Heat equation and Theta, Parts I-III

[Chris Hillman]

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:

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:

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:

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...

 

[Chris Hillman]

Heat Equation and Theta, Part IV

And here is Part IV of mathwonk's minicourse, followed by some suggested (broadly relevant) background reading:

here is the geometry of the heat equation following andreotti and mayer.recall the product space C^g x C^(g^2) can be viewed, by projecting on the second coordinate, as a family of complex g spaces, one over each gbyg matrix t.

If we mod out the space C^g over t, by the lattice {nI+mt} where n,m, run over all integer vectors, we get a family of complex tori, one over each matrix point t.

If we restrict this family over only the set H in C^(g^2), of matrices which are symmetric and have positive definite imaginary part, we can write down a convergent Fourier series defining a theta function f(z,t), hence defining a hypersurface f(z,t) = 0, in the 2 vbls (z,t),which can be viewed as a family of hypersurfaces in z, one over each t in H.

now we have a family of g-1 diml hypersurfaces, one over each point t in H. Most of these hypersurfaces are non singular, i.e. smooth, but a codimension one family of t's have a singular hypersurface over them. this codimension one set of t's is called D, the discriminant hypersurface in H, for the family.

the set C of singular points on all theta hypersurfaces, is the common zero locus of the functions f =0, and of the partials of f wrt z. since there are g+1 such functions, the common zero locus does not meet every g diml hypersurface, but does meet a closed subvariety of them.

the closed set D in H, the discriminant locus, consists of those t such that the corresponding theta divisor has at least one singular point, usually only one or two. There is a projection down p:C-->D in H, and we can look at its derivative.

I.e. the total family C of all singular points on all theta divisors, is itself usually smooth, and we can look at the derivative of p as a linear map from the tangent space to C at a singular point (z,t), down to the tangent space to D inside the tangent space to H at t. C and D have the same dimension, one less than H.

now remember we have three nested families upstairs lying over H. we have a family of smooth tori, and in that a family of not always smooth theta divisors, and in that, a family C of singular points on non smooth theta divisors.

look at the projection from these various families down to H. and at the derivative of the projections. since the tori are all smooth, the derivative of the big projection is surjective. since however the theta divisors are not all smooth, the derivative of the projection restricted to the family of theta divisors, fails to be surjective exactly at a singular point (z,t) of a singular theta divisor.

so we could also define the critical locus C, as the points upstairs where the derivative of the restricted projection from the family of theta divisors down to H, has non surjective derivative. now in general the discriminant locus downstairs is a smooth hypersurface D in H, and the image at t in D of the derivative of the restricted projection, is just the tangent space to D at t.

moreover the equation of that tangent hyperplane is the vector of first partials of the theta function wrt t. This is what comes up on one side of the heat equation.

Now at a general singular point upstairs on the singular theta divisor over t in D, the singular point is a double point, at which the first non zero terms of the taylor series are quadratic, and the matrix of this quadric is the symmetric matrix of second partials of the theta function wrt z. that is what is on the other side of the heat equation.

now notice that the tangent space to H consists of symmetric matrices, and on the other hand the tangent space to the vertical space C^g over t in H consists of g diml vectors z1,..zg. Now a quadric tangent cone to a double point of the theta divisor over t, thus is a symmetric matrix of second partials of f wrt z. i.e. a quadratic homogeneous polynomial in the vbls zi.

this quadric cone in C^g, on the other hand can be looked at as a determining a symmetric matrix and hence a tangent vector to H, i.e. as the coordinates of a vector in t space.

the heat equation says these are the same.

i.e. the geometric heat equation says the symmetric gbyg matrix determined by the quadric tangent cone to a double point of the singular theta divisor lying over the point t of D, is the same as the vector in g(g+1)/2 space determining the tangent plane at t to the discriminant locus D in H.If the theta divisor over t has several singular points, then you get several quadrics and several tangent planes at t in H. The intersection of those tangent planes gives the tangent directions in H of the locus of t's whose theta divisors have as many singular points as does the one at t.

hence if you can compute the dimension of the intersection of those tangent planes at t, you can see how large is the locus of t's having the same number, or same dimension, of singular points as the one at t.

AM showed that near a jacobian period matrix t, the locus of matrices with g-4 dimensional singularities on theta, was 3g-3, exactly the dimension of the set of jacobian matrices. thus near a jacobian, one can recognize another jacobian because the theta divisor has the same size singular locus at as t.

they made the computation by using the heat equation to equate it with the computation of how many quadrics contained a certain canonical model of the curve X defining the jacobian matrix. thus classical geometry in projective space enabled a tangent computation in the moduli space of abelian varieties.Is not this amazing?

Suggested reading for general background (I'll probably add to this once I catch my breath!):

Algebraic geometry generally:

Miles Reid, Undergraduate Algebraic Geometry, LMS Student Texts 12, Cambridge University Press, 1988.

Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, Springer, 1992. IVA for short. One of the best books of all time! Lots of pictures and computational tools.

Hal Schenck, Computational Algebraic Geometry, LMS Student Texts 58, Cambridge University Press, 2003. See chapter 7 for the "homological signature" of configurations of points. (We are interested in something spiritually analogous, only involving "embeddings" of the next complicated thing after points. Sort of...)

