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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{5}{Class assignment 5: Hypercomplex manifolds}

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\exercise
Let $(M,I,J,K)$ be a hypercomplex manifold,
$d_I:=IdI^{-1}, d_J:=JdJ^{-1},d_K:=KdK^{-1}$,
and $D:= dd_Id_Jd_K:\; \Lambda^*(M)\arrow \Lambda^{*+4}(M)$.
Prove that $D$ is independent from the choice of a
basis $I,J,K$ in quaternions.
\ez

\exercise
Let $(M,I,J,K)$ be a hypercomplex manifold, and
$\nabla$ a torsion-free connection preserving $I,J,K$
(such connection is called the Obata connection).
Consider $M$ as a complex manifold $(M,I)$.
Prove that $\nabla^{0,1}=\bar\6$ on $\Lambda^{p,0}_I(M)$,
where 
\[ 
  \bar\6:\; \Lambda^{p,0}_I(M)\arrow \Lambda^{p,1}_I(M)=
  \Lambda^{p,0}_I(M)\oplus \Lambda^{0,1}_I(M)
\]
Prove that $\nabla^{1,0}_X(\eta)= J(\6 J^{-1}(\eta)\cntrct X)$
for any $\eta\in \Lambda^{p,0}_I(M)$ and $X\in T^{1,0}_I(M)$.
\ez

\exercise
Construct a nilpotent Lie group with a non-trivial 
left invariant hypercomplex structure.
\ez

\definition
Let $(M,I,J,K)$ be a hypercomplex manifold, and $g$ a quaternionic
Hermitian metric. Consider the corresponding 
2-form $\Omega:=\omega_J+\1 \omega_K\in \Lambda^{2,0}(M,I)$.
The metric $g$ is called HKT if $\6\Omega=0$,
where $\6:\;  \Lambda^{2,0}(M,I)\arrow  \Lambda^{3,0}(M,I)$
is the standard Hodge differential.
\ed

\exercise
Let $(M,I,J,K)$ be a hypercomplex manifold, $g$ a quaternionic
Hermitian metric, and  $\Omega:=\omega_J+\1 \omega_K\in \Lambda^{2,0}(M,I)$.
Prove that $\6\Omega=0$ if and only if $d\omega_I$
has weight 1 with respect to the natural $SU(2)$-action
on 3-forms.
\ez

\exercise
Let $g$ be a K\"ahler metric on $(M,I)$, where
$(M,I,J,K)$ is hypercomplex, and $g_1$ be $g$ averaged
with the natural $SU(2)$-action. Prove that $g_1$ is HKT.
\ez


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