A departure from directly working with varieties, we’re going to do something that’s strictly topological (at first glance) but which really has deep and important connections with Hodge theory. We’re going to talk about monodromy and monodromy representations.

Let {F\rightarrow E\rightarrow B} be a fiber bundle. Though we don’t really NEED any hypotheses, as far as I’m aware, because our real goal is the study of varieties, we’ll assume that the fiber, {F} is a complete variety, and that the base, {B}, is a variety (often open). So really, we’re fiddling with families of varieties, but just topologically for the moment, so we’ll ignore any degenerations (those will come later).

In the meantime, monodromy is a trick to get some nice representations of {\pi_1(B)}. First, we’ll look at the motivating example for a lot of this:

\bf{Example}: Ignore most of the stuff above, and just look at {p:\tilde{X}\rightarrow X} a covering space of degree {n}. Then there’s a map {\pi_1(X)\rightarrow S_n} given by the action of {\pi_1(X)} by interchanging the sheets of the cover.

So this gave us a map from the fundamental group of the base into the automorphism group of the fibers. This actually happens in general, for any fiber bundle, just go around the loop and see how the transition functions change. But that’s not the most interesting case (though it can be adapted into a nice proof that {\mathcal{M}_g} is irreducible). The problem being that the transition maps may come from ridiculously huge groups, things like {\mathrm{Diff}_+(S^1)}, and while that’s a group I do rather like, it’s not so helpful for getting good representations.

But what tools can we possibly have to linearize something on a space? Homology and cohomology, of course! Here’s an example.

\bf{Example}: Look at the family {y^2z=x(x-z)(x-\lambda z)} of projective elliptic curves. We’re ignoring degenerations for the moment (we’ll complete families later) so this lives over {\mathbb{P}^1\setminus\{0,1,\infty\}}. This is also visible as {\mathbb{C}^\times\setminus\{1\}}. So we’ve got two nontrivial loops, one around zero, the other around one, and there are no relations, this is the free group of rank 2. We can envision all of this as living in the plane with four points marked as {0,1,\infty,\lambda}, and a cut from {0} to {\lambda} and a cut from {1} to {\infty}. Then the loops are what happens if we loop {\lambda} around things. We get one loop from rotating the cut between {0} and {\lambda} all the way around. The other one is a bit trickier to describe directly, but we can describe it in terms like this.

We’ll focus on the first one. Look at {H_1} of the elliptic curve. It has two generators, call them {\delta} and {\gamma}, where {\delta} is a loop around the {0\lambda} cut, and {\gamma} is a loop through the two cuts. This is a standard homology basis, and we’ll look at the action of our element of {\pi_1(\mathbb{P}^1\setminus\{0,1,\infty\}}) on the homology.

If we rotate the cut only 180 degrees, we get the square root of the correct matrix, so we’ll have to square whatever it is we get out of this, but the setup is easier to describe. Call out 180 degree turn operator {T}. Then it’s fairly easy to see that {T(\delta)=\delta}, because {\delta} is just brought back to itself. However, {T(\gamma)} is a bit more interesting. It will be {T(\gamma)=a\delta+b\gamma}, and we can compute {a,b} by intersecting with {\gamma} and {\delta}. Doing this gives {T(\gamma)=\delta+\gamma}, and so our matrix is {\left(\begin{array}{cc}1&1\\ 0&1\end{array}\right)}. Squaring it, we get the matrix {\left(\begin{array}{cc}1&2\\ 0&1\end{array}\right)}. If we do the same procedure for the other generator, we get {\left(\begin{array}{cc}1&0\\2&1\end{array}\right)}, which is a perfectly good representation of our free group, on {2\times 2} integer valued matrices!

This sort of thing will work in general, though computing the representation is a bit trickier when the base of the family is higher dimensional, because we can’t have as nice pictures in our heads. But in particular, we get a representation of {\pi_1} on the automorphism group, and on all of the homology (and thus, cohomology) spaces.

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