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I am taking the liberty of collecting mathwonk's "short course" for some followup comments/questions, since this topic is IMHO more interesting than the context in which it first appeared. (Hope this is OK under PF rules!).
Part I:
Part II:
Part III:
How annoying, Part IV won't fit (is there really a 20000 word limit on the size of a post?), TBC...
Part I:
mathwonk said:here is a little crash course in the heat equation and its use in algebraic geometry.
it has long been known that a cubic curve X in the plane has the structure of a group. this is essentially because any two points determine a line, which meets the curve again in a third point, which determines the sum of the first two.
more topologically, the complex points on a smooth plane curve form a torus, or doughnut with one hole, as you can sort of see by looking at the simplest cubic, a triangle.
Now one can see that a torus can be made into a group as follows: take the complex plane and set equal to zero all points which are linear integral combinations of two vectors with different directions, say 1 and i. I.e. C is a group and {n + mi, for all n,m, in Z} is a subgroup and you take the quotient group C/{n+mi}, which as a group is a product of two circles.
topologically it is also the product of two circles, since it formed by gluing the opposite edges of the parallelogram formed by 0, 1, i, and 1+i, hence a torus. using the weierstrass P function and its derivative, one can embed this torus in the complex plane as a cubic. thus any lattice defines a plane cubic.
now riemann or abel or someone back there, showed how to go backwards: i.e. given a complex plane cubic X, it inherits a complex and topological structure from the complex plane C^2, in which it lies, at least once it is compactifed at infinity, and hence it has two independent loops on it, one for each circle, i.e. a "homology basis" in fancy language, called say u and v.
There is also a single holomorphic differential dz, which is well defined on the torus, even though the coordinate z is not, because z is well defined up to translation by an element of the lattice {n + mi} and d of a constanT TRANSLATE IS ZERO.
so we get two complex numbers A and B by path integrating dz around u and around v, and Riemann showed these are independent complex numbers hence give a lattice {nA+mB} in C, which then determines a torus group C/{nA+mB}, which in fact is both analytically and group theoretically isomorphic to the original plane cubic X.
Now where does the heat equation come in? well first riemann showed one could normalize the complex generators A,B of the plane lattice so that one of them is always A = 1, and the other B = t, has positive imaginary part.
then one can write down a Fourier series using t which defines a "theta function". f(z,t). i.e. one first gives a quadratic non homogeneous polynomial with linear coefficient z and quadratic coefficient t, and then exponentiates it, and sums over all integer arguments. (see mumford's tata lectures on theta, where he credits me for this description, but i originally learned it from c.l.siegel.)
this gives one a function of the two variables z and t. we think of t as determining the complex structure of the curve (since from t, one can reconstruct the curve as C/{n+mt}), and z as a coordinate on the curve it self.
for fixed t, i.e. fixing the curve, the theta function is a function of z, hence on the curve, which is not well defined, since it is not doubly periodic, but its zero set is doubly periodic so it defines a well determiend zero locus on the curve which is only one point.
so we have a theta function f(t,z), a function of two complex variables (t,z, where t is thught of as determining a complex torus, and z as a point on the torus.
If in the product CxC with coiordinates (t,z) we mod out by the family of alttices {n+mt}, we get a family of tori, one over each point t, and aglobal theta function whose zeroes determine one point no each torus.
the t line is a moduli space for 1 dimensional tori, and over each number t, we have a copy if the corresponding torus and a distinguished point.
as you may know, this theta function is a characteristic solution of the heat equation, so that pde must contain some useful informaton about curves.
Part II:
mathwonk said:This really comes into its own in higher dimensions and genera. I.e. Riemann generalized this construction to assign a group to each curve of any genus > 0, as follows: he proved a curve of genus g, i.e. a doughnut with g holes, has g independent holomorphic differentials w1,...,wg, and a homology basis of 2g loops u1,...ug, v1,...vg, and thus determines a g by 2g matrix of path integrals [A, B]. he showed one can again normalize the bases wi and ui,vj, so that the matirx contains a gbyg identity matrix, and another gbyg complex matrix t, with pos. def. imaginary part, i.e. [I, t].
then he wrote down "riemanns theta function" f(t,z) of g complex variables z, and apparently g^2 complex variables t, and if one mods out C^g by the lattice of semi periods i.e. by n + mt, where now n,m are integer g-vectors, one gets a complex g dimensional torus C^g/{nI+mt].
