From ef74821fe5ba08542bebabf29d02c6ee241546b5 Mon Sep 17 00:00:00 2001 From: Alexander Luzgarev Date: Sun, 19 Jun 2016 22:44:08 +0300 Subject: [PATCH] Add Lecture 7 --- motives.pdf | Bin 579995 -> 612358 bytes motives.tex | 429 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 429 insertions(+) diff --git a/motives.pdf b/motives.pdf index 2f3d518e9488bb6db74d0f6c9d062533821dd398..c6294112656fc5103b0a762823bfc4c3daeb7ac9 100644 GIT binary patch delta 201864 zcmZtsQ;?ud6RrzS+s3qQThq2}+qR#!&1p{Cwr$(Sv^{P8@3&&b{txy($*8EiA}bFn zBJa4evKmRl>c~;a6-CAAnHkt&$>)}a*I=1h*y>R`;6b<>r{OPJ$9X=q@|X$}+QM)gEylzp{wq+@ zgm04}CYejzsB4KtSgjTaj=-S-OMylsS?L$0r4NS7F=OK*#tPmHSys1UH`=1V5Um?* zP$n}fv#{Jly2!`vC*=SY2#7(Ov>c=bpN#kvo(uocszT65L!O=HEvz_{F;N`@&Gc8K z567L{We|#g80OX;qz%;sD+9NTk5tGpa+N9Lc1_2QB7_pNPw59>i$??L$)gUjbBhwc z4km=7Zz86cx)@j<$UR4o6-TJ}8#?e-@dptaC}{tH0Js4J6f`mwmk>B|3~Z{hqfUzE 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--git a/motives.tex b/motives.tex index ace0aca..51c1744 100644 --- a/motives.tex +++ b/motives.tex @@ -62,6 +62,7 @@ \DeclareMathOperator{\pr}{pr} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\res}{res} +\DeclareMathOperator{\Trd}{Trd} %\DeclareFontFamily{OT1}{pzc}{} %\DeclareFontShape{OT1}{pzc}{m}{it}{<-> s * [1.2] pzcmi7t}{} @@ -81,6 +82,8 @@ \newtheorem{theorem}{Теорема}[subsection] \newtheorem{lemma}[theorem]{Лемма} \newtheorem{proposition}[theorem]{Утверждение} +\newtheorem{corollary}[theorem]{Следствие} +\newtheorem{conjecture}[theorem]{Гипотеза} \theoremstyle{definition} \newtheorem{example}[theorem]{Пример} @@ -2556,4 +2559,430 @@ $\mathbb{Z} \oplus \mathbb{Z}\{2^{k-1}-1\}$. (теорема Никиты Семенова). \end{remark} +% 02.04.2012 + +\subsection{Пример: $\mathsf{F}_4$} + +Над замкнутым полем группа типа $\mathsf{F}_4$~--- это автоморфизмы +эрмитовых матриц $3\times 3$ над октонионами: +$\mathsf{F}_4 = \Aut(H_3(\mathbb{O}))$. + +В общем случае приведем сначала <<конструкцию по модулю $2$>>. +Вместо $\mathbb{O}$ нужно взять другие октонионы +(они задатся формой Пфистера $\lAngle a,b,c\rAngle$), +и диагональную эрмитову форму вида $\la 1, -d, -e\ra$. +Здесь $a,b,c,d,e\in F^* / (F^*)^2$. +Если у поля $F$ нет расширений нечетной степени, то любая группа +типа $\mathsf{F}_4$ так выглядит. + +Например, над $\mathbb{R}$ есть три группы типа $\mathsf{F}_4$: +\begin{enumerate} +\item построенная по расщепимым октонионам (и тогда неважно, каковы $d,e$); +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (4) at ($\sm*(4.2, 0)$); +\draw (1)--(2); +\draw (3)--(4); +\draw (2) edge[transform canvas={yshift=2pt}] (3); +\draw (2) edge[transform canvas={yshift=-2pt}] (3); +\draw ($\sm*(2.0, 0.2)$) -- ($\sm*(2.2, 