As for “Yay” (cue youtube), the biggest reason for the “Oops” is that my thesis is finally posted to the arXiv! The next project won’t take so long.

However, for today, we’re going to take something most of us first saw in differential geometry (I first met this map in do Carmo‘s book) and translate it into algebraic geometry.

We will start in the absolute least general way possible, following do Carmo.  Let be a surface.  Then there’s a map to the unit sphere taking each point to its unit normal vector.  This is the Gauss map, and it’s a REALLY useful tool, as anyone who has gone through this book can attest.  For instance, if you want to define the curvature of a surface in , the Gauss map is essential.  For instance, the Gaussian curvature is the determinant of the derivative of , and in fact it would be redundant to go through everything about the Gauss map for surfaces in because there’s a whole chapter in do Carmo titled “The Geometry of the Gauss Map!”

We’re going to generalize and then algebraize.  First, let’s just drop the orientation on our surface.  To forget that, we can replace the normal vector with the normal line.  So then instead of getting a point in we get a pair of antipodal points, or just a point of from our surface.  Then we can see that the map is really given by taking the inclusion, then we have , and then taking the line perpendicular to the image plane.  Taking the union of these maps, we just have the map .  Then, using the Riemannian metric on , we can make this a map , and follow it up with the fact that , and project down, to get the map , and then we can rewrite it by taking each point to the line in , which gives us a map , giving the usual Gauss map.

So how can we simplify and generalize this? Step 1 is to replace the normal vector with the tangent space, which gives a point in the dual projective space.  Then we want to generalize dimension.  It’s easy to handle hypersurfaces in , we just get a map to (or rather, to its dual).  In general, if we allow non-hypersurfaces, we get maps to Grassmannians, so gives us a Gauss map .

Now, we’re going to let the target space vary.  We just need a space such that where is the tangent space at some specific point.  If we have trivial tangent bundle, we can identify all the fibers and then the derivative of our map actually gives us a map from the domain to a Grassmannian.  What are some spaces that have this property? Lie groups! It’s important that we have Lie groups, not just homogeneous spaces, because of the unique way that we can identify fibers.

Now, if we try to algebraize, the first thing we get is a Gauss map for affine varieties .  We can even get rid of and replace it with an algebraic group , but most of the algebraic groups that immediately come to mind are affine, things like etc, the classical groups.  Plus and and products of these.  But that still only gives us affine varieties, nothing projective or complete.  Fortunately, there’s one remaining option that are commonly studied: abelian varieties.  Though we can’t do much with rational varieties (as there are no maps from them into abelian varieties other than constant maps), we can get a lot of mileage out of the Gauss map on abelian varieties, as we’ll see in the next post.

We can use to construct a lattice.  Inside of , there’s naturally a lattice , but it’s not of full rank, the quotient isn’t compact.  However, if we add in another rank lattice that is independent from the natural one, that will be full rank.  So now, we take the lattice .  This is then a lattice in of full rank that varies holomorphically with the pair of Riemann surface and symplectic basis of .

This means that the quotient varies holomorphically with that data.  In fact, because a change of basis is linear, the transformation extends to all of , and so the quotient doesn’t depend on which symplectic basis we chose! Thus, this torus, which we will denote by only depends on the Riemann surface, not on the symplectic basis.

We can also define a function on out of , and it will behave decently with respect to the lattice.  Define , the Riemann theta function, to be the multivariate Fourier series .  It’s not hard to see from this definition that if we translate by an element of , the function is invariant.  And it’s holomorphic (convergence is guaranteed because is symmetric and has positive definite imaginary part) everywhere.  So it can’t be periodic in the other directions.  In those, you pick up an exponential factor.

Because of this, is periodic, and so we get a divisor on called the Theta divisor, .  The geometry of this divisor is a rich and detailed subject of study, and we’ll talk about it a bit in later posts.  For now, the main point is that for any Riemann surface , this divisor defines what is called a principal polarization.  One way to see that is because the bilinear form on the lattice is unimodular.  Another way is by directly computing that has a unique (up to scaling) global section.  The third, and in many ways most geometric, way is by looking at the map given by where is the translate of by .  This gives an isomorphism between and the dual torus of degree zero divisors on .

Next time, we’ll have an interlude with an application of the geometry of the theta divisor, and then we’ll get back to constructing things with 1-forms on curves.

Today’s word is one that I use quite a bit: algebraic variety.  The kanji is 代数多様体.  The pronounciation is “daisutayoutai.”  Really this works out as “algebraic manifold.”

In fact, 代数 appears to be a prefix that means algebraic, and is used in many other situations.  Here, the two kanji are actually pretty straightforward in why they make something algebraic.

The first is 代, which here is pronounced dai, and has a great many other pronounciations.  It also has a great many different shades of meaning.  But the obviously relevant one here is “substitute” or “replace.”

