From 05abd2de12cc150ac54454d645aaf5015658a49c Mon Sep 17 00:00:00 2001 From: Alexander Luzgarev Date: Mon, 13 Jun 2016 17:03:43 +0300 Subject: [PATCH] Add Lecture 4 --- motives.pdf | Bin 444581 -> 502335 bytes motives.tex | 476 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 476 insertions(+) diff --git a/motives.pdf b/motives.pdf index 88e8bc4deae3508ff277c759da1300221950e58c..754e60756e2ef294956ce140829c1490bd219d12 100644 GIT binary patch delta 233619 zcmZU)Q*bWq5AI#twpQC!+qP}nw%ez+ZQHiHYTLGL?f3idnSJsdB?tHXGMTwE$z-mS zB;fWf5g?H%h=|cL(y_vj%`FYD!EiJnb-;l#GH@m!qc-iptssIhGB&l~rK15eH_0&u zV*F?3#W!JrS(|E<@6bS4|M#xZ!VVFbt%=y<6*(!qg#wY80ft`2)Xv<+f{2NUnLTMA zjsh^OWoy4VhWt~j&v;gL6f~aWo3YJ{1^MPQ%_^`6;i%Ot8=Y%Zg~=X)Hs*A28TRwS zCmn&(K`J`zPAU1OoxcT@64ec{ZXhJ@Q(CEr{f@L=^mm0u-4?sQrI7n(_If<;t`nQ1=1j=aea@ zDnHF0E%^`PUM|dCI=b{H;QXbxeR(`y#@%~|yA-Co6g8_wOGTT}t-jZnip!UAU4eGH zcBaqL%vtkU@R-X7Gd5!F7kssBi}P2XFsJ8d<$XU9#}Yf+RLH=Rd8lzNKHk 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\newcommand{\la}{\langle} \newcommand{\ra}{\rangle} @@ -1156,6 +1163,7 @@ X(R) = \{P'\leq G_R \mid\mbox{существуют }S/R, g\in G(S):\; gP'g^{-1} проективного однородного многообразия ${}_{E}X$, отличного от точки, ${}_{E}X$ изотропно. +\begin{example} Пусть $G = \PGL_n$. Ее скрученная форма ${}_{E}G$ имеет вид $\Aut(A)$, а соответствующая скрученная форма проективного пространства~--- $\SB(A)$. @@ -1185,7 +1193,475 @@ $\SL_1(A) = \{g\in A\mid\Nrd(g)=1\}$. соответствует присоединенной группе (без центра); максимальная решетка $\mathbb{Z}\varpi_1\oplus\dots\oplus\mathbb{Z}\varpi_l$ соответствует односвязной группе (у нее самый большой центр). +\end{example} +% 12.03.2012 +\subsection{$\SO_{2n}$} + +Посмотрим на однородные многообразия для $\SO_{2n}$. +Диаграмма Дынкина выглядит так: +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (2p) at ($\sm*(1.9, 0)$); +\coordinate (d1) at ($\sm*(2.1, 0)$); +\coordinate (d2) at ($\sm*(2.3, 0)$); +\coordinate (d3) at ($\sm*(2.5, 0)$); +\coordinate (3m) at ($\sm*(2.7, 0)$); +\coordinate (3) at ($\sm*(3.2, 0)$); +\coordinate (4) at ($\sm*(4.6, 0)$); +\coordinate (5) at ($\sm*(5.6, 1)$); +\coordinate (6) at ($\sm*(5.6, -1)$); +\node at (1) [below=5pt,font=\scriptsize] {$1$}; +\node at (4) [below=5pt,font=\scriptsize] {$n-2$}; +\node at (5) [below=5pt,font=\scriptsize] {$n-1$}; +\node at (6) [below=5pt,font=\scriptsize] {$n$}; +\draw (1)--(2); +\draw (2)--(2p); +\draw (3m)--(3); +\draw (3)--(4); +\draw (4)--(5); +\draw (4)--(6); +\foreach \point in +{1,2,3,4,5,6} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{d1,d2,d3} + { + \fill [black] (\point) circle (0.7pt); + } +\node (c) at ($\sm*(7.1, 0)$) {$\mathsf{D}_{n}$}; +\end{tikzpicture} +\] +Весу $\varpi_1$ отвечает квадрика $\{q(v)=0\}$, что соответствует естественному +представлению $V$ группы $\SO_{2n}$. +Весу $\varpi_2$~--- представление $\Lambda^2 V$. +Соответствующее многообразие~--- множество вполне изотропных плоскостей +$\la u,v\ra$, то есть, таких, что $q|_{\la u,v\ra}=0$. +Это условие можно описать так: $q(u) = q(v) = f(u,v) = 