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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: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\proof}{{\bf \green Proof:\ }} \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{Observation \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{exercise}[section] \setcounter{exercise}{0} \renewcommand{\theexercise}{\noindent{Exercise \thesection.\arabic{exercise}}} \newcommand{\exercise}{% \setcounter{exercise}{\value{Mycounter}} \refstepcounter{exercise} \stepcounter{Mycounter} {\bf \green EXERCISE:\ }} \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{\problem}{% {\bf \green PROBLEM:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Locally conformally K\"ahler manifolds \\[15mm] \small lecture 15: complex surfaces} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE and IUM, Moscow \\[2mm] June 9, 2014 } \end{center} \newpage {\bf\blue Complex surfaces: Riemann-Roch theorem} \definition {\bf \blue A complex surface} is a compact, complex manifold of complex dimension 2. It is called {\bf \blue minimal} if it does not contain rational curves with self-intersection -1. %\claim %A rational curve with self-intersection -1 is obtained by %a blow-up of a smooth complex surface. % %\claim %Any complex surface $M$ admits {\bf \blue minimal model}, %which is a minimal complex surface birational to $M$ and %obtained by blowing down a sequence of $-1$-curves. \definition {\bf \blue Holomorphic Euler characteristic} of a coherent sheaf $F$ is $\sum_i (-1)^i \dim H^i(F)$. \theorem {\bf \blue (Riemann-Roch-Hirzebruch)} Let $L$ be a holomorphic line bundle on a complex surface $X$, and $K_X$ its canonical bundle. {\bf \red Then $\chi(L)= \chi(\calo_X)+ \frac{(L-K_X,L)}2$, and $\chi(\calo_X) = (c_1(X)^2- c_2(X))/12$}. \endproof \corollary Let $L$ be a line bundle on a complex surface, $(L,L)=d>0$. Then $\dim H^0(L^k)$ or $\dim H^0(L^{-k}\otimes K_X)$ grows at least quadratically with $k$. {\bf \blue Proof:} $\chi(L^k)= d k^2 + ak + b$, where $a, b$ are independent from $k$. However, $\chi(L^k)= h^0(L^k)- h^1(L^k) + h^2(L^k)$, hence either $h^0(L^k)$ or $h^2(L^k)$ grows quadratically. Finally, $ h^2(L^k) = h^0(L^{-k}\otimes K_X)$ by Serre's duality. \endproof \newpage {\bf\blue Douady space} \definition Let $M$ be a metric space, and $S, S'\subset M$ two subsets. The {\bf\blue Hausdorff distance} between $S$ and $S'$ is an infimum of all $\epsilon$ such that $S$ lies in $\epsilon$-neighbourhood of $S'$ and $S'$ lies in $\epsilon$-neighbourhood of $S$. \definition Given a complex subvariety $S\subset M$, {\bf \blue the Douady space} of deformations of $S$ in $M$ is the set of all complex subvarieties in the same cohomology class, equipped with topology induced by the Hausdorff metric $d_H(S, S')$. \claim {\bf \red Douady space is a complex analytic variety.} \endproof %\theorem %Let $C\subset M$ be a complex curve of genus $g$ on a surface. %Then the Douady space of deformations of $C$ is at least $d$-dimensional, where %$d=(C, K_M)+(1-g)$ when $g>0$ and %$d=(C, K_M)+(1-g)-3$ if $C$ is a rational curve. %\endproof \newpage {\bf\blue Base point set of a bundle} \definition Let $L$ be a holomorphic line bundle. Define {\bf \blue base point set} $\bps(L)$ as the set of all $x\in M$ such that any section of $L$ vanishes in $x$. \definition A {\bf\blue movable divisor} is a divisor with positive-dimensional Douady set (that is, movable in a family). \claim Let $L$ be a holomorphic line bundle, $t$ its non-zero section, and $D$ its zero divisor. {\bf\purple Then $D=D_0\cup D_1$, where $D_0\subset \bps(L)$, and $D_1$ is a union of movable divisors.