product”
of X, d times, then the symmetric product X^(d) parametrizes unordered d tuples, and
inherits a complex structure as well.
Thus the map ƒ:X--->C yields a holomorphic injection P^2*--->Π of the projective plane into X^(d). I.e. the map ƒ determines a complex subvariety of X^(d) isomorphic to a linear space Π ≈ P^2*. Now conversely, this “linear system” Π of divisors of degree d on X determines the map ƒ back again as follows:
Define ƒ:X--->Π* = P^2** =P^2, by setting ƒ(p) = the line in Π consisting of those
divisors D that contain p. Then this determines the point ƒ(p) on C in P^2, because a
point in the plane is determined by the lines through that point. [draw picture]
Thus the problem becomes one of determining when the product X^(d) contains a
holomorphic copy of P^2, or copies of P^n for models of X in other projective spaces.
The Jacobian variety J(X) and the Abel map X^(d)--->J(X).
For this problem, Riemann introduced a second functor the “Jacobian” variety J(X) =
k^g/lattice, where k^g complex g -dimensional space. J(X) is a compact g dimensional
complex group, and there is a natural holomorphic map Abel:X^(d)--->J(X), defined by
integrating a basis of the holomorphic differential forms on X over paths in X.
Abel collapses each linear system Π ≈ P^n* to a point by the maximum principle, since the
coordinate functions of k^g have a local maximum on the compact simply connected
variety Π. Conversely, each fiber of the Abel map is a linear system in X^(d).
Existence of linear systems Π on X: the Riemann - Roch theorem.
By dimension theory of holomorphic maps, every fiber of the abel map X^(d)--->J(X)
has dimension ≥ d-g. Hence every positive divisor D of degree d on X is contained in a
maximal linear system |D| , where dim|D| ≥ d-g. This is called Riemann’s inequality, or
the “weak” Riemann Roch theorem.
The Roch part analyzes the relation between D and the divisor of a differential form to
compute dim|D| more precisely. Note if D is the divisor cut by one line in the plane of C,
and E is cut by another line, then E belongs to |D|, and the difference E-D is the divisor of the meromorphic function defined by the quotient of the linear equations for the two
lines.
If D is a not necessarily positive divisor, we define |D| to consist of those positive divisors E such that E-D is the divisor of a meromorphic function on X. If there are no such positive divisors, |D| is empty and has “dimension” equal to -1. Then if K is the divisor of zeroes of a holomorphic differential form on X, the full Riemann Roch
theorem says:
dim|D| = d-g +1+dim|K-D|, where the right side = d-g when d > deg(K).
This sketch describes the abel maps and their relation to the RRT. The assignment X-->J(X) is the Torelli map, and classifies X by the numerical data in the lattice defining J(X), i.e. periods of integrals of the first kind on X. This assignment gives birth to the whole subject of "moduli" as numerical invariants of complex or geometric structure.a
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