数理解析研究所講究録
第 1922 巻 2014 年 28-33

28Spaces of equivariant maps to toric varieties
山口耕平 (Kohhei Yamaguchi)
電気通信大学情報理工学研究科(Department of Math., Univ. of Electro-Communications)
Abstract
The main purpose of this note is to announce the recent result in [13] concerning
the homotopy type of spaces of algebraic maps from a real projective space to acompact smooth real toric variety. This note is also based on the joint work withAndrzej Kozlowski and Masahiro Ohno [8].

An irreducible normal algebraic variety $X$ (over ) is called a toric
Toric varieties.
, such thatvariety if it has an algebraic action of the complex algebraic torus
some point $*\in X$ is dense in $X$ and isomorphic tothe orbitof the form
is a subset of
A strong convex rational polyhedral cone indoes not contain any line.
, such that the set
is called a fan if
A finite collection of strongly convex rational polyhedral cones inevery face of element of is belongs to and the intersection of any two elements of isa face of each other. It is well-known that a toric variety $X$ is completely characterized up
the corresponding toric variety associatedto isomorphism by the fan . We denote byis called smooth (reps. simplicial) if it is generated by a subset of ato . A cone in. It(resp. a subset of a basis of ). A fan is called complete if
basis ofisis smooth iﬀ every
is known thatis compact iﬀ is complete, and thatis isomorphic to the quotient ofsmooth [4, Theorem 1.3.12]. It is also known thatis
. [ , Theorem 12.1.10]. In particular,
by the subgroup generated bysimply connected if it is compact.
$\mathbb{C}$$\mathbb{T}_{\mathbb{C}}^{r}=(\mathbb{C}^{*})^{r}$$\mathbb{T}_{\mathbb{C}}^{r}.$$\mathbb{T}_{\mathbb{C}}^{r}\cdot*of$$\mathbb{R}^{n}$$\mathbb{R}^{n}$$\sigma=$$\sigma$$\{\sum_{k=1}^{s}a_{k}n_{k}|a_{k}\geq 0\}$$\{n_{k}\}_{k=1}^{s}\subset \mathbb{Z}^{n}$$\mathbb{R}^{n}$$\Sigma$$\Sigma$$\Sigma$$\sigma$$\Sigma$$\Sigma$$\Sigma$$X_{\Sigma}$$\mathbb{R}^{n}$$\Sigma$$\mathbb{R}^{n}$$\mathbb{Z}^{n}$$\bigcup_{\sigma\in\Sigma}\sigma=\mathbb{R}^{n}$$\Sigma$$X_{\Sigma}$$\sigma\in\Sigma$$X_{\Sigma}$$\pi_{1}(X_{\Sigma})$$\mathbb{Z}^{n}$$\bigcup_{\sigma\in\Sigma}\sigma\cap \mathbb{Z}^{n}$$4$$X_{\Sigma}$consisting
denote the subspace of
For a fan , letReal toric varieties.
is given by replacing the complex. Alternatively the spaceof all real points ofnumbers by the real numbers everywhere in the given definitions of a toric variety
with the intersection[ $18$ , Def. 6.1], and it is called a real toric variety. Note thatwhenis a toric variety embedded in$\Sigma$$X_{\Sigma}$$X_{\Sigma}$$X_{\Sigma,\pi}$$X_{\Sigma,\mathbb{R}}$$X_{\Sigma}$$\mathbb{R}$$\mathbb{C}$$X_{\Sigma,R}$$X_{\Sigma,N}=X_{\Sigma}\cap \mathbb{R}P^{N}$$\mathbb{C}P^{N}.$$X_{\Sigma}$. We shall use the symbols
Homogenous coordinates ofor . For polynomialsvariables of polynomials, and we assume that$\{z_{k}\}_{k=1}^{r}$$X_{\Sigma,\mathbb{K}}$$\mathbb{K}=\mathbb{R}$$\mathbb{C}$to denote$f_{1},$$\cdots,$$f_{s}\in$29$\mathbb{K}[z_{1}, \cdots, z_{r}]$

