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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf  Topologia das Variedades \\[15mm]
\small Cohomology, lecture 8: multiplicative structure on cohomology of $\C P^n$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small\bf Misha Verbitsky  
\\[2mm]
June 5, 2018\\[4mm]

{\tiny\bf IMPA, Tuesdays and Thursdays, 10:30, Sala 224}
}
\end{center}


\newpage

{\bf \blue Differential graded algebras}


\definition
An algebra $A$ is called {\bf\blue graded} if $A$ is 
represented as $A= \bigoplus A^i$, where $i\in \Z$,
and the product satisfies $A^i \cdot A^j
\subset A^{i+j}$. Instead of $\bigoplus A^i$
one often writes  $A^*$, 
where $*$ denotes all indices together.
Some of the spaces $A^i$ can be zero, but the 
ground field is always in $A^0$, so that it is non-empty. 

\definition
Let  $A^* = \oplus_{i\in \Z}A^i$ be a graded algebra.
For $x\in A^i$, let $\tilde x:= i$.
It is called
{\bf\blue graded commutative}, or {\bf\blue supercommutative},
if $ab = (-1)^{\tilde a \tilde b} ba$.

\definition
{\bf \blue Differential graded algebra}, or {\bf \blue DG-algebra} is a graded 
commutative algebra $A^* = \oplus_{i\in \Z}A^i$ 
with a differential $d:\; A^* \arrow A^{*+1}$, $d^2=0$,
satisfying the Leibnitz identity $d(x y) = (dx) y + (-1)^{\tilde x}x (dy)$.

\example De Rham algebra is clearly a DG-algebra.

\newpage

{\bf \blue Cohomology of a differential graded algebras}

\claim
Let $(A^*, d)$ be a DG-algebra. Then {\bf \red its cohomology
is equipped with a natural graded commutative multiplication.}

\proof 
Leibnitz identity implies that 
a product of closed forms is closed.
If $\alpha = d \beta$ and $\gamma$ is closed,
one has $\alpha \wedge \gamma = d (\beta) \wedge \gamma = d (\beta\wedge \gamma)$,
hence a product of closed and exact forms is exact. In other words,
exact forms are an ideal in the ring of closed forms, and the
quotient ring is the ring of cohomology.
\endproof

\definition
{\bf \blue The ring of cohomology of a manifold} is the ring associated with
its de Rham algebra.

%\newpage
%
%{\bf \blue Differential graded algebras}
%
%\definition
%{\bf \blue A DG-ideal}, or {\bf \blue differential ideal} in a DG-algebra $A^*$
%is a graded ideal $\bigoplus J^i\subset \bidoplus A^i$
%such that $d(J^*)\subset J^*$. 
%
%\claim
%Let $J^* \subset A^*$ be a DG-ideal. {\bf \purple Then the quotient space
%$A^*/J^*$ is also a DG-algebra.} \endproof

\newpage

{\bf \blue Symplectic manifolds}

\definition
A skew-symmetric 2-form $\omega$ on a vector space $V$ is called {\bf \blue
non-degenerate} if for any non-zero $x\in V$, there exists $y\in V$
such that $\omega(x, y)\neq 0$.

\claim Let $\omega$ be a non-degenerate form on $V$. {\bf \red Then there exists a
 basis $x_1, ..., x_{2n}\in V^*$, such that $\omega = \sum_{i=1}^{n} x_{2i-1} \wedge x_{2i}$.}
\endproof

\definition
A {\bf \blue symplectic form} on a manifold $M$ is a closed,
everywhere non-degenerate 2-form. A manifold with a symplectic form is called
{\bf \blue a symplectic manifold}.

\remark Symplectic manifold is always oriented. Indeed,
at each point where $\omega = \sum_{i=1}^{n} x_{2i-1} \wedge x_{2i}$,
one has $\omega^n= n!\; x_1 \wedge x_2 \wedge... \wedge x_{2n}$,
hence {\bf \red $\omega^n$ is a nowhere degenerate volume form.}

\claim 
Let $(M, \omega)$ be a compact manifold, $\dim_\R M=2n$. {\bf \blue Then 
the cohomology class of $\omega^i$, $i=1, 2,..., n$ is non-zero.}

\proof Since  $\omega^n$ is a volume form, it is non-zero in cohomology.
Indeed, otherwise we would have $\omega^n = d \alpha$ and $\int_M \omega^n= \int_M d\alpha=0$.
Therefore, $\omega^i$ is also non-zero in cohomology.
\endproof

\newpage

{\bf \blue Homogeneous spaces (reminder)}

\definition
{\bf \blue A Lie group} is a smooth manifold equipped with a group
structure such that the group operations are smooth.
Lie group $G$ {\bf \blue acts on a manifold $M$}
if the group action is given by the smooth map
$G \times M \arrow M$. 


\definition
Let $G$ be a Lie group acting on a manifold $M$ transitively.
Then $M$ is called {\bf \blue a homogeneous space}.
For any $x\in M$ the subgroup $\St_x(G)=\{g\in G\ \ |\ \ g(x)=x\}$
is called {\bf \blue stabilizer of a point $x$}, or
{\bf \blue isotropy subgroup}.

\claim
For any homogeneous manifold $M$ with transitive
action of $G$, {\bf \purple one has $M=G/H$,} 
where $H=\St_x(G)$ is an isotropy subgroup.

\proof The natural surjective map $G\arrow M$ putting $g$ to $g(x)$
identifies $M$ with the space of conjugacy classes $G/H$.
\endproof

\remark Let $g(x)=y$. Then $\St_x(G)^g=\St_y(G)$:
{\bf \purple all the isotropy groups are conjugate.}

\newpage

{\bf \blue Isotropy representation (reminder)}

\definition 
Let $M=G/H$ be a homogeneous space, $x\in M$ and
$\St_x(G)$ the corresponding stabilizer group.
The {\bf \blue isotropy representation}
is the natural action of $\St_x(G)$
on $T_x M$.

\definition
A tensor $\Phi$ on a homogeneous manifold $M=G/H$ 
is called {\bf \blue invariant} if it is mapped
to itself by all diffeomorphisms which come from
$g\in G$.

\remark
Let $\Phi_x$ be an isotropy invariant tensor
on $\St_x(G)$. For any $y\in M$ obtained as $y=g(x)$,
consider the tensor $\Phi_y$ on $T_y M$ obtained as
$\Phi_y:=g(\Phi)$. The choice of $g$ is not unique,
however, for another $g'\in G$ which satisfies $g'(x)=y$,
we have $g=g'h$ where $h\in \St_x(G)$. Since
$\Phi$ is $h$-invariant, {\bf \purple the tensor $\Phi_y$
is independent from the choice of $g$.}

We proved

{\bf \green Theorem 1:}
{\bf \red $G$-invariant tensors on $M=G/H$ are in bijective
correspondence with isotropy invariant tensors on
$T_x M$,} for any $x\in M$.
\endproof

\newpage

{\bf \blue Representations acting transitively on a sphere}

\theorem
Let $G$ be a group acting on a vector space $V$ with scalar product $g$.
Suppose that $G$ acts transitively on a unit sphere
$\{ x\in V \ \ |\ \ g(x)=1\}$. {\bf \red Then a $G$-invariant
bilinear symmetric form is unique up to a constant multiplier.}

\pstep Since $G$ preserves the sphere, which is a level
set of the quadratic form $g$, $g$ is $G$-invariant.

{\bf \green Step 2:} For any $G$-invariant 
quadratic form $g'$, the function $x \arrow \frac {g'(x)}{g(x)}$
is constant on spheres and invariant under homothety, hence it is
constant.
\endproof


\newpage

{\bf \blue Complex projective space}

\definition
Let $V=\C^n$ be a complex vector space equipped
with a Hermitian form $h$, and $U(n)$ the group
of complex endomorphisms of $V$ preserving $h$.
This group is called {\bf \blue the complex isometry group}.

\definition
{\bf \blue Complex projective space} $\C P^n$ is the
space of 1-dimensional subspaces (lines) in $\C^{n+1}$.

\remark
Since $U(n+1)$ acts on lines transitively, {\bf \red $\C P^n$ is
a homogeneous space, $\C P^n = \frac{U(n+1)}{U(1)\times U(n)}$,}
where $U(1)\times U(n)$ is a stabilizer of a line in $\C^{n+1}$.

\example $\C P^1$ is $S^2$.


\newpage

{\bf \blue Fubini-Study form}


\example
Consider the natural action of the unitary group
$U(n+1)$ on $\C P^n$. The stabilizer of a point
is $U(n)\times U(1)$.

\theorem
There exists an $U(n+1)$-invariant Riemann
metric on $\C P^n$. Moreover, {\red \bf such a metric is unique
up to a constant multiplier.} 

\pstep To construct a  $U(n+1)$-invariant Riemann
form on $\C P^n$, we take a $U(n)$-invariant form
on $T_x \C P^n$ and apply Theorem 1.
To show that a $U(n)$-invariant form on $T_x \C P^n$ exists,
take any Riemannian form and average it with $U(n)$-action.

{\bf \green Step 2:} Uniqueness follows because 
the isotropy group acts transitively on a sphere.
\endproof

\remark This Riemannian structure is called {\bf \blue
the Fubini-Study metric}.

\newpage

{\bf \blue Fubini-Study form}

\definition Let $v\in \C P^n$ be a point, which we 
consider as 1-dimensional subspace in $\C^{n+1}$. Then $T_v \C P^n= \Hom_\C(v, v^\bot)$,
hence it is a complex vector space. Denote by $I:\; T_v \C P^n\arrow T_v \C P^n$
the corresponding {\bf \blue complex structure operator}, $I^2=-\Id$.

\remark Since $I \in U(1) \subset U(n)\times U(1)$ (stabilizer of a point),
the Fubini-Study metric is $I$-invariant: $g(Ix, Iy) = g(x, y)$.
This gives \[ g(x, Iy) = g(Ix, I^2 y) = - g(Ix, y) = - g(y, Ix).\]
In other words, the form $\omega(x, y):= g(x, Iy)$ is anti-symmetric.

\definition
Let $g$ be a Fubini-Study metric on $\C P^n$. The 2-form
$\omega(x, y):= g(x, Iy)$ is called {\bf \blue the Fubini-Study form}.
{\bf \purple It is clearly non-degenerate.}

\claim {\bf \red The  Fubini-Study form on $\C P^n$ is symplectic.}

\proof Let $\omega$ be a Fubini-Study form. Then $d\omega$
is an isotropy-invariant 3-form on $T_x \C P^n$. However,
the isotropy group contains $-\Id$, {\bf \purple hence all isotropy-invariant
odd tensors vanish.} \endproof


\newpage

{\bf \blue Projective manifolds}

\definition
{\bf \blue A projective manifold} is a submanifold of $\C P^n$ obtained as 
the set of solutions of a system homogeneous polynomial equations\\
$P_1(z_1, ..., z_{n+1}) = P_2(z_1, ..., z_{n+1})= ... = P_k(z_1, ..., z_{n+1})=0$.

\remark
Let $Z\subset \C P^n$ be a projective manifold. Then
$T_v Z$ is the set of vectors $x\in T_v \C P^n$
such that the differentials of $P_i$ (in appropriate
affine coordinates) vanish on $x$.
Therefore, $I(T_v Z)= T_v Z$.

\corollary
Let $Z\subset \C P^n$ be a projective manifold,
and $g\restrict Z$ be restriction of the Fubini-Study form.
{\bf \red Then $\omega_Z(x, y):= g(x, Iy)$ defines a symplectic form on $Z$.}

\proof The form $\omega_Z$ is non-degenerate because
 $g\restrict Z$ is positive definite, 
and closed because restriction of a closed form is closed
(this is because restriction is pullback, and pullback
commutes with the de Rham differential). \endproof


\newpage

{\bf \blue Homogeneous and affine coordinates on $\C P^n$}

\definition
We identify $\C P^1$ with the set of $n+1$-tuples $x_0:x_1:...:x_n$
defined up to equivalence $x_0:x_1:...:x_n\sim \lambda x_0:\lambda x_1:...:\lambda x_n$,
for each $\lambda \in \C^*$. This representation
is called {\bf \blue homogeneous coordimates}. 
{\bf \blue Affine coordinates} in the chart $x_k\neq 0$ are
are $\frac {x_0}{x_k}:\frac{x_1}{x_k}:...:1:...:\frac{x_n}{x_k}$;
the space $\C P^n$ is a union of $n+1$ affine charts identified with $\C^n$,
with the complement to each chart identified with $\C P^{n-1}$.

\definition Let $\pi:\; \C^{n+1}\backslash 0 \arrow \C P^{n}$ be the natural projection.
{\bf \blue The Hopf map} $H:\; S^{2n+1} \arrow \C P^n$ is the restriction of this map
to a unit sphere.

\claim
Identify the complement of affine chart $1:x_1:...:x_n$
with $\C P^{n-1}$, using the homogeneous coordinates $0:x_1:...:x_n$.
Let $(x_0, x_1, ..., x_n) \in S^{2n+1}\subset \C^{n+1}$ be a point on the unit sphere,
distinct from $(1, 0, 0, ..., 0)$.
Then $\lim\limits_{t\rightarrow 0} (tx_0:x_1:...:x_n)= H(x_1:...:x_n)$.
\endproof

\corollary
{\bf \red $\C P^n$ is obtained from $\C P^{n-1}$ by gluing a cell
homeomorphic to a unit ball $B\subset \C P^{n}$,}
with $x\in \6 B= S^{2n+1}$ glued to $H(x)$.
\endproof

\newpage

{\bf \blue Cellular homology}

\definition
{\bf \blue A $k$-dimensional CW-complex} is defined inductively
as follows. A 1-dimensional CW-complex is a graph.
To define a $k$-dimensional CW-complex $X_k$, let $X_{k-1}$
be a $k-1$-dimensional CW-complex and let $\{B_i\}$ be
a collection of $k$-dimensional closed balls. 
For each of $B_i$ fix a continuous map
$\delta_i:\; \6 B_i \arrow X_{k-1}$.
Then $X_k$ is obtained as a quotient space
of $X_{k-1} \coprod_i B_i$ by a relation
$v \sim \delta_i(v)$ for each $v\in \6 B_i$;
in other words, $X_k$ is obtained by gluing
the cells $\{B_i\}$ to $X_{k-1}$ using the maps $\delta_i$.

\definition
Let $Z_{k-1}\subset X_{k-1}$ be a $k-1$-dimensional cell
of a CW-complex $X$, and $Z_k \subset X_{k}$ its $k$-dimensional cell.
Consider the map $\Psi_{Z_{k-1}}:\; X_{k-1}\arrow S^{k-1}$ obtained by
gluing the boundary of $Z_{k-1}$, $X_{k-2}$ and the rest
of the $k-1$-dimensional cells to a point.
Then the composition of $\delta_{Z_k}:\; S^{k-1}=\6 Z_k \arrow X_{k-1}$ and
$\Psi_{Z_{k-1}}:\; X_{k-1}\arrow S^{k-1}$ is a map from spere to a sphere.
Denote by $\delta(Z_{k}, Z_{k-1})$ its degree.
The {\bf \blue cell complex} of $X$ is the complex
$(C^*,d)$, where $C^k$ is a free abelian group generated
by $k$-cells, and $d(Z_k) = \sum_{R_i} \delta(Z_{k}, R_i)$,
where $R_i$ runs from all $k-1$-cells such that $\delta(Z_{k}, R_i)$
is non-zero (there are finitely many).


\newpage

{\bf \blue Cellular homology of $\C P^n$}

\theorem
{\bf \red Cohomology of the cell complex of $X$ are equal to its 
singular cohomology.}

\proof It was proven in lectures by M. B. \endproof

\remark In the cellular decomposition of $\C P^n$
there are $n+1$ cells, with one in each even dimension
$i=0, 2, 4, ..., 2n$. Therefore,
 $H_i(\C P^n) = \Z$ when $i$ is even,
$0 \leq i \leq 2n$, and $H_i(\C P^n) = 0$
when $i$ is odd.

\corollary 
{\bf \red The cohomology algebra of $\C P^n$ is 
the algebra of truncated polynomials: $\R[\omega]/[\omega]^{n+1}$,}
where $\omega$ is the Fubini-Study form and $[\omega]$ 
its cohomology class.

\proof
Since cohomology are dual to homology,
one has $H^i(\C P^{n}, \R) = \R$ when $i$ is even,
$0 \leq i \leq 2n$, and $H^i(\C P^n) = 0$ when $i$ is odd.
Since $\omega^i$ is never cohomologous to 0,
it generates $H^{2i}(\C P^{n}, \R)$.
\endproof










\end{document}
