\documentclass{slides}

\usepackage{amssymb, amsmath, amscd, color, epsfig, url, wrapfig}
%\usepackage[matrix,arrow]{xy}
\usepackage[table]{xcolor}

\usepackage[T1,T2A]{fontenc}

\newcommand{\green}{\color[rgb]{0,0.4,0}}
\newcommand{\purple}{\color[rgb]{0.4,0,0.4}}
\newcommand{\red}{\color[rgb]{0.7,0,0}}
\newcommand{\blue}{\color{blue}}


\def\eqref#1{(\ref{#1})}
\newcommand{\goth}{\mathfrak}
\newcommand{\g}{{\frak g}}
\newcommand{\arrow}{{\:\longrightarrow\:}}
\newcommand{\Z}{{\Bbb Z}}
\renewcommand{\C}{{\Bbb C}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\Q}{{\Bbb Q}}
\renewcommand{\H}{{\Bbb H}}
\newcommand{\6}{\partial}
\def\1{\sqrt{-1}\:}
\newcommand{\restrict}[1]{{\left|_{{#1}}\right.}}
\newcommand{\cntrct}                % contraction with a vector field
{\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}}


\def\Bbb#1{\mathbb #1}


\newcommand{\calo}{{\cal O}}
\newcommand{\cac}{{\cal C}}

% Correcting TeX...
%\let\oldtilde=\tilde
%\renewcommand{\tilde}{\widetilde}
\renewcommand{\bar}{\overline}
\renewcommand{\phi}{\varphi}
\renewcommand{\epsilon}{\varepsilon}
\renewcommand{\geq}{\geqslant}
\renewcommand{\leq}{\leqslant}

% Operatornames
\newcommand{\even}{{\rm even}}
\newcommand{\ev}{{\rm even}}
\newcommand{\odd}{{\rm odd}}
\newcommand{\const}{{\it const}}
\newcommand{\fl}{{\rm fl}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\Hol}{\operatorname{{\cal H}ol}}
\newcommand{\Tot}{\operatorname{Tot}}
\newcommand{\Id}{\operatorname{Id}}
\newcommand{\id}{\operatorname{\text{\sf id}}}
\newcommand{\Vol}{\operatorname{Vol}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Alt}{\operatorname{Alt}}
\newcommand{\Iso}{\operatorname{Iso}}
\newcommand{\Sec}{\operatorname{Sec}}
\newcommand{\Ric}{\operatorname{Ric}}
\newcommand{\Sing}{\operatorname{Sing}}
\newcommand{\Spin}{\operatorname{Spin}}
\newcommand{\codim}{\operatorname{codim}}
\newcommand{\coim}{\operatorname{coim}}

\newcommand{\coker}{\operatorname{coker}}
\newcommand{\slope}{\operatorname{slope}}
\newcommand{\rk}{\operatorname{rk}}
\newcommand{\Def}{\operatorname{Def}}
\newcommand{\Lie}{\operatorname{Lie}}
\newcommand{\Tw}{\operatorname{Tw}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\Diff}{\operatorname{Diff}}
\newcommand{\Sp}{\operatorname{Sp}}
\newcommand{\Cl}{\operatorname{Cl}}

\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}



\newcommand{\inbfpare}[1]{{%
  \mbox{\tt (}\hspace{-5pt}\mbox{\tt (} #1 % 
  \mbox{\tt )}\hspace{-5pt}\mbox{\tt )}%
}}
\newcommand{\comment}[1]{{}}

\def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}}
\def\endproof{\blacksquare}
\def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}}


\makeatletter

%\@ifundefined{Bbb}
%     {\newcommand{\Bbb}[1]{{\mathbb #1}}}%
%{}%     {\edef\Bbb#1{{\Bbb #1}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       Pagestyle                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
\newcommand{\ps@verbit}{%
  \renewcommand{\@oddhead}{%
          \scriptsize {\it \small Hyperkahler manifolds, lecture 4 \hfil
  \tiny M. Verbitsky }}
  \renewcommand{\@evenhead}{\@oddhead}
  \renewcommand{\@oddfoot}{\hfil\thepage\hfil}
  \renewcommand{\@evenfoot}{\@oddfoot}}
 