0)$) -- ($\sm*(2.0, -0.2)$); +\foreach \point in +{1,2,3,4} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + \draw [black] (\point) circle (5.0pt); + } +\end{tikzpicture} +\] +\item построенная по компактным октонионам (\term{октавам}) и $d=e=1$; +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (4) at ($\sm*(4.2, 0)$); +\draw (1)--(2); +\draw (3)--(4); +\draw (2) edge[transform canvas={yshift=2pt}] (3); +\draw (2) edge[transform canvas={yshift=-2pt}] (3); +\draw ($\sm*(2.0, 0.2)$) -- ($\sm*(2.2, 0)$) -- ($\sm*(2.0, -0.2)$); +\foreach \point in +{1,2,3,4} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\draw [black] (4) circle (5.0pt); +\end{tikzpicture} +\] +\item построенная по октавам и $d = e = -1$~--- она анизотропна +(над $\mathbb{R}$ это равносильно компактности). +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (4) at ($\sm*(4.2, 0)$); +\draw (1)--(2); +\draw (3)--(4); +\draw (2) edge[transform canvas={yshift=2pt}] (3); +\draw (2) edge[transform canvas={yshift=-2pt}] (3); +\draw ($\sm*(2.0, 0.2)$) -- ($\sm*(2.2, 0)$) -- ($\sm*(2.0, -0.2)$); +\foreach \point in +{1,2,3,4} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\end{tikzpicture} +\] +\end{enumerate} + +По общей теории скрученных форм над любым полем +группа типа $\mathsf{F}_4$~--- это группа автоморфизмов +алгебры $J$, где $J$~--- скрученная +форма йордановой алгебры $H_3(\mathbb{O})$. + +Приведем теперь <<конструкцию по модулю $3$>>. +Пусть $D$~--- центральная простая алгебра степени $3$. +Они все циклические, поэтому +$D = (a,b)_3 = \la x,y\mid x^3 = a,\; y^3 = b,\; xy = \zeta yx\ra$ +для некоторых $a,b\in F^*/(F^*)^3$. +Здесь $\zeta^3 = 1$. +Возьмем еще $c\in F^*/(F^*)^2$. +тогда на $D\oplus D\oplus D$ можно завести структуру йордановой +алгебры $J(a,b,c)$ (с помощью скаляра $c$). +Ее норма выглядит так: +\[ +N(\alpha\oplus\beta\oplus\gamma) = \Nrd(\alpha) + c\Nrd(\beta) ++ c^{-1}\Nrd(\gamma) - \Trd(\alpha\beta\gamma), +\] +где $\Trd$~--- приведенный след. +Автоморфизмы этой нормы образуют группу типа $\mathsf{E}_6$, +а подгруппа в ней, сохраняющая +единицу (то есть, $1\oplus 0 \oplus 0$)~--- это группа типа +$\mathsf{F}_4$. + +\begin{conjecture}[Ослабленный вариант гипотезы Серра--Роста] +Полученная группа типа $\mathsf{F}_4$ зависит только +от $\{a,b,c\} \in K_3^M(F)/3$ +(или, что то же самое, от +$(a)\cup (b)\cup (c) \in H^3(F,\mathbb{Z}/3\mathbb{Z})$). +\end{conjecture} +Известно, что если $J(a,b,c) = J(a',b',c')$, то +$\{a,b,c\} = \{a',b',c'\}$. + +Если у поля нет квадратичных расширений, то любая скрученная +форма $H_3(\mathbb O)$ имеет вид +$J(a,b,c)$. + +Для любого поля $F$ определен инвариант +\[ +g_3\colon