, let

(1.1)

$V_{\mathbb{K}}(f_{1}, \cdots, f_{s})$denote the aﬃne variety$V_{\mathbb{K}}(f_{1}, \cdots, f_{s})=\{x\in \mathbb{K}^{r}|f_{k}(x)=0$

for each $1\leq k\leq s\}.$denote the set of all one dimensional cones in a fan , and let
denote the generator of(called the primitive
such thatelement of ) for each $1\leq k\leq r$ . Define the aﬃne variety
byLet$\Sigma(1)=\{\rho_{1}, \cdots, \rho_{r}\}$$\Sigma$$n_{k}\in \mathbb{Z}^{n}$$\rho_{k}\cap \mathbb{Z}^{n}$$\rho_{k}\cap \mathbb{Z}^{n}=\mathbb{Z}_{\geq 0}\cdot n_{k}$$Z_{\Sigma,\mathbb{K}}\subset \mathbb{K}^{r}$$\rho_{k}$(1.2)$Z_{\Sigma,K}=V_{\mathbb{K}}(z^{\hat{\sigma}}|\sigma\in\Sigma)$

,

where
denotes the monomial
define the subgroupby$z^{\hat{\sigma}}$$z^{\hat{\sigma}}= \prod_{1\leq k\leq r,n_{k}\not\in\sigma}z_{k}\in \mathbb{Z}[z_{1}, \cdots, z_{r}]$. Let$\mathbb{T}_{\mathbb{K}}^{r}=(\mathbb{K}^{*})^{r}$and$G_{\Sigma,K}\subset \mathbb{T}_{\mathbb{K}}^{r}$(1.3)$G_{\Sigma,K}$$=$$\{(\mu_{1}, \cdots, \mu_{r})\in \mathbb{T}_{K}^{r}|\prod_{k=1}^{r}\mu_{k}^{\langle m,n_{k}\rangle}=1$

for all$m\in \mathbb{Z}^{n}\}.$

It is known that that there is an isomorphism
for
if thespans
set, where the group
acts on the complementby thecoordinate-wise multiplications and the spacedenotes its orbit space.
It is known thatacts freely on the complement
if is smooth and. In this case, forisomorphisms
there are$\mathbb{K}=\mathbb{C}$$X_{\Sigma,\mathbb{K}}\cong(\mathbb{K}^{r}\backslash Z_{\Sigma,\mathbb{K}})//G_{\Sigma,K}$$\{n_{1}, \cdots, n_{r}\}$$\mathbb{R}^{n}$$\mathbb{K}^{r}\backslash Z_{\Sigma,\mathbb{K}}$$G_{\Sigma,\mathbb{K}}$$(\mathbb{K}^{r}\backslash Z_{\Sigma,\mathbb{K}})//G_{\Sigma,K}$$G_{\Sigma,\mathbb{K}}$$\mathbb{K}=\mathbb{C}$$\mathbb{K}^{r}\backslash Z_{\Sigma,\mathbb{K}}$$\Sigma$$\mathbb{K}=\mathbb{C}$

(1.4)

and

$X_{\Sigma,\mathbb{K}}\cong(\mathbb{K}^{r}\backslash Z_{\Sigma,\mathbb{K}})/G_{\Sigma,K}$

Note that (1.4) also holds for$K=\mathbb{R}$if$\Sigma$$G_{\Sigma,K}\cong \mathbb{T}_{\mathbb{K}}^{r-n}.$

is smooth and complete [19, Lemma 7.3].

We say that a set of primitive elements
is primitive if they do not lie inany cone in but every proper subset does. It is known that$\{n_{i_{1}}, \cdots, n_{i_{k}}\}$$\Sigma$(15)$Z_{\Sigma,\mathbb{K}}=\{n_{i_{1}},\cdots,n_{i_{k}}\}:\cup$$V_{\mathbb{K}}(z_{i_{1}}, \cdots z_{i_{k}})$primitiveSo$Z_{\Sigma,K}$(1.6)

is a closed variety with real dimension $(r-r_{\min})\dim\pi^{\mathbb{K}}$ , where we set$r_{\min}= \min${$k\in \mathbb{Z}_{\geq 1}|\{n_{i_{1}},$$\cdots,$$n_{i_{k}}\}$

is primitive}.

