数理解析研究所講究録
第 1897 巻 2014 年 91-97

91

グラスマン束の次数公式
早稲田大学理工学術院

楫元Hajime Kaji
Department of Mathematics,
Faculty of Science and Engineering,
Waseda University

1. INTRODUCTION
The purpose of our work is to give degree formulae for Grassmann bundles. This article is a summary of a joint paper [4] with Tomohide Terasoma.
Let $X$ be a projective variety of dimension over a field of arbitrary
characteristic, let be a vector bundle of rank on $X$ , let
bethe Grassmann bundle of corank subbundles of on $X$ with projec:, and lettion
be the universal quotient
bundle of rank . Set, the first Chern class of , whose determinant bundle,, is isomorphic to the pull-back of the tautologicalline bundle ofby the (relative) Pl\"ucker embedding over $X$ . In
this article we call the Pl\"u cker class of. The theme discussed
here is how to calculate the self-intersection number of the Pl\"ucker class,, which is the degree of
embedded in the projective
space
is very ample,
via the Pl\"ucker embedding if
where $N:=\dim \mathbb{G}_{X}(d, \mathcal{E})=d(r-d)+n.$
The result is$n$$\mathcal{E}$

$d$

$\pi$

$\mathbb{G}_{X}(d, \mathcal{E})$$r$$\mathbb{G}_{X}(d, \mathcal{E})arrow X$$d$$\theta$$\mathcal{E}$$\pi^{*}\mathcal{E}arrow \mathcal{Q}$$:=c_{1}(\mathcal{Q})$$\mathcal{Q}$$\det \mathcal{Q}$$\mathbb{P}_{X}(\wedge^{d}\mathcal{E})$$\theta$$\mathbb{G}_{X}(d, \mathcal{E})$$\mathbb{G}_{X}(d, \mathcal{E})$$\int_{\mathbb{G}_{X}(d,\mathcal{E})}\theta^{N}$$\wedge^{d}\mathcal{E}$$\mathbb{P}(H^{0}(X, \wedge^{d}\mathcal{E}))$

Theorem 1.1. Let(1)$\theta$be the Pl\"ucker class of$\mathbb{G}_{X}(d, \mathcal{E})$. Then,$\int_{\mathbb{G}_{X}(d,\mathcal{E})}\theta^{N}=N!\sum_{|k|=n}\frac{\prod_{1\leq i\triangleleft\leq d}(k_{i}-k_{j}-i+j)}{\prod_{1\leq i\leq d}(r+k_{i}-i)!}\int_{X}\prod_{1\leq i\leq d}s_{k_{i}}(\mathcal{E})$wherei-th Segre class of$k=(k_{1}, \ldots, k_{d})\in \mathbb{Z}_{\geq 0}^{d}$

(2)

with

$|k|$

$:= \sum_{i}k_{i}$, and$s_{i}(\mathcal{E})$$\mathcal{E}.$$\int_{\mathbb{G}_{X}(d,\mathcal{E})}\theta^{N}=N!\sum_{|\lambda|=n}\frac{\prod_{1\leq i<j\leq d}(\lambda_{i}-\lambda_{j}-i+j)}{\prod_{1\leq i\leq d}(r+\lambda_{i}-i)!}\int_{X}\Delta_{\lambda}(s(\mathcal{E}))$,is the92
whereis the Schur polynomial of$\Delta_{\lambda}(s(\mathcal{E}))$$(\lambda_{1}, \ldots, \lambda_{d})$.