H^1(F, \mathsf{F}_4) \to H^3(F, \mathbb{Z}/3\mathbb{Z}). +\] +\begin{itemize} +\item Если у $F$ нет квадратичных расширений, то образ $g_3$~--- это +в точности чистые символы $(a)\cup (b)\cup (c)$. +\item В этом случае ядро тривиально. +\item Гипотеза состоит в том, что $g_3$ инъективно. +\end{itemize} +Мы построили алгебру $J(a,b,c)$ и группу $G = \Aut(J(a,b,c))$. +\begin{fact} +Группа $G$ или расщепима, или анизотопна (как и в случае группы +изометрий пфистеровых форм). +\end{fact} +В частности, если $X$~--- $G$-однородное проективное многообрази, +то оно является клеточным над общей точкой, +то есть, $X_{F(X)}$ клеточное. +Пусть $X$~--- скрученная форма $\mathsf{F}_4/P_4$. + +Отступление: J.-P. Bonnet показал, что +$M({}_{\xi}(\mathsf{G}_2/P_1)) \isom M({}_{\xi}(\mathsf{G}_2/P_2))$. +Многообразия $\mathsf{G}_2/P_1$ и $\mathsf{G}_2/P_2$ +оба имеют размерность $5$. +На самом деле, $\mathsf{G}_2/P_1$~--- квадрика. +Более того, это максимальный сосед квадрики Пфистера, и поэтому +ее мотив раскладывается на мотивы Роста. +Напомним, что $\mathsf{G}_2 = \Aut(\mathbb{O})$, +где $\mathbb{O}$ задается формой $\lAngle a, b, c\rAngle$. +Возникающий мотив Роста отвечает как раз квадрике +$\lAngle a, b, c\rAngle$. +Вопрос: что если взять $\mathsf{F}_4/P_1$ и $\mathsf{F}_4/P_4$? +У них тоже одинаковая размерность и многочлен Пуанкаре. +Теорема Зайнуллина--Николенко--Семенова гласит, +что +$M({}_{\xi}(\mathsf{F}_4/P_1)) \isom M({}_{\xi}(\mathsf{F}_4/P_2))$, +если рассматриваемая группа типа $\mathsf{F}_4$ имеет вид +$\Aut(J(a,b,c))$. + +Размерность $\mathsf{F}_4/P_4$ равна $15$. +Нарисуем диаграмму Хассе для этого многообразия. +С точностью до каких-то ребер внутри она выглядит так: +\[ +\begin{tikzpicture}[scale=1.0,thin,font=\scriptsize] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (4) at ($\sm*(4.2, 0)$); +\coordinate (5) at ($\sm*(5.2, 1)$); +\coordinate (6) at ($\sm*(6.6, 1)$); +\coordinate (7) at ($\sm*(8.0, 1)$); +\coordinate (8) at ($\sm*(9.4, 1)$); +\coordinate (9) at ($\sm*(10.8, 1)$); +\coordinate (10) at ($\sm*(12.2, 1)$); +\coordinate (11) at ($\sm*(13.6, 1)$); +\coordinate (12) at ($\sm*(15.0, 1)$); +\coordinate (5x) at ($\sm*(5.2, -1)$); +\coordinate (6x) at ($\sm*(6.6, -1)$); +\coordinate (7x) at ($\sm*(8.0, -1)$); +\coordinate (8x) at ($\sm*(9.4, -1)$); +\coordinate (9x) at ($\sm*(10.8, -1)$); +\coordinate (10x) at ($\sm*(12.2, -1)$); +\coordinate (11x) at ($\sm*(13.6, -1)$); +\coordinate (12x) at ($\sm*(15.0, -1)$); +\coordinate (13) at ($\sm*(16.0, 0)$); +\coordinate (14) at ($\sm*(17.4, 0)$); +\coordinate (15) at ($\sm*(18.8, 0)$); +\coordinate (16) at ($\sm*(20.2, 0)$); +\node at ($\sm*(0, -2)$) {$0$}; +\node