differential geometry

As in the previous two posts, let {S/k} be a smooth, projective surface over an algebraically closed field {k}. In the previous posts, we set up an intersection theory for divisors, which was a symmetric bilinear form

\displaystyle \mathrm{Pic}(S) \times \mathrm{Pic}(S) \rightarrow \mathbb{Z},

that gave the “natural” answer for the intersection of two transversely intersecting curves. Specifically, we had

\displaystyle \mathcal{L} . \mathcal{L}' = \chi(\mathcal{O}_S) - \chi(\mathcal{L}^{-1}) - \chi(\mathcal{L}'^{-1}) + \chi( \mathcal{L}^{-1} \otimes \mathcal{L}'^{-1});

the bilinearity of this map had to do with the fact that the Euler characteristic was a quadratic function on the Picard group. The purpose of this post is to prove a few more general and classical facts about this intersection pairing. As usual, Hartshorne’s Algebraic geometry and Mumford’s Lectures on curves on an algebraic surface are very helpful sources for this material; I also found Abhinav Kumar’s lecture notes useful.

1. The Riemann-Roch theorem

The Euler characteristic of a line bundle {\mathcal{L}} on {S} is a “topological” invariant: it is unchanged under deformations. Given an algebraic family of line bundles {\mathcal{L}_t} on {S} — in other words, a scheme {T} and a line bundle on {S \times_{k} T} which restricts on the fibers to {\mathcal{L}_t} — the Euler characteristics {\chi(\mathcal{L}_t)} are constant. This is one of the parts of the semicontinuity theorem on the cohomology of a flat family of sheaves. Over the complex numbers, one can see this by observing that the Euler characteristic of a line bundle is the index of an elliptic operator — more specifically, the index of the Dolbeault complex associated to {\mathcal{L}} — and can therefore be computed in purely topological terms via the Hirzebruch-Riemann-Roch formula.

In algebraic geometry, the fact that the Euler characteristic is a topological invariant is reflected in the following result, which computes it solely in terms of intersection numbers:

Theorem 1 Let {\mathcal{L}} be a line bundle on {S}. Then

\displaystyle \chi(\mathcal{L}) = \frac{1}{2} \mathcal{L}.( \mathcal{L} - K) + \chi(\mathcal{O}_S), \ \ \ \ \ (1)

where {K} is the canonical divisor on {S}. (more…)

This is the second post intended to understand some of the ideas in Milnor’s “Note on curvature and the fundamental group.” This is the paper that introduces the idea of growth rates for groups and proves that the fundamental group in negative curvature has exponential growth (as well as a dual result on polynomial growth in nonnegative curvature). In the previous post, we described volume comparison results in negative curvature: we showed in particular that a curvature bounded above by c < 0 meant that the volumes of expanding balls grow exponentially in the radius. In this post, we’ll explain how this translates into a result about the fundamental group.

1. Growth rates of groups

Let {G} be a finitely generated group, and let {S} be a finite set of generators such that {S^{-1} = S}. We define the norm

\displaystyle \left \lVert\cdot \right \rVert_S: G \rightarrow \mathbb{Z}_{\geq 0}

such that {\left \lVert g \right \rVert_S} is the length of the smallest word in {S} that evaluates to {g}. We note that

\displaystyle d(g, h) \stackrel{\mathrm{def}}{=} \left \lVert gh^{-1} \right \rVert_S

defines a metric on {G}. The metric depends on the choice of {S}, but only up to scaling by a positive constant. That is, given {S, S'}, there exists a positive constant {p} such that {\left \lVert \cdot \right \rVert_S \leq p \left \lVert\cdot \right \rVert_{S'}}. The metric space structure on {G} is thus defined up to quasi-isometry. (more…)

Let {M} be a compact Riemannian manifold of (strictly) negative curvature, so that {M} is a {K(\pi_1 M, 1)}. In the previous post, we saw that the group {\pi_1 M} was significantly restricted: for example, every solvable subgroup of {\pi_1 M} had to be infinite cyclic. The goal of this post and the next is to understand the result of Milnor that the group {\pi_1 M} is of exponential growth in an arithmetic sense.

Milnor’s (wonderful) idea is to translate this into a problem in geometry: that is, to relate the growth of the group {\pi_1 M} to the volume growth of expanding balls in the universal cover {\widetilde{M}}. As I understand, this idea has proved enormously influential on future work on the fundamental groups of Riemannian manifolds with restricted curvature. Note that Milnor’s result also highlights the difference between positive and negative curvature: in positive curvature, the fundamental group of every compact manifold is finite. Most of this material is from Chavel’s Riemannian geometry: a modern introduction.