$\mathcal{E}$for a partition$\lambda=$, the push-forward of the
In fact, we give two formulae for ch
Chern character of $\det Q$ by , explicitly (Theorem 2.1), under the assumption that $X$ is a scheme of finite type over a field : The above result
is a direct consequence of those formulae.
here are the ones satisfying
The Segre classes
are respectively the Segre series andandas in [1], [5], where
differs by
the Chern polynomial of in . Note that our Segre classthe $sign(-1)^{i}$ from the one in [2].
Theorem 1.1 with $n=0$ yeilds the degree formula of Grassmann varieties, as follows:$\pi_{*}$$(\det \mathcal{Q})$

$\pi$

$k$

$s(\mathcal{E}, t)c(\mathcal{E}, -t)=1$$s_{i}(\mathcal{E})$$c(\mathcal{E}, t)$$s(\mathcal{E}, t)$$\mathcal{E}$$t$$s_{i}(\mathcal{E})$

Corollary 1.2 [2, Example 14.7.11 (iii)] . The degree of the Grassmannvarietyof codimension subspaces of an -dimensional vector spacewith respect to the Pl\"ucker embedding is given by
$)$

$($

$d$

$\mathbb{G}(d, r)$$r$$\deg \mathbb{G}(d, r)=\frac{(d(r-d))!\prod_{1\leq k\leq d-1}k!}{\prod_{1\leq k\leq d}(r-k)!}.$

2. MAIN RESULTS
Theorem 1.1 follows from more general results, as follows: Setting $m!$ $:=$$\Gamma(m+1)$ for, one has $1/m!=0$ if $m<0$ . To simplify the notation,, set
for a finite set of integers$m\in \mathbb{Z}$$\{a_{i}\}_{0\leq i\leq d-1}$$\{a_{i}\}!:=\prod_{l}a_{l}!, \triangle(a_{i}):=\prod_{i<j}(a_{i}-a_{j})$.

Theorem 2.1. Assume that $X$ is a scheme of finite type over a field
be the Grassmann bundle of corank subbundles of a vector
Let
, let
bundle of rank on $X$ with projection :be the Chernbe the universal quotient bundle of rank , and let ch. Denote bycharacter ofis the push-forward by . Then
(1)
$k.$

$d$

$\mathbb{G}_{X}(d, \mathcal{E})$$\mathcal{E}$

$r$

$\pi$

$\mathbb{G}_{X}(d, \mathcal{E})arrow X$$d$$\pi^{*}\mathcal{E}arrow \mathcal{Q}$$(\det \mathcal{Q})$$\pi_{*}:A^{*}(\mathbb{G}_{X}(d, \mathcal{E}))\otimes \mathbb{Q}arrow A^{*-d(r-d)}(X)\otimes \mathbb{Q}$$\det \mathcal{Q}$$\pi$$\pi_{*}ch(\det \mathcal{Q})=\sum_{k}\frac{\triangle(k_{i}-i)}{\{r+k_{i}-i\}.!}\prod_{1\leq l\leq d}s_{k_{l}}(\mathcal{E})$wherek$\mathcal{E}$$=(k_{1}, \ldots, k_{d})\in \mathbb{Z}_{\geq 0}^{d}$, and$s_{i}(\mathcal{E})$,is the i-th Segre class of

93(2)$\pi_{*}ch(\det \mathcal{Q})=\sum_{\lambda}\frac{\triangle(\lambda_{i}-i)}{\{r+\lambda_{i}-i\}!}\Delta_{\lambda}(s(\mathcal{E}))$

where$\mathcal{E}$$\Delta_{\lambda}(s(\mathcal{E}))$$:=\det[s_{\lambda_{i}+j-i}(\mathcal{E})]_{1\leq i,j\leq d}$

for a partition$\lambda=(\lambda_{1}, \ldots, \lambda_{d})$3. SKETCH$($.

$oF)^{2}$,is the Schur polynomial of

PROOF

Let $X$ be a scheme of finite type over a field , and let be a vector
the partial flag bundle ofbundle of rank on $X$ . Denote by$X$
on , parametrising flags of subbundles of corank 1 up to in . Thendecomposes as a
it is easily shown that the projection:
successive composition of projective space bundles,$k$$\mathcal{E}$$\mathcal{E}$$\mathbb{F}_{X}^{d}(\mathcal{E})$

