\documentclass[11pt]{article} %\addtolength{\topmargin}{-10mm} %\addtolength{\textheight}{20mm} %\addtolength{\oddsidemargin}{-5mm} %\addtolength{\textwidth}{10mm} \input{defs-listki-en.tex} % version 1.0, 23.20.2019 \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}