The second kanji, 数, is pronounced kazu su here, again among many other things.  The meanings are “number,” “law,” “figures,” and the like.  (According to a commenter, only “number” and “to count” are common meanings, the others may be archaic)

So together, they mean something like “substituting numbers” or “laws of substitution,” which mean algebra.  As last time, we discussed the word “tayoutai,” or manifold, this means that daisutayoutai translates as “algebraic manifold.”

The issue of what the proper thing to integrate really boils down to understanding a bit of notation that most people ignore or at least don’t think much about in calculus:

On a first pass, is just something you write so that the profession doesn’t mark down your score.  On a second, it’s an infinitesimal that you can manipulate if you’re careful.  But really, it’s something else entirely.  One thing that can be agreed on is that if is a differentiable (holomorphic) function, then .  This is the familiar chain rule, if we say we can divide, we get that .  But it turns out, unsurprisingly in many ways, that the chain rule is the key to everything.

Let be any Riemann surface, it doesn’t have to be compact.  It can be covered by charts each of which is biholomorphic to an open subset of the complex plane, and with transition maps.  We define a holomorphic 1 form to be on each chart an object , where is holomorphic, such that if is a transition function, then we have that .  This means that transforms between and , so our actual object is coordinate independent, because it is well-defined on the overlaps.  THIS is what you want to integrate around curves.

And now, we integrate in exactly the usual way.  We pick a path to integrate along, parametrize, and then substitute into the 1-form and do the integral.  Of course, the story is hardly simple. First off, there’s the question of existence of holomorphic differentials! If we are really worried, we can define meromorphic 1-forms in the same way, but assuming that the function or is meromorphic rather than actually being holomorphic, but there turn out to be plenty of holomorphic 1-forms to use.  In fact, it’s a consequence of the Riemann-Roch theorem (and many other methods of proof) that the dimension of the space of holomorphic 1-forms on a compact Riemann surface, denoted by , is equal to the genus, and we can just take this to be the definition of the genus, if we so choose.

Now, of course, there’s a problem.  As discussed before, we get the same integral if we change to any homologous loop.  So integrals are path dependent.  If we integrate along a curve from to , we get a different answer than if we first go in a loop, then integrate along the path from to .  So integrals from one point to another are only well-defined up to the loops, and the key to understanding integration on a compact Riemann surface is to understand the integrals along loops.

So, we pick a basis for .  And instead of picking an arbitrary basis, we pick a symplectic one.  We can always find a basis that makes the intersection form , and we label the loops and so that , and .  Then we note that the pairing is non-degenerate, so we can take a basis for the 1-forms dual to the ‘s, .  This leaves only the values on the ‘s unknown, and we arrange them as and call the matrix the period matrix of the Riemann surface with respect to the basis .

The Period Matrix will take a starring role in the next few posts, as we develop a few of its properties and use it to understand curves themselves a bit better directly, as well as in more complex constructions.

Our first word is “manifold.”  Every geometer of every stripe has studied manifolds at some point, and lots of other concepts require manifolds before they can even be defined, so that’s where we’re starting.

First, the individual kanji, with links to description pages: 多, 様, and 体.  We put these three together, and get “tayoutai” which means “manifold.”

The first of these kanji, 多, can be pronounced as “ta,” which seems to be the most common reading.  Looking at the actual structure of the kanji, this seems reasonable, as “ta” is written タ in katakana, and the kanji is two copies of this kana.  As for meaning, it means “many” or “much” or “frequent,” and is the beginning of a direct translation of manifold.

The second character, 樣, is apparently not a general use character, it only occurs in special situations.  Here, it is pronounced “you” (which we should note is two syllables “yo-u”) though can also be pronounced “sama.”  Here it seems to be being used to indicate will or “way of” in some sense.

And finally, we get to the last character: 体.  Here it is pronounced “tai” (again, ta-i), though it has others.  It means “body” or “object.”

So, altogether, “tayoutai” or 多様体 means something like “many ways object” or “diverse bodies,” and translates to “manifold,” which is fairly direct.

We start with contour integration in the complex plane.  As described to undergrads, the objects involved are a curve and a function that is holomorphic in some neighborhood of , and the operation of integration gives us back a complex number depending on both.

Let’s be a bit more careful.  Let be complex numbers and a meromorphic function on the plane.  Then the integral from to of is not, itself, well-defined: there are many possible numbers we can get.  In some sense, there’s a “fundamental value” (note: this is rhetorical, there is no preferred value or path, though in many situations, there’s an “obvious” choice), but then we can also get that number plus any integer linear combination of the residues of at the poles.

In fact, this is getting down to the core of what integration is.  First we need to think about the domain: let be the open subset of $\mathbb{C}$ where is actually a holomorphic function.  Then we need to understand loops in , but only up to homology, which counts how the loops go around each puncture, and only that information, which is what we need to actually compute the contribution of the residues.  So, at the moment, integration appears to be a pairing .