\pagestyle{verbit}


   \setlength\paperheight {10in}%
    \setlength\paperwidth  {13.5in}
\setlength{\textwidth}{0.8\paperwidth}
\setlength{\textheight}{0.8\paperheight}

 \setlength{\pdfpageheight}{\paperheight}
 \setlength{\pdfpagewidth}{\paperwidth}
\addtolength{\topmargin}{-20mm}
\addtolength{\leftmargin}{-25mm}
\addtolength{\rightmargin}{-25mm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Lemma, sublemma, corollary, proposition, theorem,             %
% definition,example defined there:                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcounter{section}
\newcounter{Mycounter}[section]
\newcounter{lemma}[section]
\setcounter{lemma}{0}
\renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}}
\newcommand{\lemma}{%
     \setcounter{lemma}{\value{Mycounter}}
     \refstepcounter{lemma}
     \stepcounter{Mycounter}
     {\bf \green LEMMA:\ }}

\newcounter{claim}[section]
\setcounter{claim}{0}
\renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}}
\newcommand{\claim}{%
     \setcounter{claim}{\value{Mycounter}}
     \refstepcounter{claim}
     \stepcounter{Mycounter}
     {\bf \green CLAIM:\ }}

\newcounter{corollary}[section]
\setcounter{corollary}{0}
\renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}}
\newcommand{\corollary}{%
     \setcounter{corollary}{\value{Mycounter}}
     \refstepcounter{corollary}
     \stepcounter{Mycounter}
     {\bf \green COROLLARY:\ }}

\newcounter{theorem}[section]
\setcounter{theorem}{0}
\renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}}
\newcommand{\theorem}{%
     \setcounter{theorem}{\value{Mycounter}}
     \refstepcounter{theorem}
     \stepcounter{Mycounter}
     {\bf \green THEOREM:\ }}

\newcounter{conjecture}[section]
\setcounter{conjecture}{0}
\renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}}
\newcommand{\conjecture}{%
     \setcounter{conjecture}{\value{Mycounter}}
     \refstepcounter{conjecture}
     \stepcounter{Mycounter}
     {\bf \green CONJECTURE:\ }}

\newcounter{proposition}[section]
\setcounter{proposition}{0}
\renewcommand{\theproposition}
       {\noindent{Proposition \thesection.\arabic{proposition}}}
\newcommand{\proposition}{%
     \setcounter{proposition}{\value{Mycounter}}
     \refstepcounter{proposition}
     \stepcounter{Mycounter}
     {\bf \green PROPOSITION:\ }}

\newcounter{definition}[section]
\setcounter{definition}{0}
\renewcommand{\thedefinition}
       {\noindent{Definition~\thesection.\arabic{definition}}}
\newcommand{\definition}{%
     \setcounter{definition}{\value{Mycounter}}
     \refstepcounter{definition}
     \stepcounter{Mycounter}
     {\bf \green DEFINITION:\ }}


\newcounter{example}[section]
\setcounter{example}{0}
\renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}}
\newcommand{\example}{%
     \setcounter{example}{\value{Mycounter}}
     \refstepcounter{example}
     \stepcounter{Mycounter}
     {\bf \green EXAMPLE:\ }}

\newcounter{remark}[section]
\setcounter{remark}{0}
\renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}}
\newcommand{\remark}{%
     \setcounter{remark}{\value{Mycounter}}
     \refstepcounter{remark}
     \stepcounter{Mycounter}
     {\bf \green REMARK:\ }}


\newcounter{observation}[section]
\setcounter{observation}{0}
\renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}}
\newcommand{\observation}{%
     \setcounter{observation}{\value{Mycounter}}
     \refstepcounter{observation}
     \stepcounter{Mycounter}
     {\bf \green OBSERVATION:\ }}

\newcounter{question}[section]
\setcounter{question}{0}
\renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}}
\newcommand{\question}{%
     \setcounter{question}{\value{Mycounter}}
     \refstepcounter{question}
     \stepcounter{Mycounter}
     {\bf \green QUESTION:\ }}

