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\begin{document}

\listok{0}{Complex algebraic geometry \\ (Hodge theory), exam}
\lhead{\small Hodge theory, HSE, May 2018, exam.}

{\scriptsize
Each student receives a random selection of test problems
(the output of the randomizer is printed on a separate sheet). The
number of the problems is $\max(5,15-r)$, where $r=[t/10]$ and
$t$ is the total score for the handouts. Each problem is
worth 1, 2 or 3 points. The final score
for the term is $s= p +r$, where $p$ is the total number of points
for the exam. The exam is oral. 
}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Differential operators}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise[2 points]
Let $M$ be a compact Riemann manifold
and $d+d^*:\; \Lambda^{\even}(M)\arrow\Lambda^{\odd}(M)$
sum of de Rham differential and its adjoint. Prove that
$d+d^*$ is elliptic, and its index is the Euler characteristic of $M$.
\ez



\exercise[2 points]
Let $M$ be a compact Riemannian manifold
and $D:\; C^\infty M \arrow C^\infty M$ a second order elliptic
operator. Prove that the index of $\Delta$  vanishes.
\ez


\exercise
Let $(M,\omega,I)$ be a complex Hermitian manifold,
$L(\eta):= \eta\wedge\omega$, $\Lambda:=L^*$  the Hodge operators,
and  $d$, $d^c:= I dI^{-1}$ the corresponding differentials.
Consider the operator  $\Delta_\omega:= d\delta+\delta d$,
where $\delta:=[d^c, \Lambda]$.
\enum
\ite Prove that $\Delta_\omega$ commutes with $d$ and $d^c$.
\ite[2 points] Prove that $\Delta_\omega:\; \Lambda^*(M)\arrow\Lambda^*(M)$
cannot be surjective if $M$ is compact.
\ee
\ez

\exercise
Construct an elliptic operator $D:\; F \arrow G$
of order 3 or show that such operators don't exist.
\ez

\exercise
Consider the standard action of $SO(n+1)$
on $S^n$, and let $D$ be an $SO(n+1)$-invariant 
second order differential operator. Prove that
$D(f)= af + b \Delta(f)$, where $\Delta$ is the
usual Laplacian associated with the standard metric,
and $a, b\in \R$.
\ez

\exercise
Let $M$ be a $n$-dimensional manifold, $n>1$.
Prove that the set $S$ of order $i$ elliptic operators
on $C^\infty M$ is empty for any odd $i$.
Prove that for an even $i$ the set $S$ has
two connected components which are convex.
\ez

\exercise
Let $f$ be a smooth function on a compact Riemannian manifold,
such that $\Delta(f)=\lambda f$, where $\lambda\in C^\infty M$ is a negative
function. Prove that $f=0$.
\ez

\exercise[3 points]
Let $f\in C^\infty (S^{n-1})$ be an eigenvector
of the Laplacian operator on a sphere $S^{n-1} \subset \R^n$
with the usual Riemannian metric. Prove that $f$
can be expressed as a polynomial of coordinate functions on $\R^n$.
\ez

\exercise
Let $V = C^0([0,1])$ be the space of continuous
functions on $[0,1]$ with the uniform topology,
and $K\subset V$ the space of smooth functions
 $f:\; [0,1]\arrow \R$ with $|f'|<1$.
Prove that the closure of $K$ is the set of
1-Lipschitz maps $[0,1]\arrow \R$.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Almost complex manifolds, connections, symplectic structures}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\exercise
Let $f$ be a holomorphic function on an almost complex manifold.
Suppose that $|f|$ is constant. Prove that $f$ is constant.
\ez

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

\exercise[2 points]
Let $\omega$ be a non-degenerate 2-form on a Riemannian manifold,
and $\nabla$ its Levi-Civita connection. Assume that $\nabla(\omega)=0$.
Prove that $M$ admits a complex structure $I$ such that $\nabla(I)=0$.
\ez

\exercise
Let $M$ be a complex manifold. Construct a torsion-free connection
$\nabla$ such that $\nabla(I)=0$.
\ez

\exercise[2 points]
Let $(M,I)$ be an almost complex manifold, and\\ 
$N:\; \Lambda ^2 T^{1,0}M\arrow T^{0,1}M$
its Nijenhuis tensor. Assume that $N$ is surjective. Prove that any 
holomorphic function on $(M,I)$ is constant.
\ez