} {\bf\green Proof:} Let $x\in D\backslash \bps(L)$. Then there exists a continuous family of divisors such that $D_t\not\ni x$, hence the component of $D$ containing $x$ is movable. \endproof \claim Let $C, C'$ be movable divisors without common components on a surface. {\bf \purple Then $C\cap C'$ is a finite set.} \endproof \newpage {\bf\blue Finite correspondences} \definition Let $Z\subset M\times M'$ be an irreducible subvariety. Denote by $\pi$, $\pi'$ the corresponding projections. It is called {\bf \blue a birationally finite correspondence} if $\pi^{-1}(m)$ and $\pi'{}^{-1}(m')$ is finite for general $m, m'$. {\bf \green Proposition 1:} Let $L$ be a holomorphic line bundle on a surface, $c_1(L)^2>0$, $h^0(L)>2$. Consider a subvariety $Z\subset M \times {\Bbb P}H^0(L)^*$, \[ Z= \{x\in M, t \in H^0(L)^*\ \ |\ \ V_x\subset \ker t\}, \] where $V_x:= \{ h\in H^0(L)\ \ |\ \ h\restrict x=0\}$ is the space of all sections vanishing in $x$. {\bf \red Then $Z$ is a birationally finite correspondence} between $M$ and $\pi_2(Z)$, where $\pi_2:\; Z \arrow {\Bbb P}H^0(L)^*$ is a projection. {\bf \green Proof. Step 1:} Let $t\in H^0(L)\backslash 0$, and let $Z_t\subset M$ be the zero divisor of $t$, and $H_t \subset {\Bbb P}H^0(L)^*$ the dual hypersurface. Then $\pi_1(\pi_2^{-1}(H_t))=Z_t$. For any $t_1\neq t_2 \in H^0(L)\backslash 0$, denote by $W_{t_1, t_2}$ the intersection $H_{t_1}\cap H_{t_2}$. Then $\pi_1(\pi_2^{-1}(W_{t_1,t_2}))=Z_{t_1}\cap Z_{t_2}$. {\bf \green Step 2:} The intersection $Z_{t_1}\cap Z_{t_2}$ is a union of base point divisors and intersection of movable divisors. Since it is 0-dimensional outside of $\bps(L)$, the correspondence is finite outside of $\bps(L)$. \endproof \newpage {\bf\blue Finite correspondences (2)} \theorem Let $M$ be a complex surface, admitting a birationally finite correspondence to $M'$. Then {\bf \red $M'$ is K\"ahler (projective) if and only if $M$ is K\"ahler (projective).} The proof is based on currents and Hahn-Banach separation theorem. \remark This is true {\bf \red only for surfaces}! \newpage % {\bf\blue Finite correspondences, projectivity, % K\"ahlerness and currents} {\bf\blue Currents} \definition Let $F$ be a Hermitian bundle with connection $\nabla$, on a Riemannian manifold $M$ with Levi-Civita connection, and \[ \|f\|_{C^k}:=\sup_{x\in M}\left(|f| + |\nabla f| + ... + |\nabla^kf|\right) \] the corresponding {\bf \blue $C^k$-norm} defined on smooth sections with compact support. {\bf \purple The $C^k$-topology is independent from the choice of connection and metrics.} \definition {\bf \blue A generalized function} is a functional on top forms with compact support, which is continuous in one of $C^i$-topologies. \definition {\bf \blue A $k$-current} is a functional on $(\dim M-k)$-forms with compact support, which is continuous in one of $C^i$-topologies. \remark Currents are forms with coefficients in generalized functions. \newpage {\bf\blue Currents on complex manifolds} \definition The space of currents is equipped with {\bf \blue weak topology} (a sequence of currents converges if it converges on all forms with compact support). The space of currents with this topology is a {\bf \red Montel space} (barrelled, locally convex, all bounded subsets are precompact). Montel spaces are {\bf \red reflexive} (the map to its double dual with strong topology is an isomorphism). \claim De Rham differential is continuous on currents, and the Poincare lemma holds. Hence, {\bf \purple the cohomology of currents are the same as cohomology of smooth forms.} \definition On an complex manifold, {\bf \blue $(p,q)$-currents} are $(p,q)$-forms with coefficients in generalized functions \remark {\bf\red In the literature, this is sometimes called $(n-p,n-q)$-currents}. \claim The Dolbeault lemma holds on $(p,q)$-currents, and {\bf \purple the $\bar\6$-cohomology are the same as for forms.