0$, где $f$~--- +поляризация формы $q$: $f(u,v) = q(u+v) - q(u) - q(v)$. +Аналогично (с помощью вполне изотропных подпространств различной размерности) +описываются случаи $\varpi_3,\dots,\varpi_{n-2}$. + +Весам $\varpi_{n-1}$ и $\varpi_n$ соответствуют вполне изотропные подпространства +размерности $n$. +Дело в том, что многообразие вполне изотропных подпространств размерности $n$ +имеет две компоненты связности. Для того, чтобы объяснить этот эффект, +выберем базис $e_1,\dots,e_n,e_{-n},\dots,e_{-1}$, относительно +которого матрица Грама формы $q$ имеет вид +\[ +\begin{pmatrix} +0 & 0 & \dots & 0 & 1\\ +0 & 0 & \dots & 1 & 0\\ +\vdots & \vdots & \ddots & \vdots & \vdots\\ +0 & 1 & \dots & 0 & 0\\ +1 & 0 & \dots & 0 & 0 +\end{pmatrix} +\] +Оказывается, подпространства $\la e_1,\dots,e_{n-1},e_n\ra$ +и $\la e_1,\dots,e_{n-1},e_{-n}\ra$ вполне изотропны, но не переводятся +друг в друга действием $\SO_{2n}$. +Первое соответствует весу $\varpi_{n-1}$, а второе~--- весу $\varpi_n$. +Куда же делось многообразие вполне изотропных подпространств размерности $n-1$? +Оно не максимальное однородное (соответствует не максимальной параболической +подгруппе), и соответствует весу $\varpi_{n-1} + \varpi_n$. +Действительно, +\[ +\Stab(\la e_1,\dots,e_{n-1}\ra) = +\Stab(\la e_1,\dots,e_{n-1},e_n\ra) \cap +\Stab(\la e_1,\dots,e_{n-1},e_{-n}\ra). +\] +Вообще, немаксимальные многообразия соответствуют флагам. +Посмотрим на вес $\varpi_{i_1} + \dots + \varpi_{i_k}$. +Флаг для него~--- это набор подпространств таких размерностей: +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (2p) at ($\sm*(1.9, 0)$); +\coordinate (d1) at ($\sm*(2.1, 0)$); +\coordinate (d2) at ($\sm*(2.3, 0)$); +\coordinate (d3) at ($\sm*(2.5, 0)$); +\coordinate (3m) at ($\sm*(2.7, 0)$); +\coordinate (3) at ($\sm*(3.2, 0)$); +\coordinate (4) at ($\sm*(4.6, 0)$); +\coordinate (5) at ($\sm*(5.6, 1)$); +\coordinate (6) at ($\sm*(5.6, -1)$); +\node at (1) [below=5pt,font=\scriptsize] {$1$}; +\node at (2) [below=5pt,font=\scriptsize] {$2$}; +\node at (3) [below=5pt,font=\scriptsize] {$n-3$}; +\node at (4) [below=5pt,font=\scriptsize] {$n-2$}; +\node at (5) [below=5pt,font=\scriptsize] {$n$}; +\node at (6) [below=5pt,font=\scriptsize] {$n$}; +\draw (1)--(2); +\draw (2)--(2p); +\draw (3m)--(3); +\draw (3)--(4); +\draw (4)--(5); +\draw (4)--(6); +\foreach \point in +{1,2,3,4,5,6} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{d1,d2,d3} + { + \fill [black] (\point) circle (0.7pt); + } +\node (c) at ($\sm*(7.1, 0)$) {$\mathsf{D}_{n}$}; +\end{tikzpicture} +\] +с правильной инцидентностью. +А именно, для каждой из двух цепочек от первой вершины до двух последних +инцидентность~--- это включение, а для весов $\varpi_{n-1}$ и $\varpi_n$ +инцидентность означает, что пересечение соответствующих подпространств +размерности $n$ имеет размерность $n-1$. + +Перед нами пример \emph{геометрии}. +Гораздо более простой пример~--- случай системы $\mathsf{A}_2$. +Там всего два фундаментальных веса: +$\varpi_1$ соответствует точкам (и параболическим подгруппам