\exercise[2 points]
Let $V$ be a 4-dimensional vector space equipped with
a scalar product. Construct a natural homeomorphism between
$S^2 \coprod S^2$ and the space  of all orthogonal complex structures
on $V$.
\ez

\exercise
Construct a $G$-invariant Hermitian structure on $G/H$
and prove that it is K\"ahler. 
\enum
\ite[3 points] $G=SO(2n)$, $H=U(n)$.
\ite[2 points] $G=U(p,q)$, $H=U(p)\times U(q)$.
\ite[2 points] $G=U(p+q)$, $H=U(p)\times U(q)$.
\ee
\ez

\exercise[2 points]
Let $G$ be a compact, connected  Lie group with 
a left invariant complex structure
and a left invariant K\"ahler metric.
Prove that $G$ is commutative.
\ez

\exercise[2 points]
Let $\omega$ be a non-degenerate 2-form on a real manifold
$M$. Prove that there exists an almost complex Hermitian structure
such that $\omega$ is its Hermitian form.
\ez




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hodge theory and its applications}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\exercise
Let $\omega$ be a non-degenerate 2-form on a
$2n$-dimensional smooth manifold, and $d(\omega^k)=0$ for
some $k$ satisfying $0<k<n-1$. Prove that $d\omega=0$.
\ez

\exercise
Bi-invariant forms on Lie groups are forms
which are invariant under the left and right group action.
Let $G$ be a compact Lie group equipped with a
bi-invariant metric. 
\enum
\ite[2 points] Prove that all bi-invariant differential forms on $G$ are 
harmonic.
\ite Prove that all harmonic forms are bi-invariant.
\ee
\ez

\exercise
Let $M$ be a closed ball in $\R^n$ with a Riemannian
metric $g$ which smoothly extends to its boundary, and
$\alpha\in \Lambda^k(M)$ a differential form, also
smoothly extending to its boundary. Prove that
$\alpha\in \im \Delta$, where  $\Delta$ is the
Laplace operator associated with $g$.
\ez

\exercise[2 points]
Let $M$ be a compact K\"ahler manifold,
$d$, $d^c:= I dI^{-1}$ the usual differential, and
$\alpha \in \ker dd^c$. Prove that 
for any closed $(p,q)$-form $\beta$
one has $\int_M \alpha \wedge d\alpha\wedge\beta=0$.
\ez

\exercise
Let $(M, \omega)$ be a compact K\"ahler manifold,
and  $\phi\in C^\infty M$ a solution of the Monge-Ampere
equation $(\omega+ dd^c \phi)^n= e^f \omega^n$,
where $f\in C^\infty(M)$. Prove that $\omega + dd^c \phi$
is also a K\"ahler form.
\ez

\exercise
Let $F$ be an exact holomorphic $n$-form on an
$n$-dimensional compact complex manifold.
Prove that $F=0$.
\ez

\exercise
Let $M$ be a compact complex manifold, $\dim_\C M=2$.
Prove that all holomorphic forms on $M$ are closed.
\ez

\exercise
Let $\theta$ be a closed holomorphic 1-form on a 
simply connected compact complex manifold (not necessarily K\"ahler).
Prove that $\theta=0$.
\ez

\exercise[2 points]
Let $\eta$ be a closed (1,1)-form with compact support on
$\C^n$, where $n>1$. Prove that $\eta= dd^c f$, 
where $f$ is a smooth function on $\C^n$ with compact support.
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Geometry and topology of K\"ahler manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise[2 points]
Let $M=C P^4 \times \C P^4 \times \C P^4$.
Prove that $M$ does not admit a K\"ahler structure
with non-standard orientation.
\ez

\exercise
Let $M$ be a compact K\"ahler manifold, 
$\dim_\C M =4$. Prove that $M$ does not admit
a K\"ahler structure with opposite orientation
or find a counterexample.
\ez

\exercise[2 points]
Let $M$ be a compact complex manifold, and $\pi_1(M)\cong \Gamma$
where $\Gamma$ is a group of upper triangular 
integer matrices 4x4 with 1 on diagonal.
Prove that $M$ does not admit a K\"ahler structure.
\ez

\exercise[3 points]
Let  $M=\C P^2 \sharp \C P^2$ be a connected sum of $\C P^2$ with itself.
Prove that $M$ does not admit a K\"ahler structure.
\ez

