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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}}


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\listok{6}{Class assignment 6: Stable bundles and Yang-Mills connections}

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\exercise
Find an example of compact K\"ahler manifold $M$ such
that its tangent bundle is
\enum
\ite unstable
\ite unstable for any K\"ahler structure on $M$.
\ee
\ez

\exercise
Let $F$ and $G$ be coherent sheaves on a compact K\"ahler manifold $M$.
Assume that $F$ and $G$ admit filtrations $0=F_0\subset F_1\subset ... \subset F_n=F$
$0=G_0\subset G_1\subset ... \subset G_n=G$ with semistable subquotients,
and $\slope(F_i/F_{i-1})< \slope(G_j/G_{j-1})$ for all $i, j$.
Prove that $\Hom(F, G)=0$.
\ez

\exercise
Let $M$ be a compact K\"ahler manifold with unstable
tangent bundle. Suppose that $\slope(TM)\geq 0$.
Prove that there exists a foliation
${\cal F}$ on $M$ such that its tangent sheaf
is destabilizing.
\ez

\exercise
Let $(B, \nabla)$ be a holomorphic Hermitian vector bundle
on compact K\"ahler manifold
with Chern connection and curvature $\Theta\in \Lambda^{1,1}(M)\otimes\End B$ 
such that $\Lambda\Theta=f \Id_B$, where $f$ is a function. Prove that $f=\const$. 
\ez

\exercise
Let $L$ be a holomorphic line bundle on a compact Hermitian
manifold $(M,I,\omega)$ (not necessarily K\"ahler).
Prove that there exists a constant $c\in \1\R$ and a metric $h$ on $L$
such that the curvature of the Chern connection on $L$ satisfies
$\Lambda_\omega(\Theta)=c$. Prove that $h$ is unique up to a constant.
\ez



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