Joe Harris, Algebraic Geometry, Springer, 1992. Better emphasis of the role symmetry than Hartshorne, IMO. (Hartshorne isn't everywhere dense, but I am trying very hard not to frighten off the undergraduates... this is meant to be an invitation, not an initiation, as in "rite of hazing"!)

Complex curves and abelian varieties:

C. G. Gibson, Elementary Geometry of Algebraic Curves, Cambridge University Press, 1998. If you read no other book this year, read this! (Or IVA, cited above.) Not to be missed: "Cramer's paradox" is almost discussed in section 17.1.

Frances Kirwan, [I}Complex Algebraic Curves[/I], LMS Student Texts 23, Cambridge University Press, 1992. Riemann surfaces, holomorphic differentials, Riemann-Roch theorem.

Gareth Jones and David Singerman, Complex Functions, Cambridge University Press, 1987. Elliptic curves, the modular group, the Lorentz group, and so on.

Herb Clemens, A Scrapbook of Complex Curve Theory, Plenum, 1980. Offers a bit about moduli spaces of cubic curves.

Waldschmidt et al., From Number Theory to Physics, Springer, 1995, has a number of chapters dealing with abelian varieties, theta functions, and so on.

Zeta functions (cause, you know, "zeta" rhymes with "theta") and all that:

Jameson, The Prime Number Theorem, LMS Student Texts 53, Cambridge University Press, 2003. No seriously, great background for the intersection of algebra, number theory, and analysis.

Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, LMS Student Texts 50, Cambridge University Press, 2001. Ditto, density theorems in chapter 16, plus it has an appendix on Fourier Analysis if you don't know that that is.

Bedford et al, Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces, Oxford University Press, 1992. See the wonderful short introduction "Ergodic theory and subshifts of finite type" by Michael Keene, plus chapters by Series, Pollicott, and Mayer related to dynamical zeta functions.

Lie theory, representation theory, and invariant theory (cause everyone should know some)

Carter et al., Lectures on Lie Groups and Lie Algebras, LMS Student Texts 32, Cambridge University Press, 1995. Great stuff on the classification of complex Lie algebras, Weyl groups, Lie groups, and algebraic groups.

Bernd Sturmfels, Algorithms in Invariant Theory, Springer, 1993. More great stuff on invariants of finite groups and how they can be computed. Goes beyond the chapter in IVA.

Brian J. Cantell, Introduction to Symmetry Analysis, Cambridge University Press, 2002. One of many fine books on Lie's theory of differential equations which I might mention. Everyone should know that Lie theory arose as the background needed for Lie's attempt to pursue the idea that whenever you can solve a differential equation, you can do so because of some underlying symmetry.

(This might be a good place to mention that when Lie met Klein in Berlin c. 1869, he soon had to confess that he hadn't paid enough attention when he took a course on finite group theory from none other than Sylow! So if anyone is feeling a tad daunted, well, many have felt that, and the successful math students work through the terror! In 1869, Klein and Lie both found Galois theory [recently rediscovered] and the work of Darboux in differential geometry and differential equations a bit daunting. So they traveled to Paris and promptly got caught by the outbreak of the Franco-Prussian war. For some reason Lie thought this would be a good time to enact his lifelong ambition of hiking from France to Norway, via the Italian Alps. In the nude. You can guess what happened next!...fortunately, the gendarmes didn't shoot him, they put him on the train with a note that they were deporting a Norwegian lunatic.)

Peter W. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, 1995. More about Lie groups and invariant theory.

Fulton and Harris, Representation Theory, Springer, 1991. For some basic stuff about representations of the symmetric group, Young diagrams, etc.

Number Theory Generally:

Burr et al., The Unreasonable Effectiveness of Number Theory, American Mathematical Society, 1991. Articles by leading researchers, surveying beautiful connections between number theory and dynamical systems, Rogers-Ramanujan identities in statistical mechanics, diffraction gratings, acoustics of concert halls, coding theory, random number generators, computer graphics, and more. The article by Rogers should be of particular interest because this is another place where theta functions arise.

Thomas M. Thompson, From error-correcting codes through sphere packings to simple groups, MAA, 1983. Great background for the next listed item:

Conway andSloane, Sphere Packings, Lattices and Groups, Springer, 1998. More neat applications of theta functions.

Baezetics (cause John Baez is a subject all his own):

John Baez, TWF (Weeks 62-66, 157,178-188, 193, 201, 213-218, 230, 241, 243): http://www.math.ucr.edu/home/baez/TWF.html. Incidence geometry, Schubert cells, cohomology of homogeneous spaces, ADE for Lie algebras, simply laced Dynkin diagrams, Young diagrams, nonabelian Hodge theory, quantum calculus and q-deformations, the MacKay correspondence, the Monster, cohomology of groups, Eilenberg-Mac Lane spaces, categorification, Klein's quartic, zeta and L-functions, Langlands program, ADE for catastrophes, cobordism, Cartan geometry, and much more, such as hamsters in physics. (Note: if the webglimpse search tool is not working, try googling with "baez TWF site:ucr.edu keyword" where keyword is a phrase like ADE.)

Note that the LMS Student Texts are designed for UG students and are as inexpensive as possible--- mathwonk, have you ever considered writing one of these? I think moduli spaces of curves and geometric invariant theory would make a wonderful and timely topic!