\exercise
For any given $n>2$ find a $2n$-dimensional connected simply connected manifold
with  $b_{2i}\neq 0$, $i =0 ,1, ..., n$ not admitting a symplectic 
structure.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Projective manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise[2 points]
Let $M$ be a compact, non-projective K\"ahler manifold,\\
$\dim H^{2,0}(M)=1$, and $\phi:\; M \arrow M$ 
 a holomorphic involution without fixed points.
Prove that $\phi$ acts trivially on $H^{2,0}(M)$.
\ez

\exercise[3 points]
Let $M$ be a compact, non-projective K\"ahler manifold, \\
$\dim H^{2,0}(M)=1$, and $\Omega$ a generator of $H^{2,0}(M)$.
Consider a submanifold $Z\subset M$ such that $\Omega\restrict Z=0.$
Prove that $Z$ is projective.
\ez

\exercise
Let $M$ be a compact K\"ahler manifold, $H^{1,1}(M)$
one-dimensional, and  $\phi:\; M \arrow M$ a holomorphic
automorphism. Prove that $\phi$ acts trivially on $H^{1,1}(M)$.
\ez

\exercise[2 points]
Let $M$ be a projective manifold, and
$\phi:\ M \arrow M$ an automorphism. Prove that
there exists a $\phi$-invariant K\"ahler metric
or find a counterexample.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Line bundles and plurisubharmonic functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise[2 points] 
Let $M$ be a compact complex surface (manifold of complex
dimension 2), $\pi:\; M \arrow S$ a holomorphic map to a curve,
and $C$ a smooth fiber of $\pi$. Prove that 
$\deg K_M\restrict C=2g-2$, where $K_M$ is the canonical 
bundle of $M$ and $g$ the genus of $C$.
\ez

\exercise[2 points]
Let $f_1, f_2$ be holomorphic functions on a K\"ahler manifold
without common zeros. Prove that the function
 $\Delta (\log (|f_1|^2+|f_2|^2))$ is non-negative.
\ez

\exercise[2 points]
Let $B$ be a non-trivial line bundle on 
a compact complex manifold, and $h$ a metric
on $B$ with negative curvature of the Chern connection.
Prove that $B$ has no non-zero holomorphic sections.
\ez


\exercise [2 points]
Let $B$ be a Hermitian line bundle with positive
curvature, $f$ a holomorphic section of $B$, and $Z$ the
set of its zeros. Prove that $dd^c(|f|^{-1})$
is a K\"ahler form.
\ez


\exercise[2 points]
Let $f$ be a holomorphic function on a K\"ahler manifold.
Prove that the function $\Delta |f|^2$ is non-negative.
\ez

\exercise
Let $f$ be a smooth function with compact support
on an $n$-dimensional K\"ahler manifold $(M, \omega)$.
Prove that the integral $\int_M f\wedge dd^c f \wedge \omega^{n-1}$
is non-negative, and vanishes only when $f=\const$.
\ez



\exercise
(local $dd^c$-lemma)
Let $\eta$ be a closed (1,1)-form on $\C^n$.
Prove that $\eta=dd^c \alpha$.
\ez


\definition
A connection is called {\bf compatible with the holomorphic structure}
if $\nabla^{0,1}=\bar\6$.
\ed

\exercise
Let $B$ be a line bundle, and $\nabla$
a connection such that $\nabla(b)$ is holomorphic for any  holomorphic
section $b\in \Gamma(U,B)$ of $B$. Prove that the curvature of $\nabla$
is a $(2,0)$-form.
\ez

\exercise
Let $M$ be a complex manifold, $\nabla$ a torsion-free
connection preserving $I$, and $\phi$ a real function
Prove that the 2-form
$\operatorname{Hess}(\phi):=\nabla^2(\phi)$
is symmetric. Assume that $\operatorname{Hess}(\phi)$
is positive definite. Prove that $dd^c \phi$
is a K\"ahler form on $M$.
\ez




\exercise
Let $\operatorname{Pic}$ be the group of holomorphic line 
bundles on a compact K\"ahler manifold,
with the group structure defined by tensor multiplication.
Prove that the conected component of $\operatorname{Pic}$ is 
a compact torus.
\ez

\exercise[2 points]
Prove that a line bundle of degree 0 on a complex curve.
admits a Hermitian metric with flat Chern connection.
\ez


\end{document}