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\lhead{\tiny Hyperk\"ahler manifolds, HSE, Fall 2019} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  Class assignment \ \ \sc\version}
\rhead{{\tiny  Hyperk\"ahler manifolds, Misha Verbitsky}}


\begin{document}

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\listok{2}{Class assignment 2: almost complex structures}

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\exercise
Construct a left-invariant, integrable almost complex structure
on the Lie group $SU(2)\times SU(2)$.
\ez

\exercise
Let $M$ be a compact complex surface.
\enum
\ite
Prove that all holomorphic differential forms on $M$ are closed.
\ite
Find a compact complex 3-manifold which admits a 
non-closed holomorphic differential form.
\ee
\ez

\definition
Let $M$ be an $n$-dimensional complex manifold.
Fixing a volume form, we may identify
$\Lambda^{1,1}(TM)$ (pseudo-Hermitian forms on $T^*M$
and $\Lambda^{n-1,n-1}(M)$. A form $\eta\in \Lambda^{n-1,n-1}(M)$
is called {\bf (strictly)  positive} if all eigenvalues of the
corresponding (1,1)-form on $T^*M$ are (strictly) positive.
\ed

\exercise
\enum
\ite Prove that any strictly positive $(n-1,n-1)$-form $\eta$ is
equal to $\omega^{n-1}$ for some Hermitian
form $\omega$ on $M$.
\ite Find a counterexample when $\eta$ is not necessarily positive.
\ee
\ez



\exercise
Let $(M,I)$ be a smooth almost complex manifold equipped with a transitive
action of a group $G$. Assume that $I$ is $G$-invariant
(such a manifold is called {\bf  homogeneous}).
Assume, moreover, that for some
$x\in M$ there exists an element $\tau_x\in G$ fixing $x$.
Consider the induced action of $\tau_x$ on $T_xM$; denote this
operator by $\tau$.
\enum
\ite Suppose that $\tau=\lambda\Id$, where $\lambda\in \R$. 
Prove that for all $\lambda\neq 1$, the almost complex
structure $I$ is integrable.

\ite Construct examples of such $(M,I)$, $G$ and $\tau_x$ for each
$\lambda\in \R$.

\ite
Construct a homogeneous almost complex manifold 
which is not integrable.

\ite Suppose that $\tau$ is not a scalar, but all 
its eigenvalues $\alpha_i$ satisfy $9<|\alpha_i|<10$. 
Prove that the almost complex
structure $I$ is integrable.
\ee
\ez


\end{document}