International Congress of Mathematicians

Update on 3-folds 515in the hyperbolic plane, the perimeter of a disc of radius r would be 27r(sinh jj)F,bigger than the Euclidean value, and growing exponentially with r.Riemann in particular generalised Gauss' ideas on surfaces to a space given locallyby an n-tuple (xi,..., x n ) of real parameters (a "many-fold extended quantity"or manifold), with distance arising from a local arc length ds given by a quadraticform ds 2 = ^gijdxidxj. The curvature is then a function of the second derivativesof the metric function #y. Riemann's differential geometry works with manifoldsthat are not homogeneous, e.g., having positive, zero, or negative curvature at differentpoints. It was a key ingredient in Einstein's general relativity (1915), whichtreats gravitation as curvature of space-time.1.3. Riemann surfacesThe story moves on from real manifolds (e.g., surfaces depending on 2 realvariables) to Riemann surfaces, parametrised instead by a single complex variable.The point here is Cauchy's discovery (c. 1815) that differentiable functions of acomplex variable are better behaved than real functions, and much more amenableto algebraic treatment. Riemann discovered that a compact Riemann surface C hasan embedding C ^y P^ into complex projective space whose image is defined by aset of homogeneous polynomial equations.A projective algebraic curve C C P^ is nonsingular if at every point P £ Cwe can choose N — 1 local equations fi,..., /jv-i so that the Jacobian matrix -?^fihas maximal rank N — 1. It follows from the implicit function theorem that oneof the linear coordinates z = zi of f N can be chosen as a local analytic coordinateon C. In other words, a compact Riemann surface is analytically isomorphic to anonsingular complex projective curve.1.4. The genus of an algebraic curveThe canonical class Kc = û^ = T c of a curve C is the holomorphic linebundle of 1-forms on C; it has transition functions on U n U' the Jacobian of thecoordinate change ^-, where z,z' are local analytic coordinates on U,U'. If z is arational function on C that is an analytic coordinate on an open set U C C then a1-form on U is f(z)dz with / a regular function on U. That is, ii c = Ö • dz, or dzis a basis of ii c on U.The genus g(C) can be defined in several ways: topologically, a compact Riemannsurface is a sphere with g handles (see Figure 3). It has Euler numberg = 0, sphere 9=1, torus g > 2, general typeFigure 3: The genus of a Riemann surfacee(C) = 2 — 2g, which equals deg To- The most useful formula for our purpose is