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  \tiny M. Verbitsky }}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Hodge theory \\[15mm]
\small lecture 16: Currents and the Poincar\'e-Dolbeault-Grothendieck lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]
{\scriptsize NRU HSE, Moscow} \\[15mm]
{\small Misha Verbitsky, March 21, 2018 } 
\end{center}


\newpage

{\bf \blue Generalized functions}

\definition
Let $V$ be a vector space equipped with a collection of
norms (or seminorms) $|\cdot|_i$, $i=0, 1, 2, ...$
and a topology which is given by the metric
\[
d(x,y) = \sum_{i=0}^\infty 2^{-i}\min(|x-y|_i, 1),
\]
assumed to be non-degenerate. The space $V$ is called
{\bf \blue a Fr\'echet space} if this metric is complete.

\remark Completeness {\bf \purple is equivalent to convergence of
any sequence $\{a_i\}$ which is fundamental with respect to
all the (semi-)norms $|\cdot|_i$.}

\remark {\bf \purple A sequence converges 
in the Fr\'echet topology given by $d$ $\Leftrightarrow$ it converges in any of the  
(semi-)norms $|\cdot|_i$.}

\definition
Let $M$ be a Riemannian manifold, and
$\nabla^i:\; C^{\infty}(M) \arrow
\Lambda^1(M)^{\otimes i}$ the iterated connection.
{\bf \blue Topology $C^k$} on the space
$C^\infty_c(M)$ of functions with compact support
is defined by the norm
\[ |\phi|_{C^k}:= \sup_M \sum_{i=0}^k |\nabla^i\phi|.\]


\newpage

{\bf \blue Generalized functions (2)}

\definition
{\bf \blue The space of test-functions with compact support}
is the space of functions with compact support and a metric
\[
d(x,y) = \sum_{i=0}^\infty 2^{-i}\min(|x-y|_{C^i}, 1).
\]
of uniform convergence of all derivatives.

\exercise
Prove that {\bf \purple the space of test-functions with support
in a compact set $K\subset M$ is a Fr\'echet space.}

\definition
{\bf \blue Generalized function} (also called {\bf \blue distribution})
is a functional on the space of test-function which is continuous
in one of the $C^i$-topologies on the space $C^\infty(M)_{K}$
of functions with support in any compact $K\subset M$.

\example
{\bf \blue Delta-function} $\delta_z$ is a functional
mapping $\phi\in C^\infty_c(M)$ to $\phi(z)$, for a given
point $z\in M$. {\bf \purple Delta-function is continuous
in the topology $C^0$, its derivative is continuous in $C^1$
and so on}.


\newpage

{\bf \blue Currents on complex manifolds}

\remark The {\bf \blue $C^i$-topology}
is defined on the space of sections of any
vector bundle $B$ over using the same formula. It
depends on the choice of the metric on $M$ 
and on $B$, but {\bf \red the induced topology 
is clearly independent from this choice.}

\definition
{\bf \blue The space of test-forms of type $(p,q)$
on a complex manifold} is the space $\Lambda^{p,q}_c(M)$
with compact support, equipped with the Fr\'echet topology
as on the test-functions. 

\definition
A {\bf \blue $(p,q)$-current} on a complex $n$-dimensional
manifold is a functional $\theta$ on the space $\Lambda^{n-p,n-q}_c(M)$
of forms with compact support, such that for any compact
set $K\subset M$ there exists $i\geq 0$ such that
$\theta$ is continuous in $C^i$-topology on forms
with support in $K$.

\remark
{\bf \red A smooth $(p,q)$-form $\psi$ defines a $(p,q)$-current:}
given a test-form $\alpha \in \Lambda^{n-p,n-q}_c(M)$,
consider the functional  $\alpha \arrow \int_M \psi \wedge\alpha$.
This gives an embedding $\Lambda^{p,q}(M)\hookrightarrow {\cal D}^{p,q}(M)$
from forms to currents.

\remark 
{\bf \purple Currents are $(p,q)$-forms with coefficients in
generalized functions.}


\newpage

{\bf \blue Cohomology of currents}

\definition
Define {\bf \blue the de Rham differential on the space of currents}
using the formula
$ \langle d\psi, \alpha\rangle:= - (-1)^{\tilde \psi}\langle \psi, d\alpha\rangle.$
{\bf \purple This definition is compatible with the embedding
$\Lambda^{p,q}(M)\hookrightarrow {\cal D}^{p,q}(M)$  from forms to currents:}
\[
\int_M d\psi \wedge \alpha = \int_M d(\psi \wedge \alpha) -
(-1)^{\tilde \psi} \int_M \psi \wedge d \alpha =  -
(-1)^{\tilde \psi} \int_M \psi \wedge d \alpha
\]
by Stokes' formula. 