$r$

$d$

$\mathcal{E}$$p:\mathbb{F}_{X}^{d}(\mathcal{E})arrow X$$\mathbb{P}(\mathcal{E}_{i})/\mathbb{P}(\mathcal{E}_{i-1})(i\geq 1)$$p:\mathbb{F}_{X}^{d}(\mathcal{E})=\mathbb{P}(\mathcal{E}_{d-1})arrow \mathbb{P}(\mathcal{E}_{d-2})arrow\cdotsarrow \mathbb{P}(\mathcal{E}_{1})arrow \mathbb{P}(\mathcal{E}_{0})arrow X,$

is the kernel of the canonical surjection fromwith, to the tautological line bundle. Set
rk $\mathcal{E}_{i}=r-i(i\geq 0)$ : In fact,
is given as. Then, the intersection ring offollows:
, and
where
the pull-back of to$\mathcal{E}_{0}$$:=\mathcal{E}$$\mathcal{E}_{i}$$\mathcal{E}_{i+1}$$\mathcal{O}_{\mathbb{P}(\mathcal{E}_{i})}(1)$$\mathbb{P}(\mathcal{E}_{i})$$\mathbb{P}(\mathcal{E}_{i})\simeq \mathbb{F}_{X}^{i+1}(\mathcal{E})(1\leq i\leqd-1)$$\xi_{i}$$A^{*}(\mathbb{F}_{X}^{d}(\mathcal{E}))$$:=c_{1}(\mathcal{O}_{\mathbb{P}(\mathcal{E}_{i})}(1))$$A^{*}( \mathbb{F}_{X}^{d}(\mathcal{E}))=\frac{A^{*}(X)[\xi_{0},\xi_{1}.’\ldots,\xi_{d-1}]}{(P_{0}(\xi_{0}),P_{1}(\xi_{1}),..,P_{d-1}(\xi_{d-1}))}$(3.1)$= \bigoplus_{0\leq i_{l}\leq r-l-1 ,(0\leq l\leq d-1)}0^{i_{0}i_{1}}\cdots,$

where
and the symbol of pull-back tois omitted. Denote by$arrow A^{*-c}(X)$ the push-forward by, where$P_{i}(\xi_{i}):=\backslash \xi_{i}^{r-i}-c_{1}(\mathcal{E}_{i})\xi_{i}^{r-i-1}+\cdots+(-1)^{r-i}c_{r-i}(\mathcal{E}_{i})\in A^{*}(\mathbb{P}(\mathcal{E}_{i}))[\xi_{i}],$$\mathbb{F}_{X}^{d}(\mathcal{E})$$c:= \sum_{0\leq i\leq d-1}^{\backslash }(r-i-1)$$p$

ative dimension ofin(3.1), one has$\mathbb{F}_{X}^{d}(\mathcal{E}\cdot)/X$. Then, for$A^{*}(\mathbb{F}_{X}^{d}(\mathcal{E}))(\alpha_{i_{0}i_{1}\cdots i_{d-1}}\in A^{*}(X))$(3.2)$p_{*}:A^{*}(\mathbb{F}_{X}^{d}(\mathcal{E}))$, the rel-$\alpha=\sum\alpha_{i_{0}i_{1}\cdots i_{d-1}}\overline{\xi_{0^{i_{0}}}}\overline{\xi_{1^{i_{1}}}}\cdots\overline{\xi_{d-1^{d-1}}}$

with respect to the decomposition in$p_{*}\alpha=\alpha_{r-1,r-2,\ldots,r-d}.$Indeed,if and only if $i_{l}=r-l-1$ for eachLetbe the Grassmann bundle of corank subbundles$X$
be the universal quotient bundle of rankof on , and leton , parametrising flags ofof. Consider the flag bundle,
subbundles of corank 1 up to $d-1$ in . Then, as in the case of$l.$$\sum_{l}i_{l}\geq c$$G$$\mathcal{E}$

$d$

$d$

$:=\mathbb{G}_{X}(d, \mathcal{E})$$\pi^{*}\mathcal{E}arrow \mathcal{Q}$.