HE ALSO SHOWED THAT THE period matrix t is symmetric so there are really only (g)(g+1)/2 variables t. thus the riemann theta function is a holomorphic function on the product space of points (z,t) in C^g x C^(gxg). Again we can mod out this product to form a family of complex tori, and the theta function determines a family of hyperurfaces, one in each torus. these hypersurfaces are called theta divisors.
Now the inverse problem above is of interest. I.e. given a g diemnsional complex torus, when does it arise as above from a genus g complex curve? This is called the Schottky problem. presumably if so, it should be visible from looking at the theta divisor of the corresponding torus.
Now curves depend on 3g-3 parameters, so In genera 1,2, and 3, essentially all "indecomposable" tori do arise from curves, but in genus 4, curves only have 9 parameters and 4 dimensional complex tori have (4)(5)/2 = 10.
so there is one condition that should specify whether or not a complex 4- torus comes from a genus 4 curve. Riemann shoiwed that tori coming from curves in fact have "singular" theta divisors, i.e. if the torus comes from a curve, there is a kink or node on the theta divisor. This raises the opposite question, do all 4-tori with singular theta divisors come form curves? (those which do are called jacobians, so we are trying to recognize jacobians among all complex tori.)
In his thesis at Columbia, Allan Mayer showed about 1960 that at least locally near a 4 diml jacobian, there is a nbhd where this is true. he did it by observing that jacobians J form a hypersurface of codimension one in the space of all 4 dimensional complex tori, and J is contained in the set N of tori with singular theta divisors, so all he had to do was show that N is also a hyperurface of copdimension one.
But the cauchy data for the heat equation implies that if all theta functions satisfying the ehat equation had singular zero loci, then the theta function would be the identically zero solution of the ebnat equation, and it isnt.
this story goes on. Mayer and Andreotti showed in 1967 that in all genera, jacobians are acomponent of N. then in 1977, Beauville showed that in genus 4, N has exactly one other component, thus completely describing 4 dimensional jacobians geometrically.
More recently Robert Varley and I gave a shorter proof of this corollary of Beauville's more extensive work.
Varley and then i used the ehat equation to show that also in genus 5, N has exactly 2 components, and computed the multiplicity of jacobians J on the correspoing component of N, but did not uniquely specify J there.
if you look at the heat equation you see it equates a second derivative of theta wrt z to a first derivative wrt t. As Andreoti and mayer showed, this gives a geometric relation between tangent directions in the moduli space of tori, with quadratic tangent cones to th theta divisor in the torus itself.
Later Welters gave a completely algebraic proof of this version of the heat equation, so that it makes sense in characteristic p geometry, and Varley and I used that version to generalize a famous result of Mark Green on theta divisors of complex Jacobians, to characteristic p > 2.
thus the heat equation has a completely geometric interpretation that can be used to reason about it, independently of knowing analysis or pde.
Part III:
mathwonk said:The Schottky problem of characterizing Jacobians among all complex algebraic tori, also called abelian varieties, was originally an analytic or algebraic question, that of giving actual equations in some appropriate coordinates, such as the matrices t, for the moduli space of abelian varieties that vanish exactly on jacobians.
The problem was given its impetus 100 years ago by Schottky who wrote down some relations which he proved were indeed satisfied by jacobians, but it was hard to show even that these relation were not identically zero, much less that they vanished only on jacobians.
In the 1970's Igusa annunced he could prove in genus 4, that (the closure of) jacobians was the only component of the zero locus of the one genus 4 Schottky relation, and in about 1981 he wrote down the proof. He used a differential equation satisfied by hyperellipic jacobians, to deduce that every possible component of the Schottky locus must pass through the "boundary" locus of degenerate 4 dimensional abelian varieties, i.e. products of 4 elliptic curves (genus one curves).