at ($\sm*(1.4, -2)$) {$1$}; +\node at ($\sm*(2.8, -2)$) {$2$}; +\node at ($\sm*(4.2, -2)$) {$3$}; +\node at ($\sm*(5.2, -2)$) {$4$}; +\node at ($\sm*(6.6, -2)$) {$5$}; +\node at ($\sm*(8.0, -2)$) {$6$}; +\node at ($\sm*(9.4, -2)$) {$7$}; +\node at ($\sm*(10.8, -2)$) {$8$}; +\node at ($\sm*(12.2, -2)$) {$9$}; +\node at ($\sm*(13.6, -2)$) {$10$}; +\node at ($\sm*(15.0, -2)$) {$11$}; +\node at ($\sm*(16.0, -2)$) {$12$}; +\node at ($\sm*(17.4, -2)$) {$13$}; +\node at ($\sm*(18.8, -2)$) {$14$}; +\node at ($\sm*(20.2, -2)$) {$15$}; +\node at (2) [above=3pt] {$h$}; +\node at (3) [above=3pt] {$h^2$}; +\node at (4) [above=3pt] {$h^3$}; +\node at (5) [above=3pt] {$\rho$}; +\node at (9) [above=3pt] {$\rho^2$}; +\draw (1)--(2)--(3)--(4)--(5)--(6)--(7)--(8)--(9)--(10)--(11)--(12)--(13)--(14)--(15)--(16); +\draw (4)--(5x)--(6x)--(7x)--(8x)--(9x)--(10x)--(11x)--(12x)--(13); +\foreach \point in +{1,2,3,4,5,6,7,8,9,10,11,12,5x,6x,7x,8x,9x,10x,11x,12x,13,14,15,16} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\end{tikzpicture} +\] + +\begin{enumerate} +\item Берем образующую в $\CH^1(\mathsf{F}_4/P_4)$. +Она рациональная, то есть, лежит в образе отображения +\[ +\CH^1({}_{\xi}(\mathsf{F}_4/P_4)) \xrightarrow{\res} +\CH^1(\mathsf{F}_4/P_4). +\] +В $\mathsf{F}_4$ решетка весов совпадает с решеткой корней. +В частности, $\varpi_4$~--- корень. +\item Тогда $h^2$, $h^3$~--- образующие $\CH^2$ и $\CH^3$. +\item Далее, $h^4$ и $\rho$~--- базис для $\CH^4$. +\item Кроме того, $\rho^2 h^7 = [\pt]\pmod{3}$. +\item $X_{F(X)}$ клеточное. +\item Если $\Aut(J(a,b,c))$ расщепляется над расширением +степени, взаимно простой с $3$, то она и была расщепимой +(это теорема типа Спрингера). +\end{enumerate} + +Начинаем применять метод общей точки: +\[ +\begin{tikzcd} +\CH^*(X\times X) \arrow[->>]{r} \arrow{d}{\res} & +\CH^*(X_{F(X)}) \arrow{d}{\isom} \\ +\CH^*(X_{\ol{F}}\times X_{\ol{F}}) \arrow[->>]{r} & +\CH^*(X_{\ol{F}(X)}). +\end{tikzcd} +\] +Элемент $\ol{\rho}\in\CH^*(X_{\ol{F}(X)})$ поднимается +до элемента в $\CH^*(X\times X)$. +Поэтому в $\CH^*(X_{\ol{F}}\times X_{\ol{F}})$ есть +рациональный элемент вида +\[ +\alpha = \rho\times 1 + ? \cdot h^3\times h ++ ?\cdot h^2\times h^2 + ?\cdot h\times h^3 ++ ?\cdot 1\times h^4 + c\cdot 1\times \rho. +\] +Подправив $\alpha$ на рациональные элементы (степени $h$), +можно добиться, что останется только +$\rho\times 1 + c\cdot 1\times\rho$. +Поскольку $X$ над кубическим расширением становится клеточным, +можно считать, что $c = 0$ или $c = \pm 1$. +Почему $c\neq 0$? +Если $\rho\times 1$ рационален, то и $1\times\rho$ рационален, +откуда $\rho^2\times\rho^2$ и $h^7\rho^2\times h^7\rho^2$ +рационален, а потому