Spaces of continuous maps. For connected spaces $X$ and , let Map $(X, Y)$ be thespace of all continuous maps : $Xarrow Y$ and Map* $(X, Y)$ the corresponding subspace ofall based continuous maps. If $m\geq 2$ and $g\in Map^{*}(\mathbb{R}P^{m-1}, X)$ , let $F(\mathbb{R}P^{m}, X;g)$ denotethe subspace ofgiven by$Y$$f$

$Map^{*}(\mathbb{R}P^{m}, X)$(1.7)$F(\mathbb{R}P^{m}, X;9)=\{f\in Map^{*}(\mathbb{R}P^{m}, X):f|\mathbb{R}P^{m-1}=g\},$by putting $x_{m}=0$ . It is known that there is a homotopy
where we identify
equivalence $F(\mathbb{R}P^{m}, X;g)\simeq\Omega^{m}X$ if it is not an empty set.
$\mathbb{R}P^{m-1}\subset \mathbb{R}P^{m}$

30
From now on, we assume that the following two conditions are satisfied:

Assumptions.

be a complete smooth fan in(1.7.1) Let
dimension cones in , and all primitive elements
denotes the primitive element of
where$\Sigma$$\mathbb{R}^{n},$$\Sigma(1)=\{\rho_{1}, \cdots, \rho_{r}\}$$\Sigma$$n_{k}\in \mathbb{Z}^{n}$$\{n_{1}, \cdots, n_{r}\}$be the set of all oneof the fan spans$\Sigma$$\mathbb{R}^{n},$for $1\leq k\leq r.$$\rho_{k}$(1.7.2) Let $D=(d_{1}, \cdots, d_{r})$ be an -tuple of positive integers such that $\sum_{k=1}^{r}d_{k}n_{k}=0.$$r$

Then we can identify
sponding element of$X_{\Sigma,K}=(\mathbb{K}^{r}\backslash Z_{\Sigma,\mathbb{K}})/G_{\Sigma,\mathbb{K}}$$X_{\Sigma,K}$

for each

and we denote by$[a_{1}, \cdots, a_{r}]$the corre-$(a_{1}, \cdots, a_{r})\in \mathbb{K}^{r}\backslash Z_{\Sigma,\mathbb{K}}.$Spaces of polynomials representing algebraic maps. Let
denote the -vector subspace consisting of all homogeneous polynomials of degree . Let$A_{D}(m)$ denote the spaceand letdenote the subspace$\mathcal{H}_{d,m}^{K}\subset \mathbb{K}[z_{0}, .

for any$A_{D,\Sigma}^{\mathbb{K}}=\{(f_{1}, \cdots, f_{r})\in A_{D}^{\mathbb{K}}(m)|F(x)\not\in Z_{\Sigma,\mathbb{K}}$where we set $F(x)=(f_{1}(x), \cdots, f_{r}(x))$ .
, we choose $x_{0}=[1,$Becauseby. Define the subspaceof$(1, 1, \cdots, 1)\in \mathbb{K}^{r}\backslash Z_{\Sigma,\mathbb{K}}$, z_{m}]$$\cdots$,1$]$$x\in \mathbb{R}^{m+1}\backslash \{0\}\},$$\in X_{\Sigma,\mathbb{K}}$as the base-point$A_{D}(m, X_{\Sigma,K})\subset A_{D,\Sigma}^{\mathbb{K}}(m)$$X_{\Sigma,K}$

(1.9)

.