\newcommand{\exercise}{{\bf \green EXERCISE:\ }} 
\newcommand{\pstep}{{\bf \green Proof. Step 1:\ }} 
\newcommand{\proof}{{\bf \green Proof:\ }} 


\begin{document}
\setcounter{page}{1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Hyperkahler manifolds, \\[15mm]
\small lecture 4: Bochner vanishing}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]
{\scriptsize NRU HSE, Moscow} \\[15mm]
{\small Misha Verbitsky, September 28, 2019 } \\ [15mm]
\url{http://bogomolov-lab.ru/KURSY/HK-2019/}
\end{center}

\newpage

{\bf \blue Clifford algebras (reminder)}

\definition
{\bf \blue The Clifford algebra} of a vector space $V$ with
a scalar product $q$ is an algebra generated by $V$ with 
a relation $xy+yx = -2q(x,y) 1$, that is, a quotient of
$T^{\otimes} V:= k \oplus V \oplus
V\otimes V \oplus ... \oplus T^{\otimes i} V$ by an ideal
generated by $xy+ yx= -2g(x,y)$ for all $x,y\in V$.

\newcommand{\Mat}{\operatorname{Mat}}

\theorem {\bf \blue (Bott periodicity over $\C$)}\\
Clifford algebra $\Cl(V, q)$ of a complex vector space
$V=\C^n$ with $q$ non-degenerate {\bf \red is isomorphic to
$\Mat\left(\C^{n/2}\right)$ ($n$ even) and 
$\Mat\left(\C^{\frac{n-1}{2}}\right)\oplus \Mat\left(\C^{\frac{n-1}{2}}\right)$
($n$ odd).}

\newpage

{\bf \blue $\Spin(n,n)$: an explicit
  construction}

Let $W= U \oplus V$ be a vector space
with $U, V$ dual and the quadratic form
pairing $(u, v)$ and $(u', v')$ as follows
$q((u, v), (u', v')) = \langle u, v'\rangle + \langle u', v\rangle$.

\definition
Consider {\bf \blue the exterior multiplication}
operator \\$e_u:\; \Lambda^*(U)\arrow \Lambda^{*+1}(U)$
with $e_u(\alpha) = u\wedge \alpha$ and {\bf \blue the convolution
operator} $i_v:\; \Lambda^*(U)\arrow \Lambda^{*-1}(U)$, with
$i_v(\alpha)(v_1, ..., v_k)=\alpha(v,v_1, ..., v_k)$.

\claim
These operators satisfy the following relations:
{\bf \purple $i_v, i_{v'}$ anticommute for all $v, v'$;
$e_u, e_{u'}$ anticommute for all $v, v'$;
finally, $\{i_v, e_u\}=\langle u, v\rangle \cdot \Id$,}
where $\{\cdot, \cdot\}$ (as usual) denotes the supercommutator,
$\{a, b\}= ab - (-1)^{\tilde a \tilde b} ba.$
\endproof

\remark These are the same relation as in Clifford
algebra! {\bf \purple 
This defines a map $\Cl(W) \arrow \Mat(\Lambda^*(U))$.}

\exercise
Fix a basis $u_i$ in $U$, and let $v_j$ be the dual basis
in $V$. For any pair of monomials $A, B$ in
$\Lambda^*(U)$, {\bf \purple find a product of a sequence
of some $i_{v_i}$, $e_{u_j}$ which maps $A$ to $B$ and puts all
other monomials to 0.}

\claim {\bf \red The natural map 
$\Cl(W) \arrow \Mat(\Lambda^*(U))$
is an isomorphism.}

\proof
See the previous exercise.
\endproof


\newpage

{\bf \blue Spinorial group $\Spin(2n)$ (reminder)}

\exercise Let $V$ be a vector space over a field
of characteristic 0.
Prove that {\bf \red the automorphism group
$\Aut(\Mat(V))$ is isomorphic to 
$PGL(V)$} (the quotient of $GL(V)$ by its center).


