Thank you. You are so kind. I know Yau must have some specific motivations in his mind. I guess “cut out by a set of algebraic equations” must mean “polynomial calculation time” for him. Now he’s talking about embedding an elliptic curve in to R^3, which of course would make Wiles horrified: “Where is the Modell-Weil group, man?”

But seriously, for a Riemann surface to explicitly find a set of algebraic equations to cut it out, is a difficult problem. For example let’s assume the Riemann surface is the canonical curve in P^{g-1}, which is defined by an canonical ideal. This ideal then by Hilbert’s syzygy theorem has a resolution, this means we can have a finite set of generators (i.e. equations for the curve), and a finite set of relations among the generators, and relations on relations, and so on. There is a well-known conjecture by your colleague Mark Green, about 25 years ago, which asserts this resolution is determined by whether the Riemann surface pocesses certain special divisors. But even this conjecture does not tell how to find these explicit set of equations.

Fano manifolds are projective and therefore they admid a Kahler metric.
I believe that the conjecture is wether they admit a Kahler-Einstein metric.

From what I understand of Yau’s talk, it is the geometric structure which is generating the torsion-free connection on some canonical bundle. This structure might be a Riemannian metric (in which case the connection is the Levi-Civita connection on the tangent bundle, which is torsion-free by definition), or it might be a preferred coordinate system in which the coordinate transformations between charts are linear, affine or projective (in which case the tangent or affine bundle locally trivialises to a flat bundle on Euclidean space, and one uses the standard connection for that bundle, which is clearly torsion-free by Clairaut’s theorem and is also independent of the chart by the assumption on the coordinate transformations). The affine Kahler manifolds that Yau discussed in his talk are of this latter type.

The other example Yau mentioned was that of complex structures, i.e. local complex coordinate charts of an almost complex manifold. Here the analogue of the torsion tensor would be the Nijenhuis tensor. I presume that this tensor can indeed be interpreted as the torsion of some natural connection on some natural bundle (perhaps the complexified tangent bundle?), which is manifestly torsion-free when local complex coordinates exist, but I don’t have the details.

By affine bundle, I think Yau means the direct sum of the tangent bundle and the trivial R bundle, thus sections of the affine bundle are formal sums of a vector field and a scalar field.

Dear sz: I’ll ask Yau tomorrow about Andreotti’s construction. I gather that Yau was for some reason particularly interested in constructing representatives of a given conformal class of Riemann surfaces as embedded surfaces in R^3 rather than P^{g-1}, though I don’t exactly know why.

Your comment

“Torelli’s theorem abstractly shows that a surface is determined by the periods of its holomorphic differential forms, but does not show how to explicitly construct a concrete surface from these periods.”

is not right, such explicit construction do exist, it’s given by Aldo Andreotti in his 1958 paper “On a theorem of Torelli”. Roughly the periods determines an abelian variety, the Jacobians J of the Riemann surface, and integration along a based path gives the Abel embedding X->J, whose (g-1) iteration defines the theta divisor \Theta. The Gauss map (since the Jacobian is flat) defined by this theta divisor has an image which is the dual of the canonical curve in P^{g-1}. From this we can reconstruct the original Riemann surface.

Such explicit constructive Torelli however is completely missing for higher dimensions, notably for K3 surfaces for example, whose Torelli theorem is proved very indirectly.

Some necessary conditions can be extracted from looking at the flat connection on the relevant bundle (usually the tangent bundle, though in affine or projective geometry it is the affine bundle which is important) induced from the special coordinate charts

What connection are you talking about? Levi-Civita? If so, at what point did you admit a metric? And surely it is not flat in general… In the case when a connection is flat, the holonomy just detects the fundamental group, but in general, it is sensitive to curvature also…

Another more “local” necessary condition, which arises ultimately from Clairaut’s theorem from undergraduate calculus, is that the connection needs to be torsion-free if it is to arise from a special coordinate system.

What is the definition of the “affine bundle”? In particular, I believed that in the absense of any other geometric structure enabling identification of bundles, that torsion can only be defined for a connection on the tangent bundle. Presumably you are claiming that a projective structure imposes such an identification?