Running in Circles!

I wrote a guest post for my friend Nick Arnosti’s blog. It’s about the kinds of curves you can trace out while running side-by-side with a friend!

Chern classes without Segre classes

In the modern approach to intersection theory (i.e. Fulton’s landmark book), a rough outline of the development is as follows:

1. Chow groups, rational equivalence, functoriality
2. Intersection with a Cartier divisor
3. Segre classes
4. Rational equivalence on bundles
5. Deformation to the normal cone (normal bundle)
6. Intersection products

(My numbers don’t exactly match the chapters of the book; I’m glossing over the significant but complicated material on cones.)

You might be surprised to notice that Chern classes, one of the most important tools in algebraic geometry, do not appear on the list!

Definition (roughly). If $E$ is a vector bundle on $X$ of rank $r$, the codimension-$i$ Chern class $c_i(E)$ is defined as follows: we let $v_1(x), \ldots, v_{r+1-i}(x)$ be general global sections of $E$. We consider the degeneracy locus

$D(v_1, \ldots, v_{r+1-i}) = \{x \in X : v_1(x), \ldots, v_{r+1-i}(x) \text{ are linearly dependent in} E(x)\}.$

The cohomology class of this locus does not depend on the choice of sections (as long as they are ‘general’) and this is one of the most important invariants of $E$. For example, the top Chern class $c_r(E)$ is the vanishing locus of a single section.

Segre classes $s_i(E)$ are defined similarly, except that we replace “linearly dependent” by “do not span the fiber”, and we take $r+i-1$ sections.

Chern classes are not absent from Fulton’s book — indeed, they get constructed abstractly in the middle of Steps 3-4, and the top Chern class (of the normal bundle) is central to the definition of intersection products. And Fulton follows normal parlance in referring to the first Chern class of a line bundle $c_1(L)$, rather than calling it (-1 times) $s_1(L)$, the first Segre class.

Yet the other Chern classes are not needed at all, despite their historical (and ongoing) significance and utility. And the construction also camouflages the natural geometric meaning of Chern classes as degeneracy loci. It’s not until much later — Chapter 14! — that this description appears.

Of course Fulton had a good reason for the approach via Segre classes. It turns out to be unavoidable: Segre classes generalize from vector bundles to cones, but Chern classes do not. He even discusses how it is strange that Segre classes turn out to be the “right” tool.

Segre classes are very nice, in fact

The formal construction of Segre classes is actually pretty satisfying: for $E \to X$ a vector bundle of rank $r+1$, and $\pi : \mathbb{P}(E) \to X$ the associated projective bundle, we define

$s_i(E) \cap \_\_ : A_k(X) \to A_{k-i}(X),$

$s_i(E) \cap \alpha := \pi_*( c_1 O_E(1)^{r+i} \cap \pi^* \alpha ).$

1. It’s clearly well-defined on rational equivalence classes, since it is built directly out of flat pullback, intersecting with a Cartier divisor, and proper pushforward. This is good because the hardest part of many older constructions — such as the rough description above — is to show well-definedness.
2. All the Segre classes are built together.
3. If $E^*$ is globally generated and $s_1, \ldots, s_{r+i}$ are general sections, it is straightforward to unwind the definition above to show that
$D(s_1, \ldots, s_r) := \{x \in X : \mathrm{span}(s_1(x), \ldots, s_{r+i}(x)) \ne E_x\}$
represents the Segre class: the locus where the sections fail to span the fiber.

I thought it would be nice to have an equally clean and satisfying definition of Chern classes, built with the same modern machinery, but using only the first few bits of it (i.e. not up to Chapter 14).

So here is how that works.

Note: this is not new math… but I thought through it on my own, so I don’t have a reference to suggest here; some of it can be extracted or guessed from Chapter 14.

A nice definition of Chern classes

We’ll start off with our “base case”. For Segre classes, the “base case” was $c_1(L)$ for $L$ a line bundle. Our base case will be the top Chern class: the vanishing locus of a section.

We need this theorem:

Theorem. Let $E \to X$ be a vector bundle of rank $r$. The flat pullback $\pi^* : A_k(X) \to A_{k+r}(E)$ is an isomorphism.

There is a lovely proof that combines the geometry of $X$, $E$ and $\mathbb{P}(E)$. One way to think about it is as a kind of “moving lemma for vector bundles”: it says every subvariety $V \subset E$ is rationally equivalent to a union of fibers of $E$. That is, a subscheme of the form $V' = \pi^{-1}(Y)$ for some $Y \subset X$. (More generally, a formal sum of such subvarieties.)