$\mathbb{F}_{G}^{(d-1}(\mathcal{Q})$$\mathcal{Q}$$\mathcal{Q}$$G$$\mathbb{F}_{X}^{d}(\mathcal{E})$94
the projection
projective space bundles$\mathbb{F}_{G}^{d-1}(\mathcal{Q})arrow G$

decomposes as a successive composition of:$\mathbb{P}(\mathcal{Q}_{i+1})/\mathbb{P}(\mathcal{Q}_{i})(i\geq 1)$$q:\mathbb{F}_{G}^{d-1}(\mathcal{Q})=\mathbb{P}(\mathcal{Q}_{d-2})arrow \mathbb{P}(\mathcal{Q}_{d-2})arrow\cdotsarrow \mathbb{P}(\mathcal{Q}_{1})arrow \mathbb{P}(\mathcal{Q}_{0})arrow G,$

where
, and
is the kernel of the canonical surjection fromthe pull-back of, to the tautological line bundletowith$\mathcal{Q}_{i}=d-i(i\geq0)$rk: In fact,.
It follows from the construction of vector bundles that is a coranksubbundle of, which induces a morphism, :onby
the universal property of the Grassmann bundle . Then it turns out thatis naturally isomorphic toover via , as is easily verified
by using the universal property of flag bundles (see [5, \S 6], [7, \S \S 0-1]): We
identify them via the natural isomorphism. Under thisidentification, it follows that$\mathcal{Q}_{0}$$:=\mathcal{Q}$$\mathcal{Q}_{i+1}$$\mathcal{Q}_{i}$$\mathbb{P}(\mathcal{Q}_{i})$$\mathcal{O}_{\mathbb{P}(\mathcal{Q}_{i})}(1)$$\mathbb{P}(\mathcal{Q}_{i})\simeq \mathbb{F}_{G}^{i+1}(\mathcal{Q})(1\leq i\leqd-2)$$p^{*}\mathcal{E}$$\mathbb{F}_{X}^{d}(\mathcal{E})$$d$$\mathcal{E}_{d}$$\mathcal{E}_{i}$$r$$\mathbb{F}_{X}^{d}(\mathcal{E})arrow G$$G$$\mathbb{F}_{G}^{d-1}(\mathcal{Q})$$\mathbb{F}_{X}^{d}(\mathcal{E})$

$G$

$r$

$\mathbb{F}_{G}^{d-1}(\mathcal{Q})\simeq \mathbb{F}_{X}^{d}(\mathcal{E})$$\xi_{i}=c_{1}(\mathcal{O}_{\mathbb{P}(\mathcal{E}_{i})}(1))=c_{1}(\mathcal{O}_{\mathbb{P}(Q_{i})}(1))$

in, where the symbol of pull-back tois omitted, as before.
For the Pl\"ucker class, one has$A^{*}(\mathbb{F}_{X}^{d}(\mathcal{E}))=A^{*}(\mathbb{F}_{G}^{d-1}(\mathcal{Q}))$$\mathbb{F}_{G}^{d-1}(\mathcal{Q})$$\theta=c_{1}(\mathcal{Q})$

Lemma 3.1.
(1)$q^{*}\theta=\xi_{0}+\cdots+\xi_{d-1}$(2)
in$\theta^{N}=q_{*}(\xi_{0}^{d-1}\xi_{1}^{d-2}\cdots\xi_{d-2}q^{*}\theta^{N})$in $A^{*}(G)$ .
.

$A^{*}(\mathbb{F}_{X}^{d}(\mathcal{E}))=A^{*}(\mathbb{F}_{G}^{d-1}(\mathcal{Q}))$