типа $\varpi_1$), +а $\varpi_2$~--- прямым (и параболическим подгруппам типа $\varpi_2$). +Более подробно, посмотрим на трехмерное векторное пространство $F^3$. +Ненулевой вектор $u$ порождает одномерное подпространство +$\la u\ra\subseteq F^3$, и его стабилизатор +$\Stab_{\SL_3}(\la u\ra)$~--- это параболическая подгруппа типа $\varpi_1$: +\[ +\begin{pmatrix} +* & * & * \\ 0 & * & * \\ 0 & * & * +\end{pmatrix} +\mbox{ --- стабилизатор вектора } +\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}. +\] +Для описания прямых можно воспользоваться двойственностью и перейти +к пространству $(F^3)^*$. +Ненулевой ковектор $\ph\in(F^3)^*$ порождает одномерное подпространство +$\la\ph\ra\subseteq (F^3)^*$, и его стабилизатор +$\Stab_{\SL_3}(\la \ph \ra)$~--- это параболическая подгруппа типа +$\varpi_2$: +\[ +\begin{pmatrix} +* & * & * \\ * & * & * \\ 0 & 0 & * +\end{pmatrix} +\mbox{ --- стабилизатор ковектора } +\begin{pmatrix} 0 & 0 & 1\end{pmatrix}. +\] +Отношение инцидентности между ними такое: +точка лежит на прямой тогда и только тогда, когда $\ph(u) = 0$. +В терминах параболических подгрупп: +$\Stab(\la u\la) \cap \Stab(\la \ph \ra)$ содержит борелевскую подгруппу +(то есть, параболическую подгруппу типа $\varpi_1 + \varpi_2$). + +Если мы теперь посмотрим на геометрию, заданную абстрактными аксиомами +проективной плоскости (с аксиомой Дезарга, обеспечивающей ассоциативность, +но без аксиомы Паппа, обеспечивающей коммутативность), +мы получим группу $\SL_1(A)$, где $A$~--- центральная простая алгебра +степени $3$. + +\subsection{Вычисление колец Чжоу}\label{ssect:chow-map-definition} + +Пусть $E \in H^1 (F, G)$, и задано однородное проективное $G$-многообразие $X$. +Рассмотрим скрученное многообразие ${}_E X$; нас интересуют инварианты этого +многообразия в смысле алгебраической геометрии. +Например, $\CH^*({}_E X)$. + +Вложение поля $F$ в его алгебраическое замыкание $\ol{F}$ дает морфизм +схем $\Spec\ol{F} \to \Spec F$. +Пулбэком получается многообразие $X_{\ol{F}}$: +\[ +\begin{tikzcd} +X_{\ol{F}} \arrow{r} \arrow{d} & X \arrow{d} \\ +\Spec\ol{F} \arrow{r} & \Spec F +\end{tikzcd} +\] +Отсюда получаем гомоморфизм +\[ +\CH^*({}_{E}X) \to \CH^*(({}_{E}X)_{\ol{F}}) = \CH^*(X_{\ol{F}}). +\] +Нас интересует образ этого гомоморфизма: кручение содержится в его ядре, +за счет чего легче жить. +Первый шаг~--- вычисление $\CH^*(X_{\ol{F}})$. + +\subsection{Пример: проективное пространство}\label{ssect:chow-ring-of-pn} + +\begin{example}\label{example:projective-space} +Рассмотрим $\mathbb{P}^n \times \mathbb{P}^n$ с диагональным действием +$\SL_{n+1}$. +Это действие не транзитивно: есть диагональ $\mathbb{P}^n$. +Как выглядит дополнение к диагонали? +Мы утверждаем, что оно расслаивается над $\Gr(1,2;n+1)$ со слоем +$\mathbb{A}^1$. +Здесь $\Gr(1,2;n+1)$~--- многообразие флагов, состоящих из прямой и плоскости, +в $(n+1)$-мерном пространстве. +\[ +\begin{tikzcd} +\mathbb{P}^n \times \mathbb{P}^n & \arrow{d}{\mathbb{A}^1} & +\mathbb{P}^n \arrow[left hook->]{ll} \\ + & \Gr(1,2;n+1) +\end{tikzcd} +\] +Это