$A_{D,\Sigma}^{\mathbb{K}}(m)\subset A_{D}^{\mathbb{K}}(m)$$A_{D}^{\mathbb{K}}(m)=\mathcal{H}_{d_{1},m}^{\mathbb{K}}\cross \mathcal{H}_{d_{2)}m}^{\mathbb{K}}\cross\cdots\cross \mathcal{H}_{d_{r},m}^{\mathbb{K}}$

(1.8)

.

$d$

$\mathbb{K}$$A_{D}(m, X_{\Sigma,\mathbb{K}})=\{(f_{1}, \cdots, f_{r})\in A_{D,\Sigma}^{\mathbb{K}}(m)|(f_{1}(e_{1}), \cdots, f_{r}(e_{1}))=(1,1, \cdots 1)$

where $e_{1}=(1,0, \cdots, 0)\in \mathbb{R}^{m+1}$ , and let us choose $[e_{1}]=[1 : 0:\cdots : 0]$ as the base-pointby:. Define the natural mapof$\mathbb{R}P^{m}$(1.10)$j_{D,K}’$$A_{D,\Sigma}^{\mathbb{K}}(m)arrow Map(\mathbb{R}P^{m}, X_{\Sigma,\mathbb{K}})$for$j_{D,K}’(f_{1}, \cdots, f_{r})([x])=[f_{1}(x), \cdots, f_{r}(x)]$is connected, the image ofSince the space, which is denoted by
Map$A_{D,\Sigma}^{\mathbb{K}}(m)$Note thatset$j_{D,K}’$

lies in a connected component of. This gives the natural map$Map_{D}(\mathbb{R}P^{m}, X_{\Sigma,K})$$(\mathbb{R}P^{m}, X_{\Sigma,\mathbb{K}})$(1.11)$x=(x_{0}, \cdots, x_{m})\in \mathbb{R}^{m+1}\backslash \{0\}.$$j_{D,\mathbb{K}}’$:$A_{D,\Sigma}^{K}(m)arrow Map_{D}(\mathbb{R}P^{m}, X_{\Sigma,K})$$j_{D,\mathbb{K}}’(f_{1}, \cdots, f_{r})\in Map^{*}(\mathbb{R}P^{m}, X_{\Sigma,K})$if.

$Map_{D}^{*}(\mathbb{R}P^{m}, X_{\Sigma,\mathbb{K}})=Map^{*}(\mathbb{R}P^{m}, X_{\Sigma,\mathbb{K}})\cap Map_{D}(\mathbb{R}P^{m}, X_{\Sigma,\mathbb{K}})$(1.12)$i_{D,K}=j_{D,\mathbb{K}}’|A_{D}(m, X_{\Sigma,\mathbb{K}})$:. Hence, if we, we have the natural map$(f_{1}, \cdots, f_{r})\in A_{D}^{\mathbb{K}}(m, X_{\Sigma})$$A_{D}(m, X_{\Sigma,\mathbb{K}})arrow Map_{D}^{*}(\mathbb{R}P^{m}, X_{\Sigma,\mathbb{K}})$.

Suppose that $m\geq 2$ and let us choose a fixed element $(g_{1}, \cdots, g_{r})\in A_{D}(m-1, X_{\Sigma,K})$ .
. Then define the subspace
For each $1\leq k\leq r$ , letby$B_{k}^{\mathbb{K}}=\{g_{k}+z_{m}h:h\in \mathcal{H}_{d_{k}-1,m}^{\mathbb{K}}\}$$A_{D}(m, X_{\Sigma,\mathbb{K}};g)\subset A_{D}(m, X_{\Sigma,\mathbb{K}})$(1.13)$A_{D}(m, X_{\Sigma,\mathbb{K}};g)=A_{D}(m, X_{\Sigma,\mathbb{K}})\cap(B_{1}^{\mathbb{K}}\cross B_{2}^{\mathbb{K}}\cross\cdots\cross B_{r}^{\mathbb{K}})$.

31
It is easy to see that
denotes the map in$Map_{D}^{*}(\mathbb{R}P^{m-1}, X_{\Sigma,\mathbb{K}})$(1.14)$g([x_{0}:.

.

