From 4c963c8f0e55d4bc5c371bbc93d5402b0bcdcd42 Mon Sep 17 00:00:00 2001 From: Alexander Luzgarev Date: Tue, 21 Jun 2016 23:03:35 +0300 Subject: [PATCH] Add Lecture 8 --- motives.pdf | Bin 612358 -> 641365 bytes motives.tex | 269 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 269 insertions(+) diff --git a/motives.pdf b/motives.pdf index c6294112656fc5103b0a762823bfc4c3daeb7ac9..8c6afbcbd25207eb42f9c576a828a9f15df46aa1 100644 GIT binary patch delta 214561 zcmZU(Ly#s66Qx@=yKLLGZQHiZx6ozVt}ffQZQHhazyHR>Y-X1kdDeNJIGLe!WXp*Z zf5;R?#p#&n*9UCMhwY zG@eS_!6IC+#~?0*1BiTsB9Le`XAv78PC~fm>hwvEfaqH`X%$rmjgg%Xw2ysDBjncFzU@jq?D3$x2;EQ|<+H`|Vm_fW^cUL(z$$l2#Lv zjY$e=E0PMyQyznP?W01Ig*3Ja6KhsQL-os{zQn_T07T_s!TG5<3;HF4RfCfd0P#TV z;#LLSgqO*NMC{&1gk;_n(ZGPhhGItUpuAq*mD+3JD$11RJj>Hf$!!;ohrs$j=I z5=9}wlFag)!6x|;jf%ioxT=PHx~Lo63|&Fed$$G30la1TMWp*$|B%Gu2Pj(-{sI?3 zL5x~bAX!yeY=$`f8KsO)(-tke0JtG3iI{Pc4g3xFW1N?soC%p>#?D5H?l&8-uWZh& z+oih{ts87mCN(OvSS7??5#bOPwf+0MXPT|~!d@s`7tKiz<_C4-J~3E7w`pZX0Vb9m z>l?*a*pF1>+5!V3fui3BFVPnGuTd7LCDsHnuJf#ZwNHmO6$zzwTmec*2mqH~%I27T zm>qa%h#6S8u-QNXYJt=shD{!VIQ|HoU5Si7lWLd%DZPJo2*wS88HAG`U_ZuV>wZOj z1)Hl9MzYUmz(lMr-Or^o&mE%HJbp;}fEOF;jsmYd%`aM2w9f}+?>dGA0|PAl#N`;c zC=fu3$=_ds^afUc6!0gJ8z8>I<5*0(<>N~`!J_n2=h=;P{CY4xeP!&{osRLO>ic

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\DeclareMathOperator{\res}{res} \DeclareMathOperator{\Trd}{Trd} +\DeclareMathOperator{\sing}{sing} %\DeclareFontFamily{OT1}{pzc}{} %\DeclareFontShape{OT1}{pzc}{m}{it}{<-> s * [1.2] pzcmi7t}{} @@ -2984,5 +2985,273 @@ $\Mor(X,Y) = \CH^{\dim Y}(X\times Y) / p$. про конечные подгруппы $\mathsf{E}_8$). \end{remark} +% 09.04.2012 + +\subsection{$J$-инвариант} + +Остается открытым вопрос, каков размер $R_p({}_{\xi}G)$. +Мы знаем, что над замыканием получается +$R_p({}_{\xi}G)_{\ol{F}} = \bigoplus\mathbb{Z}/p\{\dots\}$. +Как посчитать эти сдвиги? +Можно закодировать их многочленом: по прямой сумме +$\bigoplus_i\mathbb{Z}/p\{i\}^{\oplus a_i}$ построим +\term{полином Пуанкаре} $P(R_p(G),t) = \sum_i a_i t^i$. +Понятно, что этот многочлен контролирует +образ отображения +$\CH^*(X)/p \to \CH^*(X_{\ol{F}})/p$. + +Сейчас мы построим набор целых чисел $J_p({}_{\xi}G)$~--- +\emph{$J$-инвариант}~--- со следующими свойствами: +\begin{enumerate} +\item $P(R_p({}_{\xi}G),t)$ выражается через $J_p({}_{\xi}G)$; +\item $J_p({}_{\xi}G)$ контролирует, какие ${}_{\xi}G$-однородные +проективные многообразия действительно являются клеточными +над общей точкой; +\item для многих исключительных групп $J_p({}_{\xi}G)$ выражается +через индекс Титса. +\end{enumerate} +Пусть $B\leq G$~--- борелевская подгруппа. +Рассмотрим <<последовательность>> \emph{чего-то} +\[ +B \to G \to G/B \to BB = \pt//B. +\] +Переходя к кольцам Чжоу, получаем точную последовательность +градуированных колец вида +\[ +\CH^*(BB) \to \CH^*(G/B) \to \CH^*(G) \to \CH^*(B). +\] +При этом $\CH^*(B) = \mathbb{Z}$. +Заметим, что $\CH^*(G/B) \to \CH^*(G)$~--- сюръекция. +Поэтому есть точная последовательность +\[ +\CH^*(BB) \to \CH^*(G/B) \to \CH^*(G) \to 0. +\] +Борелевская подгруппа гомотопически эквивалентна тору: +$B \sim \mathbb{G}_m\times \dots \times \mathbb{G}_m$. +Покажем, что $\CH^*(BB) = S^*(X^*(T))$. + +Что такое $B\mathbb{G}_m$? +Первое приближение: +$\mathbb{P}^n = (\mathbb{A}^{n+1}\setminus\{0\})/\mathbb{G}_m$. +Правильный ответ: $B\mathbb{G}_m = \varinjlim_n\mathbb{P}^n$. +Мы уже знаем, что $\CH^*(\mathbb{P}^n) = \mathbb{Z}[x] / (x^{n+1})$. +Поэтому $\CH^*(B\mathbb{G}_m) = \mathbb{Z}[x]$~--- +проективный предел в категории градуированных колец +(но не в категории колец). + +Пусть $\chi_1,\dots,\chi_l$~--- базис решетки $X^*(T)$. +Если $G$ односвязна, можно взять $\varpi_1,\dots,\varpi_l$. +Если $G$ присоединенная, можно взять $\alpha_1,\dots,\alpha_l$. +Тогда $S^*(X^*(T)) \isom \mathbb{Z}[\chi_1,\dots,\chi_l]$. + +Чтобы описать отображение $\CH^*(BB) \to \CH^*(G/B)$, +достаточно задать образы элементов $\chi_i$. +Положим $\chi_i\mapsto [L_{\chi_i}]$ (класс линейного +расслоения $L_{\chi_i}$ в группе Пикара). +Здесь $L_{\chi_i}$~--- линейное расслоение на $G/B$, +построенное следующим образом. +Характер $\chi_i$ является отображением +$B\to \mathbb{G}_m$. +Тогда $L_{\chi_i} = G\times_{B}\mathbb{A}^1$, +где на $\mathbb{A}^1$ задано действие с помощью $\chi_i$. +Каноническое отображение +$G\times_{B}\mathbb{A}^1 \to G\times_{B}\pt = G/B$ +превращает $L_{\chi_i}$ в линейное расслоение. + +Пока что $G$ была расщепима. +Оказывается, если подкрутить все на $\xi$, все +$[L_{\chi_i}]$ останутся рациональными. + +\begin{example} +Рассмотрим $\mathbb{P}^1 = \SL_2/B$ и обозначим +характер тора через $\xi$. +Нас интересует действие $B$ на $\SL_2\times_{B}\mathbb{A}^1$, +при котором +матрица $\begin{pmatrix}\alpha & * \\ 0 & \alpha^{-1}\end{pmatrix}$ +действует на $\mathbb{A}^1$ умножением на $\alpha$. +Упражнение: $L_{\xi} = \mathcal{O}(-1)$, +$L_{\xi^{-1}} = \mathcal{O}(1)$. + +В общем случае в $G/B$ есть клетки Шуберта коразмерности $1$. +Пусть $G$ односвязна. +Эти клетки соответствуют фундаментальным характерам: +клетке $\chi_i$ сопоставляется $i$-ая клетка Шуберта +коразмерности $1$, равная $c_1(L(\chi_i))$. +\end{example} + +Теперь мы взяли $\xi\in H^1(F,G)$. +Тогда ${}_{\xi}(G/B)$~--- многообразие, клеточное над общей точкой +(при переходе к его полю функций у $G$ появляется борелевская +подгруппа, поэтому можно написать фильтрацию). +Нас интересует образ отображения +\[ +\CH^*({}_{\xi}(G/B)) \xrightarrow{\res} \CH^*(G/B). +\] +Все, что приходит с $\CH^*(BB)$, лежит в образе $\res$, +поскольку линейные расслоения можно скрутить: +$\res([{}_{\xi}L_{\chi}]) = [L_{\chi}]$. +Точность сохранится при факторизации по $p$: +\[ +\CH^*(BB)/p \to \CH^*(G/B)/p \to \CH^*(G)/p \to 0. +\] +При этом $\CH^*(BB)/p \isom \mathbb{Z}/p[x_1,\dots,x_l]$. +Умножение $G\times G\to G$ дает нам отображение +$\CH^*(G) \to \CH^*(G) \otimes \CH^*(G)$, +которое превращает $\CH^*(G)$ в алгебру Хопфа. +Таким образом, $\CH^*(G)/p$~--- алгебра Хопфа над $\mathbb{Z}/p$, +градуированная, конечномерная, связная, коммутативная. +\begin{theorem} +Все такие алгебры Хопфа изоморфны (как алгебры) +$\mathbb{Z} / p [x_1,\dots,x_r] / (x_i^{p^{k_i}})$. +\end{theorem} +Если $X$~--- клеточное над $\mathbb{C}$, можно сравнить +$\CH^*(X)$ и $H^*_{\sing}(X)$. +Оказывается, $\CH^i(X) = H^{2i}_{\sing}(X)$. +Для $G$ над $\mathbb{C}$ есть нетривиальные элементы +в $H^{2i+1}_{\sing}(G)$ +Например, $G = \SL_2(\mathbb{C})$, и $\SL_2(\mathbb{C})$ +гомотопически эквивалентно $S^3$. +\begin{remark} +Если $p\neq 2,3,5$, и $G$ не изогенична $\SL_n$, +то $\CH^*(G)/p = \mathbb{Z}/p$. +Случай $p=5$ возникает только для $\mathsf{E}_8$, +а случай $p=3$~--- только для $\mathsf{F}_4$, +$\mathsf{E}_6$, $\mathsf{E}_7$, $\mathsf{E}_8$ +(это делители числа Кокстера). +\end{remark} +\emph{Таблица Каца} дает для каждой $G$ и для каждого $p$ +значения $k_i$ и степени элементов $x_i$. +Обозначим $d_i = \deg(x_i)$. + +Заметим, что ${}_{\xi}(G/B) = ({}_{\xi}G)/B$; это дает короткую +точную последовательность +\[ +{}_{\xi}G \to ({}_{\xi}G)/B \to BB, +\] +из которой получаем стрелку $\CH^*(BB)/p \to \CH({}_{\xi}(G/B))/p$. +Рассмотрим коммутативную диаграму +\[ +\begin{tikzcd} +& \CH({}_{\xi}(G/B))/p \arrow{d} \arrow{dr}{\ph} \\ +\CH^*(BB)/p \arrow{ur} \arrow{r} +& \CH^*(G/B) \arrow[->>]{r} +& \CH^*(G)/p \arrow[equal]{d} \\ +& & (\mathbb{Z}/p)[x_i] / (x^{p^{k_i}}). +\end{tikzcd} +\] +Нас интересует образ вертикальной стрелки +в $\CH^*(G/B)/p$. +Обозначим через $j_i$ наименьшее целое число такое, что +$x_i^{p^{j_i}} + \mbox{члены меньшего порядка} \in \im(\ph)$. +Порядок мы понимаем в смысле Deglex; +$x_1\leq x_2\leq\dots\leq x_r$, если +$d_1\leq d_2\leq\dots\leq d_r$. +Можно рассмотреть $\CH^(G)/p$ как комодуль над собой, +и тогда $\im(\ph)$ будет подкомодулем. + +Заметим, что $0\leq j_i \leq k_i$, так как $x_i^{p^{k_i}} = 0$. +Равенство $j_i = 0$ равносильно тому, что +$x_i + \mbox{члены меньшего порядка}\in\im(\ph)$. +Набор чисел $(j_i)$ обозначим через $J_p(\xi)$ +(он действительно зависит только от $\xi$, но не от ${}_{\xi}G$). +Тогда полином Пуанкаре выглядит так: +\[ +P(R_p(G), t) = \prod_{i=1}^r\frac{1-t^{p^{j_i}\cdot d_i}}{1-t^{d_i}}. +\] +\begin{example} +Рассмотрим группу типа $\mathsf{F}_4$, $p=3$. +Тогда $\CH^*(G) = (\mathbb{Z}/3)[x_1]/(x_1^3)$, +где $d_1 = \deg x_1 = 4$. +Таким образом, $k_1=1$, и для $j_1$ есть два варианта: $0$ и $1$. +\begin{itemize} +\item случай $J_p(\xi) = (0)$ неинтересен +(см. замечание: полином Пуанкаре равен $1$; +\item в случае $J_p(\xi) = (1)$ получаем полином Пуанкаре +\[\frac{1-t^{3\cdot 4}}{1-t^4} = 1 + t^4 + t^8.