\definition
{\bf \green (Elie Cartan, 1913)}\\
{\bf \blue Spinor representation}
of the Lie algebra $\goth{so}(2n)$
is its representation on $\C^{2^n}$ induced 
by the isomorphism $\goth{pgl}(2^n)= \goth{sl}(2^n)=\Aut(\Cl(2n))$.
{\bf \blue Spinor representation}
of the Lie algebra $\goth{so}(2n+1)$
is any of two representations of $\goth{so}(2n+1)$
on $\C^{2^n}$ induced 
by the isomorphism $\goth{pgl}(2^n)=\goth{sl}(2^n)=\Aut(\Cl^\pm(2n+1))$,
where $\Cl^{pm}(V) = \Mat(2^n)$ is one of two components
of 
$\Cl(2n+1) =
\Mat\left(\C^{n}\right)\oplus \Mat\left(\C^{n}\right)$,


\definition
{\bf \blue Spinor group} $\Spin(k)$ is a 
double cover of $SO(k)$ obtained as a
Lie group of $\goth{so}(V)$ acting
on its spinorial representation.


\newpage

{\bf\blue Principal bundles (reminder)}

\definition
Let $G$ be a Lie group.
{\bf\blue Principal $G$-bundle} over a manifold $M$
is a smooth fibration $P\mapsto M$ with a smooth $G$-action
which acts freely and transitively on fibers.

\example
{\bf\blue Frame bundle} on a smooth $n$-manifold $M$
is the bundle of all frames (basises) in $T_x M$, for 
all $x\in M$. 

\definition
Let $H\arrow G$ be a group homomorphism, and
$P$ a principal $H$-bundle. Then the quotient
$P_G:=P \times G/H$ (with $H$ acting on both components
in a natural way) is called {\bf \blue an associated 
principal bundle}, and $P$ is called {\bf \blue
reduction of the principal $G$-bundle
$P_G$ to the group $H$}.

\definition
Let $G$ be a Lie group, and $G\arrow GL(n,\R)$
a group homomorphism. {\bf \blue A $G$-structure
on a manifold $M$} is a reduction of the principal
frame bundle to $G$.

\definition
Let $G$ be a Lie group, $V$ its representation,
and $P$ a principal $G$-bundle on $M$.
The quotient $P \times V/G$ is a vector bundle
over $M$, called {\bf\blue the associated vector bundle}.

\newpage

{\bf \blue Spin-structures and spinor bundles (reminder)}

\definition {\bf \blue A spin-structure} on an oriented $n$-manifold $M$
is a reduction of its structure group to $\Spin(n)$.
A manifold is called {\bf \blue spin} if it admits
a spin-structure.

\remark {\bf \purple This happens precisely when the second 
Stiefel-Whitney class $w_2(M)$ vanishes.}

\definition
{\bf \blue A bundle of spinors} 
on a spin-manifold $M$ is a vector bundle
associated to the principal $\Spin(n)$-bundle
and a spin representation.

\remark
The Levi-Civita connection {\bf \purple is naturally extended
from a connection on the bundle of orthogonal
frames to its double cover.} This defines
the Levi-Civita connection on the spinor bundle.


\newpage

{\bf \blue Spin-structures on Calabi-Yau manifolds}

\remark
Let $(M, I, g)$ be a K\"ahler manifold.
The Hermitian form defines a pairing between 
the spaces $T^{1,0}(M)$ and $T^{0,1}(M)$, which
are isotropic. {\bf \red Therefore, 
 $\Cl(TM\otimes \C, g)=\Mat(\Lambda^{*,0}(M))$.}

\remark
The real structure on $\Cl(TM\otimes \C)$ exchanges 
$i_v$ and $e_{\bar v}$, hence
it exchanges $\Lambda^{p,0}(M)$ and
$\Lambda^{n-p, 0}(M)$. Therefore,
{\bf \purple the bundle $\Lambda^{*,0}(M)$ can be
identified with spinors only for
Calabi-Yau manifolds.}

\claim
For any Calabi-Yau manifold, 
{\bf \red there is a spin structure
such that $\Lambda^{*,0}(M)$ is a spinorial
representation.}

\proof
To construct such a structure, we need to exhibit
a real structure $\tau$ on $\Lambda^{*,0}(M)$ which is
compatible with the real structure on $\Cl(\otimes \C)$,
that is, exchanging $i_v$ and $e_{\bar v}$.
For any $(p,0)$-form $\alpha$,
let $\tau(\alpha):= \overline{\frac{*\alpha}{\bar \Theta}}$,
where $\Theta\in \Lambda^{n,0}(M)$ is a parallel section
which trivializes the canonical bundle.
\endproof


\newpage


{\bf \blue Spinor bundles and Dirac operator (reminder)}


\definition
Consider the map $TM \otimes \Spin \arrow \Spin$
induced by the Clifford multiplication.
One defines {\bf \blue the Dirac operator} 
$D:\; \Spin \arrow \Spin$
as a composition of 
$\nabla:\; \Spin \arrow\Lambda^1 M\otimes\Spin = TM \otimes\Spin$
and the multiplication.