We can therefore define the super-important inverse map $(\pi^*)^{-1} : A_{k+r}(E) \to A_k(X)$, called a Gysin map. It is effectively a kind of intersection product: with the notation above, $V'$ intersects the zero section $s : X \to E$ transversely, and indeed $(\pi^*)^{-1}(V') = [V' \cap s(X)]$.

In fact — conversely — if $V' \subset E$ is any subvariety that intersects the zero section transversely, then $(\pi^*)^{-1}([V']) = [V' \cap s(X)]$. To see this, just rescale $V'$ by a scalar $t$ and let $t \to \infty$. This doesn’t change where it intersects $s(X)$, but in the limit it exactly gives a union of fibers.

So we have an operation, well-defined on rational equivalence classes, which agrees with “intersecting with $s(X)$” whenever this intersection is transverse. For this reason, we rename $(\pi^*)^{-1}$ to $s^*$.

The top Chern class (See Fulton, Example 3.3.2).

Definition. The top Chern class is the homomorphism $c_r(E) \cap \_\_ : A_k(X) \to A_{k+r}(X)$ given by the formula

$c_r(E) \cap \alpha = s^*s_*(\alpha)$

that is, first we push forward $\alpha$ along the zero section (a proper map), then we pull back using the Gysin map.

Equivalently, we could push forward using any section. In the transverse case, $c_r(E) \cap \alpha$ is exactly given by intersecting $\alpha$ with the zero locus of a section of $E$.

In Fulton, the formula above is a theorem, since the starting definitions come from Segre classes. We will instead take this formula as our definition. We can now define the other Chern classes, by a formula similar in spirit to that of the Segre classes.

Lower Chern classes.

For each $i \leq r$, we consider the product $X \times \mathbb{P}^{r-i}$ and the twisted vector bundle $\tilde{E} := \pi_1^*E \otimes \pi_2^* O(1)$.

Definition. We define the $i$-th Chern class $c_i(E) \cap \_\_ : A_k(X) \to A_{k-i}(X)$ by the formula

$c_i(E) \cap \alpha := \pi_{1*}(c_r(\tilde{E}) \cap \pi_1^* \alpha).$

That is, we pull up to the product, then apply the top Chern class of the twisted bundle, then push back down.

Theorem. The Chern classes $c_i(E)$ satisfy the usual formal properties (functoriality and projection formula, commutativity, vanishing).

For example, here is the proof that $c_0(E) \cap \alpha = \alpha$ and $c_i(E) = 0$ when $i < 0$:

Proof. It suffices to consider the case $\alpha = [Z]$ where $Z \subset X$ is a subvariety. By the projection formula, we may replace $X$ with $Z$, so it is enough to have $X$ be a variety and $\alpha = [X]$. Then $c_i(E) \cap [X] = 0$ when $i < 0$ because the Chow groups of $X$ vanish above the dimension of $X$. For $i=0$, we see that $c_i(E) \cap [X]$ is an integer multiple of $X$. Seeing what multiple it is a local calculation, so by passing to an open subset (using functoriality) we can assume $E$ is trivial. Then we have

$c_0(E) \cap [X] = \pi_{1*} (c_r(\pi_2^*O(1)^{\oplus r}) \cap [X \times \mathbb{P}^r]) = \pi_{1*} \pi_2^*(c_r O(1)^{\oplus r} \cap [\mathbb{P}^r])$

(since $\pi_1^*[X] = [X \times \mathbb{P}^r] = \pi_2^*[\mathbb{P}^r]$). So it is enough to show that $c_r O(1)^{\oplus r} \cap [\mathbb{P}^r]$ is the class of a point. But this is easy since the bundle is globally generated, we just intersect $r$ hyperplanes. //

The other parts of the proof are equally nice. And we have this:

Theorem. Suppose $E$ is globally generated and $s_1, \ldots, s_{r+1-i}$ are general global sections. Then $c_i(E) \cap [X] = [D]$, where $D = D(s_1, \ldots, s_{r+1-i})$ is the degeneracy locus

$D(s_1, \ldots, s_{r+1-i}) := \{x \in X : s_1(x) \wedge \cdots \wedge s_{r+1-i}(x) = 0\},$

the locus where the sections become linearly dependent.

Proof. The sections give a map from a trivial bundle, $s : \mathcal{O}_X^{\oplus r+1-i} \to E$. We lift this to $X \times \mathbb{P}^{r-i}$, and we compose with the inclusion $O(-1) \to O^{r+1-i}$ to get a map $\pi_2^*O(-1) \to \pi_1^*E$, or equivalently, a section of $\tilde{E}$. If the original sections are general, so is this single section, so $c_r(\tilde{E}) \cap [X \times \mathbb{P}^r]$ is represented by its vanishing locus.