} \newpage {\bf \blue Positive forms and currents} \definition A {\bf \blue weakly positive $(p,p)$-form} is a real $(p,p)$-form $\eta$ which satisfies $\eta(x_1,Ix_1,x_2,Ix_2,... x_{p},Ix_p)\geq 0$ for all $x_1, ... n_p \in TM$. {\bf \purple The set of weakly positive $(p,p)$-forms is a convex cone.} \definition A {\bf\blue weakly positive $(p,p)$-current} is a current taking non-negative values in weakly positive compactly supported $(n-p,n-p)$-forms. \definition A {\bf \blue positive generalized function} is a generalized function taking non-negative values on all positive volume forms. \remark Positive generalized functions are $C^0$-continuous. A positive generalized function multiplied by a positive volume form {\bf \purple gives a measure on a manifold,} and all measures are obtained this way. \corollary {\bf \purple A weakly positive $(p,p)$-current is $C^0$-continuous.} \newpage {\bf \blue Closed positive currents and psh functions} \definition Let $Z\subset M$ be a complex analytic subvariety. {\bf \blue The current of integration} $[Z]$ is the current $\alpha\arrow \int_Z\alpha$. {\bf \purple It is closed and positive} (Lelong). \remark (Poincare-Lelong formula) $\frac\1\pi dd^c\log |\phi|=[Z_\phi]$, where $Z_\phi$ is a divisor of a holomorphic function $\phi$. \definition A locally integrable function $f:\; M \arrow [\infty, \infty[$ is called {\bf \blue plurisubharmonic} (psh) if $dd^c f$ is a positive current. \claim (a local $dd^c$-lemma) {\bf \purple Locally, every positive, closed (1,1)-current is obtained as $dd^c f$,} for some psh function $f$. \exercise Prove that {\bf \red a locally integrable plurisubharmonic function on a compact complex manifold is constant.} \newpage {\bf \blue Hahn-Banach separation theorem and its applications} \theorem {\bf \blue (Hahn-Banach separation theorem)}\\ Let $V$ be a locally convex topological vector space, $A\subset V$ an open convex subset, and $W\subset V$ a closed subspace. Assume that $W\cap A=\emptyset$. Then there exists a continuous functional $\xi \in V^*$ such that $\xi(W)=0$ and $\xi(A)>0$. \endproof \theorem {\bf \blue (Harvey-Lawson)}\\ Let $M$ be a compact non-K\"ahler complex manifold. {\bf \red Then $M$ admits an exact $2n-2$-current such that its $(n-1, n-1)$-part is positive.} \remark Converse is also true: if $M$ admits such a current, $M$ is non-K\"ahler {\bf \purple (prove this).} {\bf \green Proof of Harvey-Lawson theorem. Step 1:}\\ Let $A\subset \Lambda^{1,1} M$ be the set of all strictly positive forms, and $W$ the space of all closed (1,1)-forms. {\bf \purple Hahn-Banach separation theorem produces a current $\xi^{1,1}\in D^{n-1,n-1}(M)$ such that $\xi^{1,1}(A)$ is positive and $\xi^{1,1}(W)=0$.} Clearly, $\xi^{1,1}(A)>0$ $\Leftrightarrow$ $\xi^{1,1}$ is positive. \newpage {\bf \blue Hahn-Banach separation theorem and its applications (2)} \theorem {\bf \blue (Harvey-Lawson)}\\ Let $M$ be a compact non-K\"ahler complex manifold. {\bf \red Then $M$ admits an exact $2n-2$-current such that its $(n-1, n-1)$-part is positive.} {\bf \green Proof of Harvey-Lawson theorem. Step 1:}\\ Let $A\subset \Lambda^{1,1} M$ be the set of all strictly positive forms, and $W$ the space of all closed (1,1)-forms. {\bf \purple Hahn-Banach separation theorem produces a current $\xi^{1,1}\in D^{n-1,n-1}(M)$ such that $\xi^{1,1}(A)$ is positive and $\xi^{1,1}(W)=0$.