5
$\begingroup$

I'm trying to understand page 161 of Fulton's "Young tableaux" in an explicit example.

I'm looking at flags in $\mathbb{C}^4$, which I think of as flags in $\mathbb{CP}^3$ (and I'm really just able to think of its real points). I denote by $X$ the set of all such flags.

I have fixed a flag, consisting of a point $p$ contained in a line $l$ contained in a plane $\pi$, all in $\mathbb{CP}^3$. They correspond to a flag $F_1 \subset F_2 \subset F_3$ in $\mathbb{C}^4$, with $\dim(F_i) = i$.

Now Fulton defines a class $x_1$ in $H^2(X)$ as follows: $x_1 = -c_1(U)$ where $U$ is the line bundle which to the point $E_1 \subset E_2 \subset E_3$ associates $E_1$.

I'm fully aware that there is an axiomatic definition of Chern classes which accomplish most of what one wants to use them for, but in this case I'm interested in finding an explicit representative for this cycle $x_1$ on $X$.

The most concrete definition I've seen of a Chern class is as follows. Take any section $X \to U$ and define $c_1(U)$ to be the class of its zero locus. his is well-defined as a class, at least if there are globally defined sections on $X$.

This is where I'm stuck - I'm not sure how I would produce an explicit section in this case, let alone understand its zero locus.

Finally, let me preemptively answer the hardcore algebraists which I expect to respond along the lines of "read this book which contains all the rigorous definitions": I've done that, but I still find it difficult to produce examples. The question is about finding an explicit representative (perhaps even in coordinates if it cannot be intuitively described (a la "look at all flags whose line intersect $l$ in this or that manner").

$\endgroup$
1
  • $\begingroup$ If you're not set on using the definition of Chern class you mention (as Poincare dual to a zero locus), you can try using the Chern-Weil description (which will give you the rational Chern class, which in your specific example gives you the $\mathbb Z$ Chern class since I'm pretty sure flag manifolds have no torsion in their cohomology). $\endgroup$ Commented Apr 6, 2014 at 15:19

1 Answer 1

2
$\begingroup$

Notice that $U$ is the pullback of the tautological line bundle $L$ on $\Bbb P^3$ under the obvious projection of your flag manifold $X=\Bbb G(1,2,3)$ to $\Bbb P^3$. What is $c_1(L)\in H^2(\Bbb P^3,\Bbb Z)$?

The dual bundle of $L$ is the so-called hyperplane section bundle. Its global holomorphic sections are homogenous polynomials of degree $1$, and these vanish precisely on hyperplanes. The Poincaré dual of this zero cycle is the (positive) generator of $H^2(\Bbb P^3,\Bbb Z)$.

$\endgroup$
2
  • $\begingroup$ I would have exactly the same difficulty here, I'm afraid. $\endgroup$
    – Eric
    Commented Apr 6, 2014 at 15:58
  • $\begingroup$ Sounds like you need to fill in some basic background. You need either some algebraic topology or some differential geometry. You might try the introductory material in Griffiths & Harris. But I did add a paragraph. $\endgroup$ Commented Apr 6, 2014 at 16:15

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .