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%version 1.0,\ \   17.11.2020
%version 1.1,\ \   19.11.2020, Nikita added some
%version 1.2,\ \   21.11.2020, 5.4 was wrong, 3.4 was 2 points, now 1



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\lhead{\tiny Complex geometry, HSE, exam} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\listok{0}{Complex geometry: exam}

{\scriptsize
Handouts score is given by the formula
$t=10+5a +8b$, where $a$ is the number of 
non-completed handouts with at least 1/2
of exercises credited, and $b$ the number
of completed handouts.
 
Each student receives a random selection of test
problems (the output of the randomizer is printed 
on a separate sheet). The final score for the course is $s= p +[t/10]$, 
where $p$ is the total number of points
for the exam. The exam is oral. 
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Almost complex manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\exercise[2 points]
Let $\omega$ be a non-degenerate 2-form on a Riemannian manifold,
and $\nabla$ its Levi-Civita connection. Assume that $\nabla(\omega)=0$.
Prove that $M$ admits a complex structure $I$ such that $\nabla(I)=0$.
\ez

\exercise
Let $f$ be a holomorphic function on an almost complex manifold.
Suppose that $|f|$ is constant. Prove that $f$ is constant.
\ez

\exercise
Construct a $G$-invariant complex structure on $G/H$,
or prove it does not exist.
\enum
\ite[3 points] $G=SL(6)$, $H=SO(6)$
\ite[2 points]  $G=GL(8)$, $H=SO(8)$
\ee
\ez
 
\exercise[2 points]
Let $(M,I)$ be an almost complex manifold, 
$\dim_\C M=n$, $U\subset M$ a dense, open subset,
and $\Omega \in \Lambda^{n,0}(U)$ a non-degenerate 
$(n,0)$-form. Assume that $d\Omega=0$.
Prove that the almost complex structure $I$ is integrable.
\ez

\exercise
Let $(M,I)$ be an almost complex manifold,
and $d=d^{-1,2}+d^{0,1}+d^{0,1}+d^{2,-1}$
the Hodge decomposition of its de Rham differential.
\enum
\ite
Prove that $\{d^{1,0}, d^{1,0}\}=0$, or find a
counterexample.
\ite Prove that $[d^{2,-1}, \{d^{1,0}, d^{0,1}\}]=0$.
\ee
\ez

\exercise
Let $(M,I)$ be an almost complex manifold,
and $W$ the Weil operator, acting on $(p,q)$-forms
as $W(\eta) = \1(p-q)\eta$. Prove that
$I$ is integrable if and only if $I^{-1} d I - [W,d]=0$.
\ez

\definition
{\bf Holomorphic differential}
on an almost complex manifold is a closed $(1,0)$-form.
\ed



\exercise
Let $M$ be an almost complex manifold,
and $\phi:\; M \arrow \R$ a function
which satisfies $dId(\phi)=0$. Prove
that $M$ admits a non-zero holomorphic differential.
\ez



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Symplectic, complex and K\"ahler  structures}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\exercise
Let $\omega$ be a non-degenerate 2-form on a Riemannian manifold,
and $\nabla$ its Levi-Civita connection. Assume that $\nabla(\omega)=0$.
Prove that $M$ admits a complex structure $I$ such that $\nabla(I)=0$.
\ez

\exercise
Construct a $G$-invariant Hermitian structure on $G/H$
and prove that it is K\"ahler. 
\enum
\ite[2 points] $G=SO(2n)$, $H=U(n)$.
\ite[2 points] $G=U(p,q)$, $H=U(p)\times U(q)$.
\ite[2 points] $G=U(p+q)$, $H=U(p)\times U(q)$.
\ee
\ez

\exercise[2 points]
Let $G$ be a compact, connected  Lie group with 
a left invariant complex structure
and a left invariant K\"ahler metric.
Prove that $G$ is commutative.
\ez


\definition
Let $(M,I)$ be a complex manifold, and $X\in TM$ a real
vector field. Recall that $X$ is called {\bf holomorphic}
if the corresponding diffeomorphisms are holomorphic,
or, equivalently, $\Lie_X I=0$.
\ed


\exercise[3 points] 
Let $(M,I)$ be a complex manifold,
$X$ a vector field, and $\nabla$ a torsion-free
connection which satisfies $\nabla(I)=0$.
Prove that $X$ is holomorphic
if and only if $\nabla X\in \Lambda^1(M) \otimes TM$,
considered as an endomorphism of $TM$, is complex linear.
\ez

\exercise
Let $(M,I, \omega)$ be an almost complex
Hermitian manifold, with $d\omega=0$.
Find the dimension of the Lie superalgebra
generated by  $L, \Lambda, d$, where
$L(\eta) = \omega\wedge \eta$, and 
$\Lambda = *L*$.
\ez

\definition
{\bf Holomorphic form} on a complex manifold
is a $(p,0)$-form $\eta$ which satisfies
$\bar\6\eta=0$.
\ed

\exercise
Let $\Omega$ be a holomorphic $(n-1)$-form on a compact
complex manifold $M$ with $\dim_\C M=n$. Prove that
$d\Omega=0$.
\ez



\exercise
Let $\eta$ be a real non-zero (1,1)-form on a compact
K\"ahler manifold $(M, \omega)$, $\dim_\C M=2$.
Assume that $\omega\wedge\eta=0$.
Prove that $\int_M \eta\wedge\eta<0$
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hodge theory and its applications}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Let $\eta$ be a holomorphic form on a K\"ahler
manifold. Prove that $\eta$ is harmonic.
\ez

\exercise
Let $\omega$ be a non-degenerate 2-form on a
$2n$-dimensional smooth manifold, and $d(\omega^k)=0$ for
some $k$ satisfying $0<k<n-1$. Prove that $d\omega=0$.
\ez


\exercise[2 points]
Let $M$ be a closed ball in $\R^n$ with a Riemannian
metric $g$ which smoothly extends to its boundary, and
$\alpha\in \Lambda^k(M)$ a differential form, also
smoothly extending to its boundary. Prove that
$\alpha\in \im \Delta$, where  $\Delta$ is the
Laplace operator associated with $g$.
\ez

\exercise
Let $F$ be an exact holomorphic $n$-form on an
$n$-dimensional compact complex manifold.
Prove that $F=0$.
\ez

\exercise[2 points]
Let $M$ be a compact complex manifold, $\dim_\C M=2$.
Prove that all holomorphic forms on $M$ are closed.
\ez

\exercise
Let $\theta$ be a closed holomorphic 1-form on a 
simply connected compact complex manifold (not necessarily K\"ahler).
Prove that $\theta=0$.
\ez

\exercise
Let $\eta$ be a (1,1)-form with compact support on 
$M\cong\C$. Prove that there exists $f\in  C^\infty M$ 
with compact support such that $\eta= dd^cf$,
or find a counterexample.
\ez

\exercise[2 points]
Let $M$ be a compact Riemannian manifold, 
${\cal H}^i$ the sheaf of harmonic $i$-forms,
and $\nu:\; {\cal H}^i\arrow \Lambda^i(M)$ the
tautological embedding. Prove that the
sequence
\[
0 \arrow {\cal H}^i \stackrel\nu 
\arrow \Lambda^i(M) \stackrel \Delta \arrow \Lambda^i(M)
\arrow 0
\]
is exact.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Geometry and topology of K\"ahler manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Let $M=C P^4 \times \C P^4 \times \C P^4$.
Prove that $M$ does not admit a K\"ahler structure
with non-standard orientation.
\ez

\exercise
Let $M$ be a compact K\"ahler manifold, 
$\dim_\C M =4$. Prove that $M$ does not admit
a K\"ahler structure with opposite orientation
or find a counterexample.
\ez

\exercise[2 points]
Let $M$ be a compact complex manifold, and $\pi_1(M)\cong \Gamma$
where $\Gamma$ is a group of upper triangular 
integer matrices 4x4 with 1 on diagonal.
Prove that $M$ does not admit a K\"ahler structure.
\ez


\exercise
For any given $n>2$ find a $2n$-dimensional connected simply connected manifold
with  $b_{2i}\neq 0$, $i =0 ,1, ..., n$ not admitting a symplectic 
structure.
\ez

 
\exercise
Let $\eta\in \Lambda^{1,1}(M)$ be a closed form 
on a compact K\"ahler manifold, $\dim_\C M=2$.
Assume that $\eta\wedge\omega=0$. Prove that
$\eta$ is harmonic.
\ez

\exercise
Let $M$ be a compact K\"ahler manifold.
{\bf K\"ahler cone} of $M$ is the set
$K(M)\subset H^{1,1}_\R (M)$ of all cohomology
classes of K\"ahler forms, where $H^{1,1}_\R (M)
=H^{1,1}(M)\cap H^2(M,\R)$. Prove that
$K(M)$ is open in $H^{1,1}_\R (M)$.
\ez

\exercise[2 points]
Let $T=\C^2/\Z^4$ be a complex torus.
Prove that $\rk \big[H^{1,1}(M)\cap H^2(M, \Z)\big]$
is always positive, or find a counterexample.
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Group action on manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Let $V$ be a representation of a finite group $G$
(not necessarily finitely-dimensional). Prove that
every vector $v\in V$ admits a finite decomposition
$v= \sum v_i$ where each $v_i$ belongs to a
finitely-dimensional representation of $G$.
\ez

\exercise
Bi-invariant forms on Lie groups are forms
which are invariant under the left and right group action.
Let $G$ be a compact Lie group equipped with a
bi-invariant metric. 
\enum
\ite[2 points] Prove that all bi-invariant differential forms on $G$ are 
harmonic.
\ite Prove that all harmonic forms are bi-invariant.
\ee
\ez

\exercise
Let $G$ be a finite group freely acting on a 
compact manifold $M$. Prove that $H^*(M/G)= H^*(M)^G$,
where $H^*(M)^G$ denotes invariants of the action of $G$
on cohomology.
\ez

\exercise
Let $M$ be a complex manifold,
admtting a holomorphic vector field with isolated fixed points only.
Prove the the topological Euler
characteristic of $M$ is non-negative.
\ez

\exercise
Let $(M,g, \omega)$ be a compact K\"ahler
manifold, $\nabla$ the Levi-Civita connection,
and $X\in TM$ a vector field which satisfies
$\nabla(X)=0$. 
\enum
\ite[2 points] Prove that $X$ is Killing, that is,
the corresponding diffeomorphism flow acts
by isometries.
\ite[3 points]
Prove that $X$ is holomorphic.
\ee
\ez

\exercise
Let $M$ be a compact Riemannian manifold,
and $G$ a group acting on $M$ by isometries. 
Denote by $G_0$ the image of $G$ in
$\Aut(H^*(M))$. Prove that $G_0$ is finite.
\ez


\exercise
Let $G$ be a finite group acting on $M=\C^n$ by
holomorphic maps. Prove that $M$ admits
a $G$-invariant holomorphic differential.
\ez


\end{document}
