Segre classes $s_n(c_1,\dots,c_n)$ in terms of Chern classes

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Polynomials

$n$

$s_n$

-c1

c1^2 - c2

-c1^3 + 2*c1*c2 - c3

c1^4 - 3*c1^2*c2 + c2^2 + 2*c1*c3 - c4

-c1^5 + 4*c1^3*c2 - 3*c1*c2^2 - 3*c1^2*c3 + 2*c2*c3 + 2*c1*c4 - c5

c1^6 - 5*c1^4*c2 + 6*c1^2*c2^2 + 4*c1^3*c3 - c2^3 - 6*c1*c2*c3 - 3*c1^2*c4 + c3^2 + 2*c2*c4 + 2*c1*c5 - c6

Definition

The entry is the homogeneous component $s_n(c_1,\dots,c_n)$ of the total Segre class of a complex vector bundle [1], written as a polynomial in the Chern classes $c_i$ [2]. It is defined by the inverse relation in Formula (1).

Parameters

$n$

— degree ($n\geq 1$)

Formulas

(1)

If $c(E)=1+\sum_{j\geq1}c_j(E)$ and $s(E)=1+\sum_{n\geq1}s_n(E)$, then $c(E)s(E)=1$. Equivalently, $s_n=-\sum_{j=1}^n c_j s_{n-j}$ with $s_0=1$.

(2)

If the $x_i$ are the Chern roots, then $s_n(c_1,c_2,\dots)=(-1)^n h_n(x_1,x_2,\dots)$, where $h_n$ is the complete homogeneous symmetric polynomial.

Comments

(3)

The table stores one homogeneous component per row, not the total Segre class $1+s_1+s_2+\cdots$ truncated at degree $n$.

(4)

The constant component is $s_0=1$ and is not stored as a row; the table begins at $n=1$.

(5)

There is no rank parameter in the table. For a vector bundle of rank $r$, the universal polynomial is specialised by setting $c_j=0$ for $j>r$.

(6)

The variables are Chern classes: $c_j$ is the $j$-th elementary symmetric polynomial in the Chern roots.

(7)

The component $s_n$ uses exactly the variables $c_1,\dots,c_n$; the coefficient of $c_n$ is $-1$.

Programs

(P1)

Sage

from sage.all import PolynomialRing, ZZ

def segre_component(n):
    R = PolynomialRing(ZZ, ["c%s" % i for i in range(1, n + 1)])
    c = [R(1)] + list(R.gens())
    s = [R(1)]
    for m in range(1, n + 1):
        s.append(-sum(c[j] * s[m - j] for j in range(1, m + 1)))
    return s[n]

print(segre_component(7))

Links

[1]

Wikipedia: Segre class

[2]

Wikipedia: Chern class

Similar tables

Todd polynomials — store another universal characteristic-class polynomial in the same Chern-class variables

Chern character polynomials — store the components of $\operatorname{ch}(E)=\sum_i e^{x_i}$ in the same Chern-class variables

Complete homogeneous symmetric polynomials — give the $h_n$ in the Chern roots; the entries here are $(-1)^n h_n$ rewritten in the Chern classes

Elementary symmetric polynomials — give the elementary symmetric functions $e_j$ that are renamed as Chern classes here

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every component with $1\leq n\leq 6$; $s_7$ is the first component needing more than six variables, which is the most a stored polynomial may have)

How they were obtained:

The generator computes exact integer coefficients in $\mathbb Z[c_1,\dots,c_6]$ from Formula (1).

Every stored component is checked against the Chern-root identity in Formula (2) and by substituting the Chern classes of $T\mathbb P^m$ for $1\leq n\leq m\leq 6$, which gives $s_n(T\mathbb P^m)=(-1)^n\binom{m+n}{n}h^n$, where $h$ is the hyperplane class.