Then he only had to count the number of components through that locus, which he did by explicitly computing the tangent cone at that locus and showing it was defined by an irreducible polynomial. Since every component of the Schottky locus must contribute at least one component to the normal cone, the irreducibility of the normal cone implied irreducibility of the Schottky locus.
In dimension 5 Varley and I were trying to show the Andreoti Mayer hypersurface N parametrizing 5 dimensional abelian varieties with singular theta divisor, had just 2 components, as Beauville had shown in dimension 4. So we used a modification of Igusa's idea, namely we showed all possible components of N had to pass through the locus of Jacobians having an "even vanishing theta null", and then we were reduced to finding the number of components of N that did pass through that locus.
Unlike Igusa's case we knew there were at least 2 components so we needed a way to count them. Unlike his case also, the normal tangent cone to this locus had an "multiple" component, i.e. one whose algebraic equation had multiplicity greater than one, which we needed to understand, since that can increase the number of normal cone components over the number of actual discriminant locus components.
The classical study by Lefschetz of moduli of singular hypersurfaces with only isolated singular points had been completed by Teissier and Le. Their theory showed that one could compute the multiplicity of the tangent cone at a point of the moduli variety of singular hypersurfaces, i.e. of the "dscriminant locus", using "Milnor numbers", which are a count of the homology cycles in the hyperurface that vanish into the singularity as the hypersurface acquires a pinch or singularity.
The multiplicity of the discriminant locus at a point corresponding to a hypersurface with finitely many singularities equaled the sum of the Milnor numbers at all singularities. We had to generalize this to the case of infinitely many singularities, i.e. a positive dimensional famnily of singular points.
We showed that in the isolated case the sum of the milnor numbers equaled the change in the global euler characteristic of the hypersurface as it acquired a singularity. This version made sense in the infinite singularities case. I.e. we defined the global milnor number to be the change in the euler characteristic, and then showed that we could meaure the multiplicity of the components of the normal cone by this new global milnor number.
Strangely we got multiplicity 3, at the point where we expected only two components to pass. But an interesting phenomenon for theta divisors that is not true for general hypersurfaces, is that on the component containing jacobians, there are in general two ordinary double pointsof the theta divisors. We could show this even by looking at a Jacobian, where there are infinitely many, because we could look in a normal direction and see that only two singularities persisted in a given normal direction under deformation.
To carry out this calculation, we used the geometric interpretation of the heat equation, to study the geometry of the family formed by the union of all the singular loci of all theta divisors, the so called "critical locus".
Still this only handled components that met the one we knew to contain jacobians, so we had to show in fact all divisors on the moduli space of abelian varities must meet. For this we worked out statement by mumford that the Picard group, was isomorphic to Z, and this could be comoputed from the second cohomology group, which in turn was linked to a group cohomology calculation for the "symplectic group" Sp(2g), one of the famous classical matrix groups defined by the standard symplectic form. It also required some homotopy calculations using postnikov towers that one learns about in algebraic topology.
Finally it followed that in fact there were only two global components to the discriminant locus of singular theta divisors in dimension 5, but one of them had "Milnor multiplicity" 2 and the other had multiplicity one. The latter result answered a question attributed to Igusa, by proving that a general abelian variety (of dimension 5) having a vanishing even theta null, only has one of them.
This theory of positive dimensional Milnor numbers was later generalized by Parusinski. you can learn the classical theory, isolated singularity case, from milnor's book on singularities of complex hypersurfaces. Using a different but related technique, involving degeneration to lower dimensions, a sort of geometric induction method, DeBarre later proved the discriminant locus of abelian varieties with singualr theta diviusors has 2 components in all dimensions. I believe he used a beautiful computation of the monodromy group of the Gauss map of a smooth theta divisor.
How annoying, Part IV won't fit (is there really a 20000 word limit on the size of a post?), TBC...
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