\definition
A {\bf \blue harmonic spinor} is a spinor $\psi$ such that
$D(\psi)=0$.

\theorem 
{\bf \blue (Bochner's vanishing)}\\  A harmonic spinor $\psi$
on a compact manifold with vanishing scalar curvature
$Sc:= Tr(\Ric)$ {\bf \red satisfies $\nabla\psi=0$.}

{\bf \green Proof:} Later today.

\newpage


{\bf \blue Bochner's vanishing on K\"ahler manifolds}

\remark {\bf \purple A K\"ahler manifold is spin if and only if
$c_1(M)$ is even,} or, equivalently, if there exists
a square root of a canonical bundle $K^{1/2}$.

\remark On a K\"ahler manifold of complex 
dimension $n$, {\bf \red one has a natural
isomorphism between the spinor bundle 
and $\Lambda^{*,0}(M)\otimes K^{1/2}$} (for $n$ even)
and $\Lambda^{2*,0}(M)\otimes K^{1/2}$ (for $n$ odd).

\remark On a K\"ahler manifold, the Dirac operator corresponds
to $\6 + \6^*$.

\corollary {\bf \purple On a Ricci-flat K\"ahler manifold,
all $\alpha \in \ker (\6 + \6^*)\restrict{\Lambda^{*,0}(M)}$
ara parallel.}

\remark $\ker \6 + \6^*= \ker \{\6, \6^*\}$, where $\{\cdot, \cdot\}$
denotes the anticommutator. However, $\{\6, \6^*\}= \{\bar\6, \bar\6^*\}$
as K\"ahler identities imply. Therefore, {\bf \purple on a Calabi-Yau
manifold, harmonic spinors are holomorphic forms}.

\theorem {\bf \blue (Bochner's vanishing)}
Let $M$ be a Ricci-flat K\"ahler manifold,
and $\Omega\in \Lambda^{p,0}(M)$ a holomorphic
differential form. {\bf \red Then $\nabla \Omega=0$.}
\endproof

\newpage

{\bf \blue Gaussian curvature}

\claim
Let $\nabla$ be a Levi-Civita connection
on a Riemannian manifold, and
$R\in T^*M^{\otimes 3} \otimes TM$ 
its curvature tensor. Using an
isomorphism $TM\cong T^*M$ given by the metric,
we may consider $R$ as an element in
$T^*M^{\otimes 4}$. {\bf\red Then $R$ is a section
of $\Sym^2(\Lambda^2T^*M)$,
antisymmetric in 1,2 and 3,4 indices.}

\definition
Let $V$ be a vectir space with non-degenerate
scalar product $g$. {\bf \blue A trace}
$\Tr_{12}:\;V^{\otimes^n} \arrow V^{\otimes^{n-2}}$
is defined as a map dual to the multiplication
 $A \arrow g\otimes A$. {\bf \blue The trace
in $i$-th and $j$-th indices}, denoted as
  $\Tr_{ij}:\;V^{\otimes^n} \arrow V^{\otimes^{n-2}}$,
is defined as a map which acts in
the $i$-th and $j$-th multiplier as $\Tr_{12}$
on the first two.

\definition
{\bf \blue Gaussian curvature} 
of a Riemannian manifold is a scalar
$\Tr_{13}\Tr_{24}(R)$, where $R$ is the 
Riemannian curvature.