This is the set of pairs $(x, L)$ such that $L$ is in the kernel of the map $\mathcal{O}_X^{\oplus r+1-i}|_x \to E_x$. Applying $\pi_{1*}$, we get the locus of $x$ such that this map has a kernel, i.e. the degeneracy locus $D$. //

An upcoming curve

This is a “Schubert curve” — related to some work of my own, and to some joint work with Maria Gillespie of Berkeley. To appear, eventually…!

Ramification and Monodromy

The following situation showed up this spring in my research, and although it ended up not seeming to lead anywhere, I still think there’s something deeper going on.

Consider an algebraic curve $S$ defined over $\mathbb{R}$. I should emphasize that this is a complex curve with real structure, that is, a Riemann surface with an action of complex conjugation. The fixed points of this action are the curve’s real points.

There are a handful of interesting topological questions we can ask about algebraic curves defined over $\mathbb{R}$. For instance:

• Does $S$ have any real points at all? Is $S(\mathbb{R})$ smooth? What about $S(\mathbb{C})$?
• How do the real points of $S$ sit inside its complex points?
• If $S(\mathbb{R})$ is smooth, then it is a disjoint union of circles. How many circles are there? We’ll call this quantity $\eta(S)$.

A small gluing construction

This came up in my research recently.

In geometry, “simple nodes” are the simplest, nicest kinds of curve singularities — they have length 1, they’re resolved after a single blowup, they are the only singularities that occur “generically”, and their scheme-theoretic properties seem generally well-understood and well-behaved. (For example, the moduli space $\mathcal{M}_{g,n}$ of marked points on stable curves of genus $g$ exists because of, among other things, the restriction to only nodal curves.)

Schubert Calculus V: Generalizations

Up until now, all the Schubert calculus I’ve posted about has been on the Grassmannian $G(k,n)$. But that isn’t where the story ends. The theory continues to be in development in at least two ways:

• the space of interest might be the flag variety, or more general $G/P$, for $G$ a linear algebraic group and $P$ a parabolic or Borel subgroup;
• the ring structure (cohomology theory) might vary: equivariant cohomology, quantum cohomology, K-theory.

So I’ll end this series of posts with a few words on these generalizations: what works, and what — so far — remains mysterious.

Schubert Calculus IV: Littlewood-Richardson Rules

When I first planned this mini-course, I expected this topic to be hard to motivate to geometers — why should they care about one algorithm over another for computing the Littlewood-Richardson numbers? But there’s plenty of subtlety in considering the strengths and weakness of the various Littlewood-Richardson rules out there:

• some are fast and convenient for actual computations (on a computer);
• some are “geometric”, that is, they actually describe something happening in $G(k,n)$;
• some are “symmetric”, that is, they exhibit directly the symmetries coming from the fact that the Littlewood-Richardson numbers are the structure constants of a commutative, associative ring — and/or they are invariant under transposing partitions;
• some have turned out to generalize more readily to other contexts (such as K-theory).

Schubert Calculus III: Cohomology and Enumerative Geometry

Last post set up the Schubert decomposition of $X = G(k,n)$. We’re going to use it to do some intersection theory computations.

Since Schubert varieties are indexed by partitions, this is really about “multiplying partitions” — and the ring structure will coincide with the structure for multiplying partitions in the contexts of $GL_n$ representation theory and symmetric polynomials.

All of these computations are in $A(X) \cong H^*(X)$, the Chow/cohomology ring.

Schubert Calculus II: Schubert Stratification

It’s time to set up the Schubert stratification of $G(k,n)$, a decomposition of the whole variety into locally closed pieces (Schubert cells), each of which is isomorphic to an affine space.

These cells are indexed by partitions and, by general theory, give rise to an additive basis of the Chow, cohomology, and $K$-theory rings of the Grassmannian. And, as a result, intersection theory on $G(k,n)$ has a complete, concrete description in terms of the combinatorics of the Schubert decomposition (i.e. partitions and the Littlewood-Richardson rule).

Addendum to last post: the total Chern classes of tautological bundles on G(k,n)

I realised after finishing the last post that it’s simpler than I had thought to compute the total Chern classes of the tautological bundles appearing in the short exact sequence

$0 \to \mathcal{S} \to \mathbb{C}^n \to \mathcal{Q} \to 0$

on the Grassmannian $G(k,n)$. I’m keeping it in a separate post, though, both because the last post is already very long and because I haven’t yet defined the Schubert varieties that appear in the answer.