1. Volume growth

To begin with, let’s say something about volume growth. Let {M} be a complete, simply connected Riemannian manifold whose sectional curvatures are {\leq c < 0}. If we choose {p \in M}, we know that the exponential map

\displaystyle \exp_p : T_p M \rightarrow M

is a diffeomorphism. Note that {\exp_p} sends the euclidean ball of radius {r} diffeomorphically onto the (metric) ball of radius {r} in {M}.

Our goal is to prove:

Theorem 13 The function {r \mapsto \mathrm{vol} ( B_M(p, r))} which sends {r} to the volume of the ball in {M} of radius {r} centered at {p} grows exponentially.

This theorem also highlights the sense in which negative curvature corresponds to the “spreading” of geodesics: the geodesics spread so much that the volumes of linearly expanding balls actually grow exponentially. (more…)

Let {M} be a complete Riemannian manifold. In the previous post, we saw that the condition that {M} have nonpositive sectional curvature had a reformulation in terms of the “spreading” of geodesics: that is, nearby geodesics in nonpositive curvature spread at least as much as they would in euclidean space. One consequence of this philosophy was the Cartan-Hadamard theorem: the universal cover of {M} is diffeomorphic to {\mathbb{R}^n}. In fact, for a {M} simply connected (and complete of nonpositive curvature), the exponential map

\displaystyle \exp_p : T_p M \rightarrow M

is actually a diffeomorphism.

This suggests that without the simple connectivity assumption, the “only reason” for two geodesics to meet is the fundamental group. In particular, geodesics have no conjugate points, meaning that the exponential map is always nonsingular. Moreover, again by passing to the universal cover, it follows that for any {p \in M} and {\alpha \in \pi_1 (M)} (relative to the basepoint {p}), there is a unique geodesic loop at {p} representing {\alpha}. In this post, I’d like to discuss some of the topological consequences of having a metric of nonpositive sectional curvature. Most of this material is from Do Carmo’s Riemannian Geometry.  (more…)

Let {(M, g)} be a Riemannian manifold. As before, one associates to it the curvature tensor

\displaystyle R: TM \otimes TM \otimes TM \rightarrow TM, \quad X, Y, Z \mapsto R(X, Y) Z.

In the previous post, we saw a quantitative expression of how the curvature is a measure of the deviation from the flatness of {M}. Given {M}, one can try to choose local coordinates around a point {p \in M} which make the metric look like the euclidean metric to order 2 at {p}, i.e. local coordinates such that the coefficients near {p} are given by

\displaystyle g_{ij} = \delta_{ij} + O(|x|^2).

However, we saw that the quadratic terms involve precisely the values of the curvature tensor at {p}. Even in the best coordinates, one can’t generally make the coefficients of a metric look euclidean to order 3: the obstruction is precisely the curvature at {p}. Today, I’d like to describe the interpretation of curvature in terms of geodesics. Once again, the material is standard and can be found in introductory textbooks on Riemannian geometry.

1. Curvature and geodesic deviation

There’s another way to think of curvature, which also leads to this: curvature measures how nearby geodesics spread. To think about this, suppose we have a one-parameter family {\gamma_s} of geodesics in {M}, where {\gamma = \gamma_0} is the starting point of the variation. One then has a vector field

\displaystyle V = \left( \frac{d}{ds} \gamma_s\right)_{s = 0}

along the curve {\gamma}, which measures the infinitesimal “spreading” of the one-parameter family {\gamma_s}. Now, a computation shows that {V} satisfies the equation

\displaystyle \frac{D^2}{dt^2} V(t) + R( V, \dot{\gamma}(t)) \dot{\gamma(t)} = 0,

in other words that {V} is a Jacobi field. Here {\frac{D}{dt}} is covariant differentiation along the curve {\gamma}. (more…)

I’d also like to describe some more quantitative versions of what curvature means. We saw in the previous post that curvature measures the sense in which a connection fails to be a local system, or in other words look locally like the standard connection on a trivial bundle. (This is perhaps part of the motivation of the use of curvature to construct de Rham representatives of the characteristic classes of a vector bundle, cf. Chern-Weil theory.)

1. Curvature as deviation from flatness

If you work in coordinates, it’s not immediately clear how to say to what extent a connection looks like a trivial connection, because the trivial connection can look very different if you change the frame. Curvature has the property of being tensorial and not depending on a given choice of frame.

But another way to say this is to try to express the connection in the best possible choice of coordinates, and see whether it looks like the standard one. Namely, let {V \rightarrow M} be a vector bundle with connection {\nabla}. Choose a neighborhood of {p \in M} that looks like {\mathbb{R}^n} with {p} at the point {0}. Since everything we do is local, we may just assume

\displaystyle M = \mathbb{R}^n, p = 0.