FIGURE 1$\mathbb{F}_{X}^{d}(\mathcal{E})=$

95
It follows from Lemma 3.1, the commutativity $p=\pi oq$ and (3.2) that$\pi_{*}(\theta^{N})=\pi_{*}q_{*}(\xi_{0}^{d-1}\xi_{1}^{d-2}\cdots\xi_{d-2}q^{*}\theta^{N})=\pi_{*}q_{*}(\prod_{i=0}^{d-1}\xi_{i}^{d-1-i}(\sum_{i=0}^{d-1}\xi_{i})^{N})$(3.3)$=p_{*}( \prod_{i=0}^{d-1}\xi_{i}^{d-1-i}(\sum_{i=0}^{d-1}\xi_{i})^{N})$$= coeff_{\overline{\xi_{0}},\ldots,\overline{\xi_{d-1}}}(\prod_{i=0}^{d-1}\xi_{i}^{d-1-i}(\sum_{i=0}^{d-1}\xi_{i})^{N};r-1, \ldots, r-d)$

wheredenotes the coefficient of$coeff_{\overline{\xi_{0_{\rangle}}}\ldots,\overline{\xi_{d-1}}}(\cdots ; r-1, \ldots, r-d)$. . . $–d$Now one can show that,$\cdots$in$\cdots$in$\sim-1--2$$\xi_{0}$$\xi_{d-1}$$\xi_{1}$Lemma 3.2.
$coeff_{\overline{\xi_{i}}}(\xi_{i}^{p_{i}};r-i-1)=const_{t_{i}}(t_{i}^{-p_{i}+r-i-1}s(\mathcal{E}_{i}, t_{i}))$,where $const_{t_{i}}(\cdots)$ the constant term in the Laurent expansion of$t_{i}.$

Applying Lemma 3.2 repeatedly, one obtains

Lemma 3.3.
$coeff_{\overline{\xi_{0}},(\overline{\xi_{d-1}}}(\xi_{0}^{p0}\cdots\xi_{d-1}^{p_{d-1}};r-1, \ldots, r-d)$$= const_{\underline{t}}(\triangle(t_{0}, \ldots, t_{d-1})\prod_{i=0}^{d-1}t_{i}^{-p_{i}+r-d}s(\mathcal{E}_{0}, t_{i}))$where $:=(t_{0}, \ldots, t_{d-1})$ , andVandermonde polynomial of$\underline{t}$$\triangle(t_{0}, \ldots, t_{d-1})$$(t_{0}, \ldots, t_{d-1})$$:= \prod_{0\leq i<j\leq d-1}(t_{i}-t_{j})$

,

the

.

By virtue of (3.3) and Lemma 3.3, one can show
Proposition 3.4. For a non-negative integer $N,$$\pi_{*}\theta^{N}=const_{\underline{t}}(P_{N}(\underline{t}))$whereis the Segre series of$\pi_{*}:A^{*}(\mathbb{G}_{X}(d, \mathcal{E}))arrow A^{*-d(r-d)}(X)$$\mathcal{E}$,is the push-forward byin , and$t$$P_{N}( \underline{t}):=\triangle(\underline{t})(\sum_{i=0}^{d-1}\frac{1}{t_{i}})^{N}\prod_{i=0}^{d-1}t_{i}^{-(d-1-i)+r-d}s(\mathcal{E}_{0}, t_{i})$.

$\pi,$$s(\mathcal{E}, t)$

96Now, to prove Theorem 2.1 (1), just expand the Laurent series $P_{N}(t)$ by
the multinomial theorem with the following
Lemma 3.5 ([2, Example A.9.3]).
$\det[\frac{1}{(x_{i}+j)!}]_{0\leq i,j\leq d-1}=\frac{\triangle(x_{i})}{\{x_{i}+d-1\}!}.$For Theorem 2.1 (2), we have two proofs, where we use a consequence ofCauchy identity [6, Chapter I, (4.3)] and Jacobi-’Rudi identity [2, Lemma
A.9.3], as follows:
Lemma 3.6.
$\prod_{i=0}^{d-1}s(\mathcal{E}, t_{i})=\sum_{\lambda\geq 0}\Delta_{\lambda}(s(\mathcal{E}))s_{\lambda}(\underline{t})$.