Now, we’re algebraic geometers here, despite talking about integration.  So we want to work with things that, quite honestly, are not naturally holomorphic on domains in the complex plane.  Or at least, the domain of holomorphy isn’t going to be in the plane.  For instance, though there’s no problem with , we have a bit more of an issue with .  For , we can choose to take a branch cut from 0 to infinity in order to make it well defined.  Or, we can realize that as it wants to assign two values to each nonzero number, we can take a double cover of the punctured plane to get a legitimate and nice domain.

And so this starts us studying Riemann surfaces.  We’ll be slightly informal and just say that a Riemann surface is a Hausdorff space (second countable, I believe) such that locally looks like open sets in the complex plane.  Just a manifold such that the transition maps are holomorphic to and from domains in .

In this context, everything still works: we can take any path , though we’ll restrict to homology classes of loops, because the indeterminacy turns out to be what’s interesting, and parametrize it, then use that parametrization to integrate a holomorphic function along it.  There’s only one problem.

Compact Riemann surfaces have no non-constant holomorphic functions.  We can actually prove this from basic complex analysis: Liouville’s theorem says that any bounded entire function must be constant, and we can look at a chart on our Riemann surface.  Any holomorphic function on the surface is bounded, by compactness, so it’s bounded on the chart, which transports it to a bounded entire function.  Thus, it is constant on the chart, and so on the whole Riemann surface.  The correct proof is even simpler.  The real and imaginary parts of are both real functions, and as the domain is compact, they attain a maximum.  Also, as is holomorphic, the real and imaginary parts are harmonic on each chart.  As the maximum of a harmonic function must occur on the boundary, and every point is in the interior of some chart, the real and imaginary parts must be constant, and so must be.

In the next post, we’ll talk about a solution to this problem, one that’s significantly better than just looking at meromorphic functions.

About six months ago I asked people for opinions about digital pens, and did my own research, so I decided to pick up a LiveScribe Echo based on what I’d heard, and the campus computer store has a 10 day return policy (so long as packaging is intact) so I decided to give it a try.  For those who are impatient, the short version of my review is: the digital pen is good, and I’m keeping mine, but there are a few improvements that would go a long way to mainstreaming, so I give it a 4/5.

I’ve used the pen for almost six months of my regular activity, including doodles, mathematical scratchwork, seminar notes, and writing based hobbies.  Here are the pros and cons:

Pros

• The detection is actually much better than I had hoped.  The pen captures my writing very well, from fairly small handwriting to large strokes.
• On the purchasable notebooks, the dot paper is only barely noticeable and not distracting at all.  The free ones that you can just print out aren’t as good, but you can print out 100 pages of the paper at a time that won’t cost you a thing (assuming access to an early modern laser printer)
• The software is very easy to use.  I just plug my pen in, open the LiveScribe desktop, and it gives me a list of all the notebooks I have active with the pen, and loads all the pages I’ve written (each page of each kind of notebook has a unique dot pattern, which is aperiodic, so a small sample tells the pen in what notebook you’re working, on which page, and where on the page) and then moving them to custom notebooks, like one for each date or subject, is just click and drag.
• PDF output
• The audio recording is pretty good, and it’s nice that it will synch the audio to your penstrokes, letting you see how you took notes during a lecture.
• The search feature is fairly good.  If you type in a word, it will use a sort of shotgun approach to finding instances of your having written the word, which is about as good as you can expect without going into high end handwriting recognition (and there is apparently an app you can purchase that improves the recognition a LOT, but I haven’t bought it)

Cons

• The pen is a bit too large.  I’ve heard people complain that it’s like writing with a magic marker, which is a bit of an exaggeration, and there’s no problem for short bursts, but over the length of a seminar talk, I found that my hand was slightly sore from holding it differently from a regular pen.  Now, I admit that my hand is sometimes slightly sore from a regular pen if I’m writing a lot, which I certainly was, because I had started livetexing some time ago.  However, I think that this problem will be solved by using the pen more, and long term I expect the pens to slim.
• You need to use their special dot paper.  This isn’t a MAJOR problem, except that there isn’t that much in the way of selection.  I personally do most of my work on unbound, unlined paper, or bound quad-ruled paper.  Most of the paper and notebooks they have are college ruled, which, and this may just be a personal idiosyncrasy, but I hate doing math on lined paper.  The free paper is all lined, and there are unlined journals that can be purchased, but there’s a definite lack of variety.  I would like to be able to print out my own unlined paper, or quad-ruled, which there seems to be no way to do.
• The pen rolls.  It rolls a lot.  It’s a bit too symmetric.  They should add a clip or something to the top, just something that will allow it to rest, rather than roll, on slightly slanted surfaces

Overall, I like the pen a lot, and it’s caused me to livetex a lot less, partly because it has a much better battery life than my laptop does and it’s a bit quieter to use than a keyboard.  There’s definitely room for improvement, but the things that annoyed me I’m noticing less and less often the longer I use the pen.