расслоение выглядит так: пара +$(\la u\ra, \la v\ra)\in\mathbb{P}^n \times \mathbb{P}^n$ +отправляется во флаг $\la u\ra \leq \la u,v\ra$. +Прообраз флага при этом~--- это многообразие способов дополнить прямую +до плоскости, то есть, $\mathbb{P}^1 \setminus \mathbb{P}^0 = \mathbb{A}^1$. +Более строго, нужно говорить про расслоения на $\Gr(1,2;n+1)$: +есть двумерное векторное расслоение $\tau_2$, сопоставляющее +флагу $\la u\ra \leq \la u,v\ra$ плоскость $\la u,v\ra$, +и есть одномерное векторное расслоение $\tau_1$, сопоставляющее +флагу $\la u\ra \leq \la u,v\ra$ прямую $\la u\ra$. + +Теперь зафиксируем в этом описании $u$, то есть, возьмем слой всей +картинки над точкой в первом сомножителе $\mathbb{P}^n$. +Получим картинку +\[ +\begin{tikzcd} +\mathbb{P}^n & \arrow{d}{\mathbb{A}^1} & +\pt \arrow[left hook->]{ll} \\ + & \Gr(1;n) +\end{tikzcd} +\] +Заметим, что $\Gr(1,n) = \mathbb{P}^{n-1}$. +Поэтому можно написать точную последовательность локализации: +\[ +\CH^{*-n}(\pt) \to \CH^*(\mathbb{P}^n) \to \CH^*(\mathbb{P}^{n-1}) \to 0. +\] +Средняя стрелка является гомоморфизмом колец, а первый член почти всегда +равен нулю. +Поэтому +\[ +\CH^i(\mathbb{P}^n) = \begin{cases} +\CH^i(\mathbb{P}^{n-1}), & i < n,\\ +\mathbb{Z}, & i = n,\\ +0, & i > n. +\end{cases} +\] +По индукции получаем, что у $\CH^*(\mathbb{P}^n)$ в каждой размерности +от $0$ до $n$ стоит одна копия $\mathbb{Z}$. +\end{example} + +\begin{example}\label{example:projective-space-filtration} +Опишем другой способ. +Пусть $\dim(V) = n+1$. +Рассмотрим действие группы $\SL(V)$ (или $\PGL(V)$) +на $\mathbb{P}(V^*) \times \mathbb{P}(V)$ +(соответствующее весу $\varpi_1 + \varpi_n$). +Там имеется подмногообразие $\{\ph(u) = 0\}$: +\[ +\begin{tikzcd} +\mathbb{P}(V^*) \times \mathbb{P}(V) +& \arrow{d}{\mathbb{A}^n} +& \{\ph(u) = 0\}\arrow[left hook->]{ll}\\ +& \mathbb{P}(V^*) +\end{tikzcd} +\] +Зафиксировав $\ph$, получаем +\[ +\begin{tikzcd} +\mathbb{P}^n +& \arrow{d}{\mathbb{A}^n} +& \mathbb{P}^{n-1} \arrow[left hook->]{ll}\\ +& \pt +\end{tikzcd} +\] +Значит, имеется следующая точная последовательность локализации: +\[ +\CH^{*-1}(\mathbb{P}^{n-1}) \to \CH^*(\mathbb{P}^n) \to \CH^*(\pt) \to 0. +\] +Вычисление по индукции приводит к тому же результату, что и +в предыдущем примере. +\end{example} + +\begin{fact} +Если $Z\subseteq X$~--- замкнутое подмногообразие, +и $U = X\setminus Z$, имеется точная последовательность локализации +\[ +\CH^{* - \codim_{X}Z} \to \CH^*(X) \to \CH^*(U) \to 0, +\] +где первое отображение~--- push-forward, а второе~--- pull-back +(и является гомоморфизмом колец). +\end{fact} + +\begin{example} +Тот же результат можно получить и прямым вычислением: +понять, что компонента кольца Чжоу коразмерности $i$ +порождается классом подпространства $[\mathbb{P}^{n-i}]$, +причем $[\mathbb{P}^n] = 1$. +Кроме этого, +\[ +[\mathbb{P}^{n-1}]^i = \begin{cases} +[\mathbb{P}^{n-i}], & i \leq n,\\ +0, & i > n. +\end{cases} +\] +Например, выбрав на $\mathbb{P}^n$ однородные координаты +$[x_0:\dots:x_n]$, можно взять $\mathbb{P}^{n-1} = \{x_0=0\}$, +другое $\mathbb{P}^{n-1} = \{x_1 = 0\}$ и обнаружить, +что их пересечение равно $\{x_0 = x_1 = 0\} = \mathbb{P}^{n-2}$. +\end{example} + +\begin{remark} +По сути, в примере~\ref{example:projective-space-filtration} +мы нарисовали фильтрацию +\[ +\begin{tikzcd} +\mathbb{P}^n +& \mathbb{P}^{n-1} \arrow[left hook->]{l}{\mathbb{A}^n} +& \dots \arrow[left hook->]{l}{\mathbb{A}^{n-1}} +& \pt \arrow[left hook->]{l}{\mathbb{A}^1} +\end{tikzcd} +\] +Вообще, если у многообразия $X$ существует фильтрация замкнутыми +подмногообразиями $S\supseteq X_1\supseteq X_2\supseteq\dots$ +такая, что $X_i\setminus X_{i+1} = \coprod\mathbb{A}^{k_i}$, +то $X$ называется \term{клеточным}. +В этом случае +\begin{itemize} +\item все $\CH^i$~--- свободные конечно порожденные абелевы группы (их ранг +равен количеству клеток в соответствующей разности); +\item $\CH(X)_i\isom \CH(X_L)_i$ для любого расширения $L/F$. +\end{itemize} +\end{remark} + +\subsection{Пример: многообразие Севери--Брауэра} + +Перейдем теперь к $\SB(D)$, где $D$~--- тело, $\ind D = n+1$. +Это скрученная форма $\mathbb{P}^n$: $\SB(D) = {}_{E}\mathbb{P}^n$. +В разделе~\ref{ssect:chow-map-definition} мы построили отображение +\[ +\CH^*(\SB(D)) \to \CH^*(\mathbb{P}^n_{\ol{F}}). +\] +Циклы из его образа называются \term{рациональными} +(по отношению к скручивающему торсору $E$). +В разделе~\ref{ssect:chow-ring-of-pn} мы вычислили правую часть: +там стоит копия $\mathbb{Z}$ в каждой компоненте с номерами от $0$ до $n$. +Образующая компоненты коразмерности $0$ всегда оказывается в образе. + +Предположим, что класс $[\pt]$ оказался рационален. +Это означает, что есть конечные (сепарабельные) расширения +$L_1,\dots,L_k$ такие, что +\begin{itemize} +\item над каждым $L_i$ наше многообразие имеет рациональную точку; +\item $\gcd_i([L_i:F]) = 1$. +\end{itemize} +Заметим, что первое условие равносильно тому, что +$[D_{L_i}]=0$ в $\Br(L_i) = 0$. +Применим отображение трансфера $\Br(L_i) \to \Br(F)$. +Получим, что $[L_i:F]\cdot [D]=0$ в $\Br(F)$ для всех $i$. +Из этого (а также из второго условия) +следует, что $[D] = 0$ в $\Br(F)$. + +\subsection{Пример: квадрика} + +Рассмотрим квадрику $Q = \{q=0\}$. +В $Q\times Q = \{(\la u\ra, \la v\ra)$ есть подмножество $\{f(u,v)=0\}$, +а в нем~--- диагональ $\{\la u\ra = \la v\ra\}\isom Q$. +Получаем фильтрацию +\[ +\begin{tikzcd} +Q\times Q & \arrow{d}{\mathbb{A}^{\dim Q}} & +\{f(u,v) = 0\} \arrow[left hook->]{ll} & \arrow{d}{\mathbb{A}^1} & +Q.\arrow[left hook->]{ll}\\ + & Q & & \OGr(1,2;f) +\end{tikzcd} +\] +Здесь $\OGr(1,2;f)$ означает многообразие флагов, состоящих из +вполне изотропных подпространств вида $\la u\ra \leq \la u,v\ra$. + +Расслоение $Q\times Q\setminus \{f(u,v)=0\} \to Q$ +устроено так: пара $(\la u\ra, \la v\ra)$ отправляется +в $\la u\ra$. +Проверим, что слой изоморфен $\mathbb{A}^{\dim Q}$. +Пусть $u = e_1$. +Тогда наше дополнение имеет вид $\{f(e_1,v)\neq 0\}$. +Условие $f(e_1,v)\neq 0$ равносильно тому, что коэффициент у $v$ +при базисном векторе $e_{-1}$ не равен $0$. +Поэтому можно читать, что