, similarly to the
One of our proofs is obtained just by expanding
proof of Theorem 2.1 (1). For the other, we establish a formula of Kadelltype for confluent Selberg integral, due to Terasoma, as follows (Cf. [3]):$P_{N}(\underline{t})$Proposition 3.7. Set$W_{\exp}(x, \underline{t}) :=\prod_{i=0}^{d-1}t_{i}^{x-1}\prod_{i=0}^{d-1}\exp(-t_{i})\prod_{i<j}(t_{i}-t_{j})^{2},$$I_{conf}( \lambda, x):=\int_{[0,+\infty)^{d}}s_{\lambda}(\underline{t})W_{\exp}(x,\underline{t})d\underline{t}.$Then$I_{conf}(\lambda, x)=d!\triangle(\lambda_{i}-i)\Gamma\{x+d-i+\lambda_{i}\},$

for a real number $x>0$ , where$\underline{t}:=(t_{0}, \ldots, t_{d-1})$

and$d\underline{t}:=dt_{0}\cdots dt_{d-1}.$

Remark 3.8. Symmetrising the Laurent series $P_{N}(t)$ with respect to thevariables , one sees thatis equal to the constant term of
the Laurent series,$\underline{t}$$c\circ nst_{\underline{t}}(P_{N}(t))$.

$P_{N}^{S}( \underline{t}):=\frac{(-1)^{\frac{d(d-1)}{2}}}{d!}\prod_{0\leq i\triangleleft\leq d-1}(\frac{1}{t_{i}}-\frac{1}{t_{j}})^{2}(\sum_{i=0}^{d-1}\frac{1}{t_{i}})^{N}\prod_{i=0}^{d-1}t_{i}^{r-1}s(\mathcal{E}, t_{i})$Roughly speaking, to obtain the constant term of, we calculate
the residue ofby using Proposition 3.7 (seeRemark 3.8): Indeed, we have.
$P_{N}(\underline{t})$$P_{N}^{s}(t_{0}^{-1}, \ldots, t_{d-1}^{-1})(t_{0}\cdots t_{d-1})^{-1}$$const_{\underline{t}}(P_{N}(\underline{t}))=const_{\underline{t}}(P_{N}^{s}(t_{0}^{-1}, \ldots, t_{d-1}^{-1}))$

97
4. EXAMPLE
Example 4.1.

$\deg \mathbb{G}_{\mathbb{P}^{4}}(2, T_{\mathbb{P}^{4}})=5040$. This number is exactly equal to

the factorial of 7 (pointed out by Agaoka): $5040=7!.$ guess this would
be nothing but a coincidence without rationale (what do you think?).
$I$

Acknowlegements. The author would like to thank Professor Daisuke Matsushita, the organizer of the symposium, for the invitation. The author
thanks also Professor Yoshio Agaoka for pointing out the coincidence. The
author is supported by JSPS KAKENHI Grant Number 25400053.

REFERENCES[1] T. Fujita: Classification theories of polarized varieties. London Mathematical Society Lecture Note Series, 155. Cambridge University Press, Cambridge, 1990.
[2] W. Fulton: Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete(3), 2. Springer-Verlag, Berlin, 1984.
[3] K. W. J. Kadell: $A$ proof of some -analogues of Selberg’s integral for $k=1$ . SIAM
J. Math. Anal. 19 (1988), no. 4, 944-968.
[4] H. Kaji, T. Terasoma: Degree formulae for Grassmann bundles, in preparation.
[5] D. Laksov, A. Thorup: Schubert calculus on Grassmannians and exterior powers.
Indiana Univ. Math. J. 58 (2009), no. 1, 283-300.
[6] I. G. MacDonald: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs (2nd ed.). The Clarendon Press Oxford University Press, 1995
[7] D. B. Scott: Grassmann bundles. Ann. Mat. Pura Appl. (4) 127 (1981), 101-140.
$q$

DEPARTMENT OF MATHEMATICS,
FACULTY OF SCIENCE AND ENGINEERING,WASEDA UNIVERSITY
3-4-1 OHKUBO, SHINJUKU, TOKYO 169-8555, JAPAN
$E$-mail address: kaj i@waseda.jp