\documentclass[10pt]{article}
\usepackage{url}

%version 1.0,\ \  Sep. 18, 2019


\newcommand{\version}{version 1.0,\ \   18.09.2019}
\newcommand{\firstdate}{18.09.2019}

%\addtolength{\topmargin}{-7mm}
%\addtolength{\textheight}{15mm}
%\addtolength{\oddsidemargin}{-5mm}
%\addtolength{\textwidth}{10mm}

\input{defs-listki-en.tex}


\setlength{\headheight}{15pt}
\pagestyle{fancy} 
\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}

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

\listok{1}{Class assignment 1: quaternionic Hermitian structures}

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

\exercise
Let $\nabla$ be a torsion-free connection on a manifold $M$,
and $\omega$ a differential form which satisfies $\nabla(\omega)=0$.
Prove that $\omega$ is closed.
\ez

\exercise
Let $\nabla$ be a torsion-free connection on a manifold $M$,
and $I$ an almost complex structure such that $\nabla(I)=0$.
Prove that $I$ is integrable.
\ez


\exercise
Let $M$ be an even-dimensional smooth manifold,
$A$ the (infinitely-dimensi\-onal) space of all non-degenerate
2-forms on $M$, and $B$ the space of all almost complex structures on $M$.
Prove that $A$ and $B$ are homotopy equivalent.
\ez

\definition
An almost hypercomplex structure on a manifold $M$
is a triple almost complex structures $(I,J,K)$
satisfying the quaternionic relations.
{\bf An almost hypercomplex Hermitian structure}
on $M$ is an almost complex structure $(I,J,K)$
and a Riemannian metric $h$ which is
invariant under the action of $I,J,K$.
\ed

\exercise
Let $(M,I)$ be almost complex manifold,
$A$ the (infinitely-dimensi\-onal) space of all 
non-degenerate (2,0) forms, and $B$
the space of all almost hypercomplex Hermitian structures
$(I,J,K,h)$.
Prove that $A$ and $B$ are homotopy equivalent.
\ez 

\exercise
Let $(M,I,J,K)$ be a
hypercomplex Hermitian 
manifold, and $\omega_I, \omega_J, \omega_K$ its fundamental forms.
Suppose that these forms are closed. Prove that $(M,I,J,K)$
is hyperk\"ahler.
\ez


\exercise
Let $(M,I,J,K)$ be a hyperk\"ahler manifold, $a, b, c, a^2+b^2+c^2$
real numbers, and $L=aI+bJ+cK$ the corresponding almost complex structure.
Prove that $L$ is integrable.
\ez


%\exercise
%Let $(M,I,J,K)$ be an almost hypercomplex manifold,
%and $\nabla$ a connection in $TM$ which satisfies
%$\nabla(I)=\nabla(J) =\nabla(K)=0$. Prove that $\nabla$
%is unique.
%\ez



%\exercise
%Let $\omega_1, \omega_2, \omega_3$ be
%a triple of 2-forms on a 4-manifold $M$.
%Assume that the intersection matrix 
%$a_{ij}= \frac{\omega_i \wedge \omega_j}{\omega_1^2}$
%is positive definite and has constant coefficients. Prove that $M$
%is hypercomplex Hermitian, and all $\omega_i$ are
%linear combinations of the fundamental forms.
%\ez


\exercise[*]
Let $\omega_1, \omega_2, \omega_3$ be
a triple of 2-forms on a manifold $M$ such that
any non-zero linear combination of $\omega_i$
is non-degenerate. 
\enum
\ite Prove that there exists a hypercomplex Hermitian structure
with fundamental forms $\omega_I, \omega_J, \omega_K$
such that the 3-dimensional bundles
spanned by  $\omega_I, \omega_J, \omega_K$ and 
$\omega_1, \omega_2, \omega_3\rangle$ coincide, or find a counterexample.
\ite Suppose that all $\omega_i$ are closed.
Prove that there exists a torsion-free connection
preserving $\omega_i$, or find a counterexample.
\ee
\ez


\exercise[*]
Let $\omega_1, \omega_2, \omega_3$ be
a triple of 2-forms on a manifold $M$  such that
any non-zero linear combination of $\omega_i$
is non-degenerate or has constant rank $\frac 1 2 \dim M$,
but not always non-degenerate.
Prove that $TM$ is equipped with an action of
the matrix algebra $\Mat(2, \R)$ preserving
$\langle\omega_1, \omega_2, \omega_3\rangle$.
\ez




\end{document}