\] +\end{itemize} +\end{example} +\begin{remark} +$J_p(\xi) = 0$ тогда и только тогда, когда ${}_{\xi}G$ расщепляется +расширением степени, взаимно простой с $p$. +\end{remark} + +\begin{example} +Пусть $G$~--- группа типа $\mathsf{G}_2$, $p=2$. +Тогда $\CH^*(G) = (\mathbb{Z}/2)[x_1]/(x_1^2)$, и $\deg x_1 = 3$. +Снова два случая: +\begin{itemize} +\item неинтересный: $J_p(\xi) = (0)$; +\item $J_p(\xi) = (1)$; полином Пуанкаре равен +\[\frac{1-t^{2\cdot 3}}{1-t^3} = 1 + t^3.\] +В этом случае $R_2({}_{\xi}(G))$~--- мотив Роста. +\end{itemize} +\end{example} +\begin{example} +В случаях $\mathsf{F}_4$, $\mathsf{E}_6$ при $p=2$ ответ тот же, +что и для $\mathsf{G}_2$. +\end{example} +\begin{example} +В случаях $\mathsf{E}_6^{\operatorname{sc}}$, $\mathsf{E}_7$ +при $p=3$ ответ тот же, что и для $\mathsf{F}_4$. +\end{example} +\begin{remark} +Степень полинома $P(R_p(G), t)$ равна $\sum (p^{j_i}-1)d_i$. +Оказывается, это равно $\operatorname{cd}_p(X)$ (\emph{каноническая размерность}). +Попросту говоря, это наименьшая из размерностей рациональных циклов. +\end{remark} +\begin{example} +Пусть $G$~--- группа типа $\mathsf{E}_8$, $p=5$. +Тогда $\CH^*(G) = (\mathbb{Z}/5)[x_1]/(x_1^5)$, +и $\deg x_1 = 5+1 = 6$ (вообще, $\deg x_i = p+1$, если $r=1$). +В этом случае любое ${}_{\xi}G$-однородное проективное многообразие +является клеточным над общей точкой. +В нетривиальном случае полином Пуанкаре равен +\[ +\frac{1-t^{5\cdot 6}}{1-t^6}. +\] +\end{example} + +\begin{example} +Рассмотрим группу типа $\mathsf{E}_8$, $p=2$. +Тогда +\[ +\CH^*(G) = (\mathbb Z/2)[x_1,x_2,x_3,x_4]/(x_1^8,x_2^4,x_3^2,x_4^2), +\] +$\deg x_1=3, \deg x_2 = 5, \deg x_3 = 9, \deg x_4 = 15$. +Это можно увидеть так: +элементы $x_2$ и $x_3$ получаются операцией Стинрода из $x^1$б +а именно, $x_2 = S^2(x_1)$ и $x_3 = S^4(x_2)$. +При этом $\deg S^m(x) = \deg x + m\cdot(p-1)$. +Если $m=\deg x$, то $S^m$~--- возведение в степень $p$; +если же $m > \deg x$, то $S^m(x) = 0$. +Операции Стинрода удовлетворяют следующим тождествам: +\begin{itemize} +\item $S^m$ линейны; +\item $S^m(xy) = \sum_n S^n(x)S^{m-n}(y)$; +\item Adem relations. +\end{itemize} +Из первых двух соотношений следует, что $\sum_m S^m$~--- гомоморфизм +колец. + +Что это означает для $J_p(\xi)$? +Мы знаем, что $J_p(\xi) = (j_1, j_2, j_3, j_4)$, +причем $0\leq j_1\leq 3$, $0\leq j_2\leq 2$, $0\leq j_3,j_4\leq 1$. +Из свойств $S^m$ следует, что $j_1\geq j_2\geq j_3$. +Кроме того, $j_1\leq j_2+1$ и $j_2\leq j_3+1$. +Если $j_1 = 3$, то $j_2=2$ и $j_3=1$. +Если же $j_1 = 0$, то $j_2=0$, $j_3=0$, и возникает +интересный случай, когда при этом $j_4=1$ +(заметим, что из равенства $j_1=0$ следует, +что инвариант Роста по модулю $2$ тривиален). +Тогда +\[ +P(R_2({}_{\xi}G), t) = \frac{1-t^{2\cdot 15}}{1-t^{15}} = 1 + t^{15}, +\] +как у мотива Роста. +Как мы уже упоминали, Никита Семенов доказал, что из этого следует, +что это и есть мотив Роста. +Получается некоторый инвариант в $H^5(F,\mathbb{Z}/2)$. +\end{example} + \end{document}