.

for

:x_{m-1}])=[g_{1}(x), \cdots, g_{r}(x)]$Then, one can define the map(1.15)

if $(f_{1}, \cdots, f_{r})\in A_{D}(m, X_{\Sigma,K};g)$ , where
given by$i_{D,\mathbb{K}}(f_{1}, \cdots, f_{r})|\mathbb{R}P^{m-1}=9$$i_{D,\mathbb{K}}’$:$g$$x=(x_{0}, \cdots, x_{m-1})\in \mathbb{R}^{m}\backslash \{0\}.$$A_{D}(m, X_{\Sigma,K};g)arrow F(\mathbb{R}P^{m}, X_{\Sigma,K};g)\simeq\Omega^{m}X_{\Sigma,\mathbb{K}}$by$i_{D,K}’=i_{D,\mathbb{K}}|A_{D}(m, X_{\Sigma,\mathbb{K}};g):A_{D}(m, X_{\Sigma,\mathbb{K}};g)arrow F(\mathbb{R}P^{m}, X_{\Sigma,\mathbb{K}};g)\simeq\Omega^{m}X_{\Sigma,\mathbb{K}}.$

Now consider the action of
on the space
given by the coordinate-wise multiplication and define the spaceby the quotient space$G_{\Sigma,\mathbb{K}}$$A_{D,\Sigma}^{K}(m)$$\overline{A_{D}}(m, X_{\Sigma,\mathbb{K}})$(1.16)$\overline{A_{D}}(m, X_{\Sigma,\mathbb{K}})=A_{D,\Sigma}^{K}(m)/G_{\Sigma,\mathbb{K}}.$

It is easy to see that one can define the map(1.17)

Let$d_{\min}$(1.18)$j_{D,\mathbb{K}}$:$\overline{A_{D}}(m, X_{\Sigma,\mathbb{K}})arrow Map_{D}(\mathbb{R}P^{m}, X_{\Sigma,\mathbb{K}})$$j_{D,K}([f_{1}, \cdots, f_{r}])([x_{0}, \cdots, x_{r}])=[f_{1}(x), \cdots, f_{r}(x)]$

for

by

$x\in \mathbb{R}^{m+1}\backslash \{0\}.$and $D_{\pi}(d_{1}, \cdots, d_{r};m, r)$ be the positive integer defined by$d_{\min}= \min\{d_{1}, d_{2}, \cdot\cdot , d_{r}\}, D(d_{1}, \cdots, d_{r};m)=(r_{\min}-m-1)d_{\min}-2.$

From now on we write$(X_{\Sigma,\mathbb{K}}, Z_{\Sigma,\mathbb{K}}, G_{\Sigma,K})=(X_{\Sigma}, Z_{\Sigma}, G_{\Sigma})$

The main results.

The main results of this note are stated as follows.

if$\mathbb{K}=\mathbb{C}.$

Theorem 1.1 ([13]). Let, let $\{d_{k}:1\leq k\leq r\}$ be
be a complete smooth fan in
the set of positive integers satisfying the conditions (1.7.1), (1.7.2), and let
be asmooth compact real toric variety associated to the fan . Then if $1\leq m\leq r_{\min}-2$ and, the map$\Sigma$$\mathbb{R}^{n}$$X_{\Sigma,\mathbb{R}}$$\Sigma$$D=(d_{1}, \cdots, d_{r})\in(\mathbb{Z}_{\geq 1})^{r}$$i_{D,\pi}’$:$A_{D}(m, X_{\Sigma,\pi};g)arrow F(\mathbb{R}P^{m}, X_{\Sigma,\mathbb{R}};g)\simeq\Omega^{m}X_{\Sigma,\pi}$is a homology equivalence through dimension $D(d_{1}, \cdots, d_{r};m)$ .

