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\rfoot{\tiny Hyperk\"ahler manifolds, 2019: exam}
\newcommand{\version}{version 1.0,\ \   23.10.2019}

\newcommand{\firstdate}{25.10.2019}

\begin{document}

\listok{0}{Hyperk\"ahler manifolds, exam}
\lhead{\small Hyperk\"ahler manifolds,, HSE, October 2019, exam.}

{\scriptsize
Each student receives a random selection of 15 test problems
(the output of the randomizer is printed on a separate sheet).
You are expected to be able to prove all theorems you use,
unless stated otherwise. You are free to use the Riemann-Roch
formula, provided you can state it correctly. The final
score is given by $s=b+3$, where $b$ is the sum of points you got.
}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Holonomy groups and connections}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Let $V=\C^{2n}$ be the tautological representation
of $\Sp(n)$. Prove that $\Lambda^{2k+1}(V)$ has no
$\Sp(n)$-invariant vectors.
\ez

\exercise
Let $V=\C^{2n}$ be the tautological representation
of $\Sp(n)$. Prove that any $\Sp(n)$-invariant vector in
$\Lambda^{2}(V)$ is proportional to the symplectic form.
\ez

\exercise
Let $V=\C^{2n}$ be the tautological representation
of $\Sp(n)$. Prove that $\Lambda^{3}(V)$ is an irreducible
representation of $\Sp(n)$, or find counterexamples.
\ez

\exercise
Let $M$ be a hyperk\"ahler manifold with global
holonomy $\Sp(n_1)\times \Sp(n_2)\times ... \times \Sp(n_k)$,
with $2\sum_{i=1}^k n_i = \dim_\C M$. Prove that $M$
is simply connected.
\ez


\exercise[3 points] 
Let $(M,\omega)$ be a symplectic manifold. Prove that $M$
admits a torsion-free connection $\nabla$ such that $\nabla(\omega)=0$.
\ez

\exercise[2 points]
Find an example of a compact $n$-manifold $M$ not admitting
a Riemannian metric of signature $(1,n-1)$.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hyperk\"ahler manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\remark In this section you can use the Fujiki  formula, provided you can
state it correctly.
\er

\definition
An algebraic fuction $A:\; \R^n\arrow \R$ is a 
continuous function which is given as one of the branches
of a multi-valued function $\mu$ with 
the graph $\Gamma_\mu  \subset \R^n\times \R$
which is an irreducible algebraic variety.
\ed

\exercise[2 points]
Let $A:\; \R^n\arrow \R$ be an algebraic function,
and $S\subset \Gr(k, n)$ an open subset in 
the Grassmanian, with $k>1$. Assume that $A$
is polynomial on each $l\in S$. Prove that $A$
is polynomial.
\ez

\exercise[2 points] Let $M$ be a hyperk\"ahler manifold of maximal holonomy,
and $M \arrow B$ a surjective holomorphic map to a K\"ahler manifold $B$. Prove that
$\dim B= 0, n, 2n$, where $2n=\dim_\C M$.
\ez

\exercise[3 points] Let $L\subset M$ be a complex Lagrangian submanifold
in a compact hyperk\"ahler manifold. Prove that $L$ is projective.
\ez

\exercise
Let $M$ be a hyperk\"ahler manifold of maximal holonomy,
and $M \arrow B$ a surjective holomorphic map to a 
K\"ahler manifold $B$, $\dim B < \dim M$. Prove that
$B$ is projective.
\ez

\exercise
Let $\eta\in H^2(M)$ be a non-zero class on a 
 hyperk\"ahler manifold of maximal holonomy, $\dim_\C M=2n$.
Prove that $\eta^n\neq 0$.
\ez



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Trianalytic subvarieties}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Find a compact hypercomplex manifold
$(M,I,J,K)$ such that for any induced
complex structure $L=aI+bJ+cK$, the 
manifold $(M,L)$ has non-trivial divisors.
\ez

\exercise
Let $(M,I,J,K)$ be a hyperk\"ahler manifold, and
$\phi:\; (M,I) \arrow (M,I)$ a holomorphic
automorphism which acts trivially on $H^2(M)$.
Prove that its graph is trianalytic.
\ez

\exercise[3 points]
Let $T$ be a complex 2-dimensional
torus, $\Hilb^n(T)$ its Hilbert scheme, and
$\Hilb^n(T)\stackrel a \arrow T=\Alb(\Hilb^n(T))$
the Albanese map. Prove that the fundamental
group of $a^{-1}(0)$ is finite. 
\ez

\definition
We call the universal cover of  $a^{-1}(0)$
{\bf the generalized Kummer variety}.
\ed

\exercise[2 points]
Let $\iota$ be an involution of the torus mapping $x$ to
$-x$. Prove that it gives a holomorphic involution of the
generalized Kummer variety, and its fixed point set
is trianalytic.
\ez