This gives us a particularly nice frame for {V}. Choose a basis {e_1, \dots, e_m} for {V_p} and then define sections {E_1, \dots, E_m} of {V} on {\mathbb{R}^n} at a point {q} by parallel translation along the (euclidean!) line from {p=0} to {q}. Then {E_1, \dots, E_m} is a global frame for {V} and is, by construction, parallel along each straight line through the origin. One therefore has:

\displaystyle \nabla E_i ( p) = 0, \quad 1 \leq i \leq m.

Using this choice of frame, we get an identification of {V} with the trivial bundle {\mathbb{R}^m}. In particular, we can write the connection {\nabla: V \rightarrow \Omega^1(V)} in the form

\displaystyle \nabla = d + \omega, (more…)

Over the past couple of days I have been brushing up on introductory differential geometry. I’ve blogged about this subject a fair bit in the past, but I’ve never really had a good feel for it. I’d therefore like to make this post, and the next, a “big picture” one, rather than focusing on the technical details.

1. Curvature of a connection

Let {M } be a manifold, and let {V \rightarrow M} be a vector bundle. Suppose given a connection {\nabla} on {V}. This determines, and is equivalent to, the data of parallel transport along each (smooth) curve {\gamma: [0, 1] \rightarrow M}. In other words, for each such {\gamma}, one gets an isomorphism of vector spaces

\displaystyle T_{\gamma}: V_{\gamma(0)} \simeq V_{\gamma(1)}

with certain nice properties: for example, given a concatenation of two smooth curves, the parallel transport behaves transitively. Moreover, a homotopy of curves induces a homotopy of the parallel transport operators.

In particular, if we fix a point {p \in M}, we get a map

\displaystyle \Omega_p M \rightarrow \mathrm{GL}( V_p)

that sends a loop at {p} to the induced automorphism of {V_p} given by parallel transport along it. (Here we’ll want to take {\Omega_p M} to consist of smooth loops; it is weakly homotopy equivalent to the usual loop space.) (more…)

There are a number of results in geometry which allow to conclude that a certain group vanishes or is bounded under hypotheses on the curvature. For instance, we have:

Theorem 1 If {M} is a compact manifold of positive curvature, then {H^1(M; \mathbb{R}) = 0}.

Another such result is the Kodaira vanishing theorem, which enables one to show that certain cohomology groups of an ample line bundle on a smooth projective variety vanish in characteristic zero.

I’ve been trying to gain an understanding of such results, and it seems that there is a common technique in such arguments. The first strategy is to identify the desired cohomology group (e.g. {H^p(M; \mathbb{R})}) with the kernel of a Laplacian-type operator, by Hodge theory. The second step is to bound below the relevant Laplacian-type operator. In this post, I’d like to try to explain what’s going on, in a special case. (more…)

I’d like to finish the series I started a while back on Chern-Weil theory (and then get back to exponential sums).

So, in the discussion of the Cartan formalism a few days back, we showed that given a vector bundle E with a connection on a smooth manifold, we can associate with it a curvature form, which is an \hom(E, E)-valued 2-form; this is a generalization of the Riemann curvature tensor (as some computations that I don’t feel like posting here will show). In the case of a line bundle, we saw that since \hom(E, E) was canonically trivialized, we could interpret the curvature form as a plain old 2-form, and in fact it turned out to be a representative — in de Rham cohomology — of the first Chern class of the line bundle. Now we want to see what to do for a vector bundle, where there are going to be a whole bunch of Chern classes.

For a general vector bundle, the curvature {\Theta} (of a connection) will not in itself be a form, but rather a differential form with coefficients in {\hom(E, E)}, which is generally not a trivial bundle. In order to get a differential form from this, we shall have to apply an invariant polynomial. In this post, I’ll describe the proof that one indeed gets well-defined characteristic classes (that are actually independent of the connection), and that they coincide with the usually defined topological Chern classes. (more…)

So, now with the preliminaries on connections and curvature established, and the Chern classes summarized, it’s time to see how they connect with one another. Namely, we want to say that, given a complex vector bundle, we can compute the Chern classes in de Rham cohomology by picking a connection — any connection — on it,  computing the curvature, and then applying various polynomials.

We shall start by warming up with a special case, of a line bundle, where the algebra needed is easier. Let {M} be a smooth manifold, {L \rightarrow M} a complex line bundle. Let {\nabla} be a connection on {L}, and let {\Theta} be the curvature.

Thus, {\Theta} is a global section of {\mathcal{A}^2 \otimes \hom(L, L)}; but since {L} is a line bundle, this bundle is canonically identified with {\mathcal{A}^2}. (Recall the notation that \mathcal{A}^k is the bundle (or sheaf) of smooth k-forms on the manifold M.)

Proposition 1 (Chern-Weil for line bundles) {\Theta} is a closed form, and the image in { H^2(M; \mathbb{C})} is {2\pi i} times the first Chern class of the line bundle {L}. (more…)

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