\newpage

{\bf \blue Clifford multiplication in $\Sym^2(\Lambda^2 V)$}

{\bf \green LEMMA 1:}
Let $R\in \Sym^2(\Lambda^2 V)$, where $V$ is a
space with scalar product $g$. Denote the
Clifford multiplication as 
$\sigma:\; V^{\otimes ^4}\arrow \Cl(V)$.
{\bf \red Then \[ \sigma(R)=\Tr_{13}\Tr_{24}R+ \sigma(\Alt(R)),\]}
where $\Alt:\; \Sym^2(\Lambda^2 V)\arrow \Lambda^4 V$ is
the exterior product map.

\proof Let  $x,y,z,t\in V$, and
 $R(x,y,z,t):= (xy-yx)(zt-tz)+(zt-tz)(xy-yx)$
be the corresponding element in $\Sym^2(\Lambda^2 V)$.
Then 

1. If $x,y,z,t$ are pairwise orthogonal, we have
$\tau(R(x,y,z,t)) = \tau(\Alt(R))$, because
$x,y,z,t$ anticommute in the Clifford algebra.

2. If  $x,y,z$  are pairwise orthogonal, and $y=t$,
then $xy-yx$ anticommutes with $zt-tz$, hence
$\tau(R(x,y,z,t))=0$.

3. If $x, y$ are orthogonal,  $y=t$ and $x=z$,
we have \[ \sigma(R(x,y,z,t))=\sigma((xy-yx)^2) = g(x,x)g(y,y).\]
\endproof

\newpage

{\bf \blue Laplacian and rough Laplacian}

\remark
Let  $D:\; S \arrow S$ be the Dirac operator, and
 $x_i\in TM$  an orthonormal frame. {\bf \purple Then
$D(s)= \sum_i \sigma(x_i, \nabla_{x_i}(s))$,
where  $\sigma:\; TM \otimes S\arrow S$ is Clifford
multiplication.}

\corollary
Let  $\Theta\in \Lambda^2M \otimes \End(S)$
be the curvature of $S$. Then
\[
  D^2(s)= \sum_{i,j} \sigma(x_i x_j,\nabla_{x_i}\nabla_{x_j}s)=
  \sum_{i,j} \sigma(x_i x_j,\Theta_{x_i, x_j} s) + 
   \sum_{i,j} \sigma(x_i x_j+x_j x_i, \nabla_{x_i}\nabla_{x_j}s).
\]
Since $\sigma(x_i x_j+x_j x_i, v)= g(x_i,x_j)v$, this
gives
\[
D^2(s)= \sigma(\Theta,s) + \sum_i\nabla_{x_i}\nabla_{x_i}s.
\]

\newcommand{\Rough}{\text{\fontencoding{T1}\selectfont \DH
\fontencoding{T2A}\selectfont}}
\definition
{\bf \blue Rough Laplacian} on a bundle
$B$ with connection on a Riemannian manifold is defined as
$\Rough(s):= \Tr_{12}(\nabla^2 s)$.

\remark
The previous corollary is therefore rewritten as
\[
D^2(s)= \sigma(\Theta,s) + \Rough (s).
\]

\newpage

{\bf \blue Weitzenb\"ock formula}

\newcommand{\Sc}{\operatorname{\sf Sc}}
\theorem {\bf \blue (Lichnerowicz-Weitzenb\"ock formula)}\\ 
Let $M$ be a Riemannian manifold with spin structure,
$\Rough:\; S\arrow S$ the rough Laplacian, $\Sc$ 
multiplication by the scalar product, and $D:\; S\arrow S$ 
the Dirac operator. {\bf \red Then $D^2 = \Rough + \Sc$.}

\proof
$D^2(s)= \sigma(\Theta,s) + \Rough (s)$, as shown above,
and $\sigma(\Theta,s)= \Sc(s)+ \sigma(\Alt(R))$ by Lemma 1.
The last term vanishes, because $\Alt(R)$ (Bianchi
identity).
\endproof

\remark
$g(\Rough(s), s)= \Tr_{12}(\nabla^2(s), s)= g(\nabla(s),
\nabla(s))$.
This gives $\int_í g(\Rough(s), s)= \int_M g(\nabla(s), \nabla(s))$.
Therefore {\bf \purple on a compact manifold
  $\Rough(s)=0$
implies $\nabla(s)=0$.}

\newpage

{\bf \blue Bochner vanishing for harmonic spinors}

\corollary {\bf \blue (Bochner vanishing)}\\
Let $M$ be a compact Riemannian manifold with
non-negative scalar curvature. 
{\bf \red Then $\nabla(s)=0$ for any harmonic spinor $s$.}
If, in addition, $\Sc>0$ somewhere on $M$, then $s=0$.