\exercise[2 points]
Let $(M,I,J,K)$ be a hyperk\"ahler manifold, 
and $X\subset (M,I)$ a complex subvariety of dimension
$2$. Prove that 
$\int_X \omega_I^{2} \geq
\frac{1}{2} \int_x (\Omega \wedge \bar\Omega)$.
Prove that equality is realized if and only if
$X$ trianalytic.
\ez

\exercise[2 points]
Let $X \subset T^{4n}$ be a trianalytic subvariety in a compact torus.
Prove that $X$ is a union of subtori.
\ez

\exercise
Let $X$ be a hyperk\"ahler manifold (not necessarily compact) with
$b_2(X)=1$. Prove that there exists an induced complex structure $L$ on $X$
such that $(X,L)$ contains no compact complex subvarieties.
\ez



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hypercomplex manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\exercise
Let $M$ be a compact hypercomplex manifold, $\dim_\R M=4n$ and
$\eta$ a closed $SU(2)$-invariant 2-form. Suppose that
$\eta$ has maximal rank in some point $m\in M$.
Prove that $\int_M \eta^{2n}\neq 0$.
\ez

\exercise
Let be a compact hypercomplex manifold, $\dim_\R M=4n$ and
$V$ the bundle of $SU(2)$-invariant 2-forms. Prove that
$\dim V = \dim \Sym^2_\C \Lambda^{1,0}(M,I)$.
\ez

\exercise
Let $(M,I,K,K)$ be a compact hypercomplex manifold, and
$\eta\in \Lambda^{2,0}(M,I)$ a 2-form which
satisfies $\6\eta=0$, $J\eta=\bar\eta$. 
Prove that $d\eta$ has weight 1 with respect to $SU(2)$.
\ez

\exercise[2 points]
Let  $(M,I,K,K)$ be a compact hypercomplex manifold, and
$V\subset \Lambda^{2,0}(M,I)$ a space of 2-forms
which satisfy $\6\eta=0$, $J\eta=\bar\eta$.
Prove that $V$ is finitely dimensional, or find a counterexample.
\ez


\exercise[2 points]
Let $M$ be a compact hypercomplex manifold, $\dim_\R M=4n$ and
$V$ the space of closed $SU(2)$-invariant 2-forms. Prove that
$V$ is finitely-dimensional.
\ez


\exercise
Let $M$ be a compact hypercomplex manifold,  and
$\eta$ the curvature of the Obata connection of the canonical
bundle of $(M,I)$. 
\enum
\ite
Prove that $\eta$ is $SU(2)$-invariant.
\ite[2 points] Prove that $\eta^{2n}=0$, where $\dim_\R M =4n$.
\ee
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Stable bundles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Let $B$ be a flat Hermitian bundle over a K\"ahler manifold,
and $B_1 \subset B$ a holomorphic  sub-bundle, not necessarily flat.
Denote the curvature of the Chern connection on $B_1$ by
$\Theta_{B_1}$. Prove that $\1\Tr_B\Lambda(\Theta_{B_1})\geq 0$, 
and equality happens if and only if the Chern connection of
$B$ preserves $B_1$.
\ez

\exercise
Let $(G,I)$ be a compact Lie group 
with a left-invariant complex structure. Prove that
$(G,I)$ admits a holomorphic, $G$-invariant map
to a homogeneous projective manifold.
Prove that the canonical bundle of $(G,I)$ has 
no non-zero holomorphic sections, unless $G$ is a 
compact torus.
\ez

\exercise
Let $F$ be a stable coherent sheaf on a smooth K\"ahler manifold $M$.
Prove that $F^{**}:= \Hom(\Hom(F,\calo_M), \calo_M)$ is stable.
\ez

\exercise
Let $X$ be a compact complex manifold equipped with a free action
of the cyclic group $G$, and $M:= X/G$. 
\enum
\ite[2 points] Suppose that all
stable coherent sheaves on $M$ have rank 1. Prove that all stable 
sheaves on $X$ have rank 1.
\ite  Suppose that all
stable coherent sheaves on $X$ have rank 1. Prove that all stable 
sheaves on $M$ have rank 1.
\ee
(don't use Donaldson-Uhlenbeck-Yau unless you can prove it).
\ez

\definition
A vector bundle $B$ with $c_1B=0$ is called {\bf Bogomolov stable}
when all sections $B^{\otimes n}\otimes (B^*)^{\otimes M}$
are sections of rank one subsheaves which are direct summands.
\ed

\exercise
Using Donaldson-Uhlenbeck-Yau, prove that $\mu$-stability
implies Bogomolov stability.
\ez

\exercise[2 points]
Let $B$ be a Bogomolov stable bundle on
$\C P^n$, with $c_1(B)=0$. Prove that $B$ is $\mu$-stable.
\ez


\exercise[2 points]
Prove that the tangent bundle 
$T\C P^2$ is stable (without using Donaldson-Uhlenbeck-Yau).
\ez



\end{document}
 