$\square$

Theorem 1.2 ([13]). Under the same assumptions as Theorem 1.1, if $1\leq m\leq r_{\min}-2$
and $D=(d_{1}, \cdots, d_{r})\in$ $()^{r}$ , the maps$\{\begin{array}{l}j_{D,\mathbb{R}}:\overline{A_{D}}(m, X_{\Sigma,\mathbb{R}})arrow Map_{D}(\mathbb{R}P^{m}, X_{\Sigma,\mathbb{R}})i_{D,\mathbb{R}}:

A_{D}(m, X_{\Sigma,\pi})arrow Map_{D}^{*}(\mathbb{R}P^{rn}, X_{\Sigma,\pi})\end{array}$are homology equivalences through dimension $D(d_{1}, \cdots, d_{r};m)$ .

$\square$32
Remark 1.3. (i) A map $f:Xarrow Y$ is called a homology equivalence through dimension
is an isomorphism for any $k\leq D.$if(ii) Letbe a finite group and let : $Xarrow Y$ be a -equivariant map between G$X$
spaces and . Then the map : $Xarrow Y$ is called a -equivariant homology equivalenceis an isomorphism for any $k\leq D$:
through dimension if$H\subset
G$$W^{H}=\{x\in
W|g\cdot
x=x$
for any $g\in H\}$ for a -space, whereand any subgroup$W$ anddenotes the restriction map $f^{H}=f|X^{H}.$-action on the(iii) Note that the complex conjugation on naturally induces the. Similarly, it also induces the, and it is easy to see thatspace
. Moreover, if we consider,,
actions on the space
also induces the-action-space of the trivial action, thethe spaceas.
,,-actions on the spaces$D$$f_{*}:H_{k}(X, \mathbb{Z})arrow^{\underline{}\simeq}H_{k}(Y, \mathbb{Z})$

$G$

$G$

$f$

$Y$

$G$

$f$

$D$

$f_{*}^{H}$$H_{k}(X^{H}, \mathbb{Z})arrow^{\simeq\underline{}}H_{k}(Y^{H}, \mathbb{Z})$$G$$f^{H}$$\mathbb{C}$$\mathbb{Z}/2$$(X_{\Sigma})^{Z/2}=X_{\Sigma,R}$$X_{\Sigma}$$A_{D,\mathbb{C}}(m, X_{\Sigma})$$\mathbb{R}P^{m}$$\tilde{A}_{D,\mathbb{C}}(m, X_{\Sigma})$$A_{D,\mathbb{C}}(m, X_{\Sigma};g)$$X_{\Sigma}$$\mathbb{Z}/2$$a\mathbb{Z}/2$$Map_{D}^{*}(\mathbb{R}P^{m}, X_{\Sigma})$$\mathbb{Z}/2$$\mathbb{Z}/2-$$Map_{D}(\mathbb{R}P^{m}, X_{\Sigma})$$F(\mathbb{R}P^{m}, X_{\Sigma};g)$Corollary 1.4 ([8], [13]). Under the same assumptions as Theorem 1.1, if $2\leq m\leq r_{\min}-2$
and $D=(d_{1}, \cdots, d_{r})\in$ $()^{r}$ , the maps$\{\begin{array}{l}i_{D,\mathbb{C}}’:are$\mathbb{Z}/2$

A_{D}(m, X_{\Sigma};g)arrow F(\mathbb{R}P^{m}, X_{\Sigma};g)\simeq\Omega^{m}X_{\Sigma}j_{D,\mathbb{C}}:\overline{A_{D}}(m, X_{\Sigma})arrow Map_{D}(\mathbb{R}P^{m}, X_{\Sigma})i_{D,\mathbb{C}}:

A_{D}(m, X_{\Sigma})arrow Map_{D}^{*}(\mathbb{R}P^{m}, X_{\Sigma})\end{array}$-equivariant homology equivalences through dimension $D(d_{1}, \cdots, d_{r};m)$ .

口

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Department of Mathematics, University of Electro-Communications
1-5-1 Chufugaoka, Chofu, Tokyo 182-8585, Japan-mail: kohhei@im.uec.ac.jp$E$