и $\pt\times\pt$ рационален. +Применяя пушфорвард, видим, что класс +$\pt\in\CH^*(X)$ рационален~--- противоречие. + +Поэтому на самом деле класс $\rho\times 1\pm 1\times\rho$ +рационален. +Значит, $\rho^2\times 1 \pm 2\rho\times\rho + 1\times\rho^2$ +рационален, а потому и +$\beta = \rho^2 + 1 \mp \rho\times\rho + 1\times\rho^2$ +рационален. +Будем умножать полученный цикл на $h^i\times h^j$, +где $i+j=7$. +Получится цикл +\[ +h^i\rho^2\times h^j \mp h^i\rho\times h^j\rho ++h^i\times h^j\rho^2\in\CH^{15}(X_{\ol{F}}\times X_{\ol{F}}). +\] +Возьмем его композицию с самим собой, и воспользуемся сравнением. +$h^{i+j}\rho^2 \equiv \pt \pmod{3}$. +Вообще, будем считать все по модулю $3$: +\[ +h^i\rho^2\times h^j + h^i\rho\times h^j\rho + h^i\times h^j\rho^2 +\] +Значит, этот цикл уже является проектором по модулю $3$. +Над замыканием каждое его слагаемое, конечно, является проектором~--- +но не рациональным. + +Итак, мы получили восемь ортогональных проекторов +$p_0,\dots,p_7$. +Правые части этих проекторов (24 штуки) образуют базис в группе Чжоу, +но не тот, который у нас был раньше (клетки Шуберта). +\[ +\begin{tikzpicture}[scale=1.0,thin,font=\scriptsize] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (4) at ($\sm*(4.2, 0)$); +\coordinate (5) at ($\sm*(5.2, 1)$); +\coordinate (6) at ($\sm*(6.6, 1)$); +\coordinate (7) at ($\sm*(8.0, 1)$); +\coordinate (8) at ($\sm*(9.4, 1)$); +\coordinate (9) at ($\sm*(10.8, 1)$); +\coordinate (10) at ($\sm*(12.2, 1)$); +\coordinate (11) at ($\sm*(13.6, 1)$); +\coordinate (12) at ($\sm*(15.0, 1)$); +\coordinate (5x) at ($\sm*(5.2, -1)$); +\coordinate (6x) at ($\sm*(6.6, -1)$); +\coordinate (7x) at ($\sm*(8.0, -1)$); +\coordinate (8x) at ($\sm*(9.4, -1)$); +\coordinate (9x) at ($\sm*(10.8, -1)$); +\coordinate (10x) at ($\sm*(12.2, -1)$); +\coordinate (11x) at ($\sm*(13.6, -1)$); +\coordinate (12x) at ($\sm*(15.0, -1)$); +\coordinate (13) at ($\sm*(16.0, 0)$); +\coordinate (14) at ($\sm*(17.4, 0)$); +\coordinate (15) at ($\sm*(18.8, 0)$); +\coordinate (16) at ($\sm*(20.2, 0)$); +\draw (1)--(2)--(3)--(4)--(5)--(6)--(7)--(8)--(9)--(10)--(11)--(12)--(13)--(14)--(15)--(16); +\draw (4)--(5x)--(6x)--(7x)--(8x)--(9x)--(10x)--(11x)--(12x)--(13); +\draw[dotted] (1) edge[bend left=45] (5); +\draw[dotted] (5) edge[bend left=45] (9); +\draw[dotted] (2) edge[bend left=45] (6); +\draw[dotted] (6) edge[bend left=45] (10); +\node at ($\sm*(1,1)$) {$p_0$}; +\node at ($\sm*(2.4,1)$) {$p_1$}; +\foreach \point in +{1,2,3,4,5,6,7,8,9,10,11,12,5x,6x,7x,8x,9x,10x,11x,12x,13,14,15,16} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } + +\end{tikzpicture} +\] +Можно также рассмотреть $\beta(h^i\times