\proof Lichnerowicz-Weitzenb\"ock formula gives
\[ 0=g(D^2(s),s) = g(\Rough(s), s) + \int_M \Sc\cdot g(s,s) =
\int_M g(\nabla(s), \nabla(s)) +\int_M \Sc \cdot g(s,s).
\]
The first term vanishes. Moreover, $s=0$ on the set
$U\subset M$ where $\Sc>0$. Then $s=0$ because
$\nabla(s)=0$. \endproof

\newpage

{\bf \blue Bochner's vanishing for holomorphic forms}

\theorem
On a compact Ricci-flat Calabi-Yau manifold, {\bf \red any holomorphic
$p$-form $\eta$ is parallel} with respect to the Levi-Civita connection:
$\nabla(\eta)=0$.

\proof Holomorphic forms are the same as harmonic spinors.
\endproof

\remark The form $\eta$ gives a harmonic spinor,
and {\bf \purple on a Riemannian spin manifold with
  $\Sc=0$,  any harmonic spinor
is parallel} (Bochner).

\remark 
Due to Bochner's vanishing,  {\bf \purple holonomy 
of Ricci-flat Calabi-Yau manifold
lies in $SU(n)$}, and {\bf \purple holonomy of Ricci-flat 
holomorphically symplectic manifold  lies in $Sp(n)$}
(a group of complex unitary matrices preserving a 
complex-linear symplectic form).

{\bf \green Exercise 1:}
Let $(M, \nabla)$ be a manifold with holonomy $\Sp(n)$.
Prove that {\bf \purple all parallel $(p,0)$-forms on $M$ are powers of the
holomorphic symplectic form.}


{\bf \green Exercise 2:}
Let $(M, \nabla)$ be a manifold with holonomy $SU(n)$.
Prove that {\bf \purple any holomorphic $(p,0)$-form on $M$ is a parallel
section of the canonical bundle.}

\newpage

{\bf \blue Holomorphic Euler characteristic}

\definition
{\bf\blue A holomorphic Euler characteristic} $\chi(M)$ of a 
K\"ahler manifold is a sum $\sum(-1)^p\dim H^{p,0}(M)$.

\theorem (Riemann-Roch-Hirzebruch)
For an $n$-fold,  
{\bf \red $\chi(M)$ can be expressed as a polynomial expressions of
the Chern classes,} $\chi(M)=td_{n}$
where $td_n$ is an $n$-th component of the Todd polynomial,
{\small \[
td(M) = 
1 + \frac1 {2}c_1 + \frac{1}{12}(c_1^2+c_2) + \frac{1}{24}c_1c_2 + 
\frac1{720}(-c_1^4 + 4c_1^2c_2 + c_1c_3 + 3c_2^22 - c_4) + ...
\]}

\vspace{-10mm}
\remark 
The Chern classes are obtained as
polynomial expression of the curvature (Chern-Weil).
Therefore {\bf \purple $\chi(\tilde M)= p\chi(M)$ for any
unramified $p$-fold covering $\tilde M \arrow M$.}

\remark Bochner's vanishing and exercises 1-2 imply:

1. When 
$\Hol(M) = SU(n)$, we have 
{\bf \red $\dim H^{p,0}(M)= 1$ for $p=1,n$, and 0 otherwise. }
In this case, $\chi(M)=2$ for even $n$ and
0 for odd.

2. When $\Hol(M) = Sp(n)$,we have 
{\bf \red  $\dim H^{p,0}(M)= 1$ for even $p$
$0\leq p\leq 2n$,  and 0 otherwise.} 
In this case, $\chi(M)=n+1$.

\corollary
$\pi_1(M)=0$ if $\Hol(M) = Sp(n)$, or
$\Hol(M) = SU(2n)$. If $\Hol(M) = SU(2n+1)$,
$\pi_1(M)$ is finite. 


\end{document}