\remark {\bf \blue The Dolbeault differentials} $\6=d^{1,0}$, $\bar\6=d^{0,1}$ are defined
on currents using the same formula.

\exercise
{\bf \red Prove the Poincar\'e lemma for currents.}

\definition
Let $f:\; X \arrow Y$ be a proper holomorphic map of complex manifolds,
$\dim_\C X = \dim_\C Y +k$, and $\alpha$ a $(p,q)$-current on $X$.
Define {\bf \blue the pushforward} $f_* \alpha$ using
$\langle f_*\alpha, \tau\rangle := \langle \alpha, f^*\tau\rangle$,
where $\tau$ is any test-form. Then {\bf \red $f_*\alpha$ has bidimension
$(p-k,q-k)$.} One should think of $f_*$ as of fiberwise integration.

\remark Clearly, $d f_* \alpha= f_* d\alpha$, 
$\6 f_* \alpha= f_* \6\alpha$, and so on.

\remark Pullback of currents is (generally speaking) not well-defined.

\newpage

{\bf \blue Poincar\'e-Lelong formula}

\claim {\bf \blue (Poincar\'e-Lelong formula)}\\
Consider a current on $\C$ given by $\frac 1{\pi z}d z $.
{\bf \red Then $d \left(\frac 1 {\pi z}d z\right ) = \delta_0\Vol$},
where $\delta_0$ is $\delta$-function in 0.

\proof For any function smooth $f$ on a closure of a disc $D$ and $w\in D$,
Cauchy formula gives 
\[
f(w) = \frac 1 {2\pi \1} \int_{\6 D}\frac{f(z)}{z-w} dz -
      \frac 1 \pi \int_D \frac {\bar \6 f}{z-w} \wedge dz.
\]
Applying this to a test-function $f$ with compact
support inside $D$, we obtain
\[ f(w)=- \left\langle\frac 1 {\pi z} dz, \bar \6 f\right\rangle = 
\left\langle\bar \6 \left(\frac 1 {\pi z}\right) d z, f\right\rangle
= \left\langle d\left(\frac {d z} {\pi z}\right) , f\right\rangle.
\]
(the last equality is true because  $d\eta=\bar\6\eta$
for any $(1,0)$-form on a disc). \endproof


\newpage

{\bf \blue Poincar\'e-Dolbeault-Grothendieck (dimension 1)}

\corollary
Let $\pi_1, \pi_2:\; \C^2 \arrow \C$ be coordinate projections,
and $\xi$ a (1,0)-current on $\C^2$ defined by $\xi:=\frac1{\pi(z-w)}dw$,
where $w,z$ are coordinates on $\C^2$. Consider {\bf \blue convolution with
the current $\xi$}, given by $P_\xi(\tau):= {\pi_2}_*(\pi^*_1 \tau\wedge \xi)$.
{\bf \red Then  $\bar\6 P_\xi(\alpha)=\alpha$ for any $(0,1)$-form
$\alpha$ with compact support.}

\proof
$\bar\6 P_\xi(\alpha)= {\pi_2}_*(\pi^*_1 \alpha\wedge \bar\6\xi)=
{\pi_2}_*(\pi^*_1 \alpha \wedge \delta_{\triangle})= \alpha,$
where  $\delta_{\triangle}$ is $\delta$-function of the diagonal
$\triangle$, which is defined as $\langle \kappa, \delta_{\triangle}\rangle := \int_\triangle \kappa$.
\endproof

\corollary
{\bf \red For any (0,1)-form $\alpha$ with compact support
on $\C$ there exists a function $f\in C^\infty(\C)$ such that
$\bar\6f=\alpha$.} Moreover, $f$ can be chosen in such a way that
$|f(z)| < C \frac 1 {|z|}$ for some constant $C>0$
depending on  $\int_\C |\alpha|$.

\proof
Take  $f=  P_\xi(\alpha)$. From the definition of $P_\xi$
we obtain $|f(z)|< \dist(z, S)^{-1}\int_\C |\alpha|$,
where $S=\Supp(\alpha)$. This implies the estimate.
\endproof

\remark Similarly, for any $(1,1)$-form $\alpha$ with compact
support one has $\bar \6(P_\xi(\alpha))=\alpha$, {\bf \purple with the
same asymptotic estimates on $P_\xi(\alpha)$.}



\end{document}