h^j)$ для произвольных +$i,j$. +Такие рациональные циклы индуцируют изоморфизмы +между $(X,p_0)\{i\}$ и $(X,p_i)$. + +Мы посчитали все по модулю $3$, но можно добиться и +целочисленных проекторов. +На самом деле, при каких-то разумных условиях всегда можно +поднять проекторы (и изоморфизмы) по модулю $2$ и $3$ +до целочисленных. + +Мотив Роста квадрики Пфистера $\lAngle a_1,\dots,a_k\rAngle$ +обозначается так: $R_{2,k}(\{a_1,\dots,a_k\})$. +Полученный выше мотив $(X,p_0)$ будем обозначать через +$R_{3,3}(\{a,b,c\})$. + +\begin{corollary} +\[ +M(X) = \bigoplus_{i=0}^{7} R_{3,3}(\{a,b,c\})\{i\}. +\] +\end{corollary} + +\begin{remark} +Точно так же доказывается, что +\[ +M({}_{\xi}(\mathsf{F}_4/P_1)) = \bigoplus_{i=0}^{7} +R_{3,3}(\{a,b,c\})\{i\}. +\] +Более того, мотив любого $G$-однородного проективного +многообразия $Y$ раскладывается в сумму сдвигов +мотива $R_{3,3}(\{a,b,c\})$. +Для квадрики Пфистера верно аналогичное замечание. +\end{remark} + +\begin{theorem}[Зайнуллин--Петров--Семенов] +Пусть $G$~--- полупростая алгебраическая группа над $F$, +$X$~--- $G$-однородное проективное многообразие, клеточное +над общей точкой, $p$~--- простое число. +Тогда мотив $M(X)\otimes\mathbb{Z}/p\mathbb{Z}$ изоморфен +сумме $R_p(G)$ с какими-то сдвигами, +где $R_p(G)$ не зависит от $X$ и неразложим по модулю $p$. +\end{theorem} +Нужно пояснить, что такое $M(X)\otimes\mathbb{Z}/p\mathbb{Z}$: +в конструкции категории соответствий нужно взять +$\Mor(X,Y) = \CH^{\dim Y}(X\times Y) / p$. + +\begin{remark} +Условие клеточности $X_{F(X)}$ можно заменить +на требование расщепимости $G_{F(X)}$. +\end{remark} +\begin{remark} +Мотив $R_p(G)$ над алгебраическим замыканием раскладывается +в сумму мотивов вида $\mathbb{Z}/p = M(\pt)\otimes\mathbb{Z}/p$ +со сдвигами, и для этих сдвигов есть некоторая формула. +\end{remark} +\begin{remark} +Все ситуации, описанные в теореме, перечислены в работе +Петрова--Семенова. +\end{remark} +\begin{remark} +Сами проекторы можно поднять в $\mathbb{Z}$, но не изоморфизмы +между ними. +\end{remark} +\begin{remark} +Если $G$ не содержит сомножителей типа $\mathsf{A}$ +и $p\neq 2,3,5$, то $R_p(G) = \mathbb{Z}/p$. +Случай $p=5$ возникает только для $\mathsf{E}_8$. +\end{remark} +\begin{remark} +Мотив $M(X)\otimes\mathbb{Q}$ раскладывается в +прямую сумму мотивов $M(\pt)\otimes\mathbb{Q}$ со сдвигами. +\end{remark} +\begin{remark} +Если $G=\mathsf{E}_8$, $p=2$, и инвариант Роста +тривиален, то все $G$-однородные проективные многообразия $X$ +подходят. +В этом случае $R_2(G)$~--- мотив Роста, +отвечающий $5$-символу (см. работу Никиты Семенова +про конечные подгруппы $\mathsf{E}_8$). +\end{remark} + \end{document} +