он равен $1$. +Теперь все коэффициенты $v$, кроме тех, что стоят при $e_{1}$ и $e_{-1}$, +можно брать какими угодно, а коэффициент при $e_1$ определяется +однозначно из условия изотропности $q(v) = 0$. +Иначе говоря, если $\tau$~--- тавтологическое расслоение на $Q$, +рассмотрим $(\tau^{\perp})^*$. +Его слой над точкой $\la u\ra\in Q$ равен $(\la u\ra)^{\perp})^*$. +Вот нужный нам изоморфизм: +\begin{align*} +\mathbb{A}^{\dim Q} & \to +\mathbb{P}((\la u\ra^{\perp})^*) \setminus +\mathbb{P}(\{\ph\in(\la u\ra^{\perp})^*\mid \ph(u) = 0\},\\ +v & \mapsto (\ph\colon w\mapsto f(v,w)). +\end{align*} + +Расслоение $\{f(u,v)=0\} \setminus Q \to \OGr(1,2;f)$ устроено проще: +его слой равен +$\mathbb{P}(\tau_2) \setminus \mathbb{P}(\tau_1)\isom\mathbb{A}^1$, +как и в примере~\ref{example:projective-space}. + +Теперь зафиксируем $u$; получим фильтрацию +\[ +\begin{tikzcd} +Q & \arrow{d}{\mathbb{A}^{\dim Q}} & +\{f(u,v) = 0\} \arrow[left hook->]{ll} & \arrow{d}{\mathbb{A}^1} & +\pt,\arrow[left hook->]{ll}\\ + & \pt & & Q' +\end{tikzcd} +\] +где $Q'$~--- квадрика размерности $\dim Q - 2$. +Получаем точные последовательности +\begin{gather*} +\CH^{*-1}(\{f(u,v)=0\}) \to \CH^*(Q) \to \CH^*(\pt) \to 0,\\ +\CH^{*-\dim Q + 1}(\pt) \to \CH^*(\{f(u,v)=0\}) \to \CH^*(Q') \to 0. +\end{gather*} +Теперь при помощи индукции можно доказать следующее. + +Пусть $\dim Q = n$ четно. +Тогда $\CH^i(Q)$~--- свободная абелева группа ранга $1$ +для всех $i=0,\dots,n$, кроме $i= n/2$; $\CH^{n/2}(Q)\isom\mathbb Z^2$. +Обозначим за $h = [Q'']\in\CH^1(Q)$ класс подквадрики коразмерности $1$. +Это гиперплоское сечение $Q$ в общем положении. +Тогда $1$~--- образующая $\CH^0(Q)$ +$h$~--- образующая $\CH^1(Q)$, +$h^2$~--- образующая $\CH^2(Q)$,\dots. +С другой стороны, $\pt$~--- образующая $\CH^n(Q)$, +$[\mathbb{P}^1]$~--- образующая $\CH^{n-1}(Q)$, +$[\mathbb{P}^2]$~--- образующая $\CH^{n-2}(Q)$,\dots. +Это классы изотропных подпространств соответствующих размерностей. +Наконец, $h^{n/2}$ является суммой двух образующих; в качестве +одной из них можно взять $[\mathbb{P}^{n/2}$. + +Это можно увидеть в координатной записи: +$Q$ задается уравнением $x_1 y_1 + \dots + x_{n/2+1}y_{n/2+1} = 0$. +После этого $Q''$ задается уравнением $x_{n/2+1} - y_{n/2+1} = 0$ +(это гиперплоское сечение, как и было обещано), +а следующие образующие задаются последовательным +наложением уравнений $x_{1} = 0$, +$x_{2} = 0$, и так далее. +Когда дойдем до коразмерности $n/2$, +получим два варианта: либо +\[ x_1 = \dots = x_{n/2+1} = 0, \] +либо +\[ x_1 = \dots = x_{n/2} = y_{n/2+1} = 0. \] + +\begin{example} +Пусть $n=4$, то есть, мы имеем дело с $\mathsf{D}_3$. +Перед нами четырехмерная квадрика. +Ее уравнение выглядит так: $x_1y_1 + x_2y_2 + x_3y_3 = 0$. +Уравнения двух образующих в коразмерности $4/2=2$ выглядят так: +\begin{gather*} +x_1 = x_2 = x_3 = 0,\\ +x_1 = x_2 = y_3 = 0. +\end{gather*} +Их пересечение имеет вид $x_1 = x_2 = x_3y_3 = 0$, +что равносильно $x_1 = x_2 = 0$. +Почему-то это условие равносильно $x_3 = y_3 = 0$. +\end{example} \end{document}