} Clearly, $\xi^{1,1}(A)>0$ $\Leftrightarrow$ $\xi^{1,1}$ is positive. {\bf \green Step 2:} Consider the space $W_1\subset \Lambda^2(M)$ generated by all closed furms and all (1,1)-forms. Extend $\xi^{1,1}$ to $W_1$ by taking $\xi^{1,1}(v)=0$ for all closed $v$. Since $\xi^{1,1}$ vanishes on closed (1,1)-forms, it is well defined on $W_1$, and can be extended to a continuous functional on $\Lambda^2$ (Hahn-Banach extension theorem). We obtain {\bf \purple a $(2n-2)$-current $\xi$ vanishing on closed forms and with positive (1,1)-part.} {\bf \green Step 3:} It remains to prove that $\xi$ is exact. Since $\langle \xi, d\alpha\rangle= \pm \langle d\xi, \alpha\rangle= 0$ for all $\alpha$, the current $\xi$ is closed. However, a pairing of $\xi$ with any closed form vanishes, hence {\bf \purple $\xi$ is exact by Poincare duality.} \endproof \newpage {\bf\blue Gauduchon metrics and Hahn-Banach theorem} \definition A Hermitian metric $\omega$ on complex $n$-manifold is called {\bf\blue Gauduchon} if $dd^c\omega^{n-1}=0$. \theorem {\bf \red Any compact complex manifold admits a Gauduchon metric.} {\bf \green Step 1:} Any strictly positive $(n-1, n-1)$-form is $(n-1)$-th power of a Hermitian form. Therefore, to construct a Gauduchon metric, it suffices to find a $dd^c$-closed strictly positive $(n-1, n-1)$-form. {\bf \green Step 2:} Let $A\subset \Lambda^{n-1,n-1}(M)$ be the cone of strictly positive $(n-1, n-1)$-forms, and $W:= \ker dd^c$. If these sets don't intersect, we can find $\xi \in \Lambda^{1,1}(M)$ which is positive and satisfies $\langle\xi, \ker dd^c\rangle =0$. \newpage {\bf\blue Gauduchon metrics and Hahn-Banach theorem (2)} \theorem {\bf \red Any compact complex manifold admits a Gauduchon metric.} {\bf \green Step 2:} Let $A\subset \Lambda^{n-1,n-1}(M)$ be the cone of strictly positive $(n-1, n-1)$-forms, and $W:= \ker dd^c$. If these sets don't intersect, we can find $\xi \in \Lambda^{1,1}(M)$ which is positive and satisfies $\langle\xi, \ker dd^c\rangle =0$. {\bf \green Step 3:} Since $\im d \subset \ker \ker dd^c$, this gives $\langle\xi, \im d \rangle =0$, hence $\xi$ is closed. Define {\bf \blue Aeppli cohomology} as \[ H^{p,q}_{AE}(M):= \frac {\ker dd^c\restrict {\Lambda^{p,q}(M)}} {\im d+ \im d^c}. \] The Poincare pairing $H^{p,q}_{AE}(M)\times H^{n-p,n-q}_{BC}(M)\arrow \C$, where $H^{*,*}_{BC}(M)$ is Bott-Chern cohomology, is non-degenerate {\bf \red (prove this)!} Since $\xi$ is orthogonal to $H^{p,q}_{AE}(M)$ under this pairing, its class in $H^{1,1}_{BC}(M)$ vanishes, hence $\xi = dd^c f\geq 0$. {\bf \green Step 4:} There are no globally plurisubharmonic generalized functions on a compact manifold {\bf \purple (prove it!)}. \endproof \remark In fact, {\bf \red there exists a Gauduchon metric in each conformal class, and it is unique up to a constant.} This is proven using elliptic equations and E. Hopf's strict maximum principle. \newpage {\bf\blue Finite correspondences and K\"ahlerness} \theorem Let $M_1$ be a complex surface, admitting a birationally finite correspondence to $M_2$. Then {\bf \red $M_1$ is K\"ahler if and only if $M_2$ is K\"ahler.} {\bf \green Proof. Step 1:} Let $\pi_1, \pi_2:\; Z \arrow M_i$ be the projection maps, $\omega_2$ be a K\"ahler form on $M_2$, and $\zeta:= \pi_1{}_*\pi_2^*\omega$. Then $\zeta$ is a positive, closed (1,1)-current which is smooth and strictly positive at each point $z\in M_1$ where the correspondence $Z$ is finite. {\bf \green Step 2:} Since $\zeta$ is infinite only around the points $z\in M_1$ where $\pi_1^{-1}(z)$ is positive-dimensional, $\zeta$ is smooth outside of a closed, finite set $\sing(\zeta)$. {\bf \green Step 3:} Let $S_\epsilon$ be an epsilon-neighbourhood of $\sing(\zeta)$. Using the local $dd^c$-lemma in $S_\epsilon$, we could write $\zeta = dd^c f$. Since a maximum of plurisubharmonic functions is plurisubharmonic, we can replace $\zeta$ by a current which is equal to $\zeta$ outside $S_\epsilon$ and equal to $dd^c\max(f, -C)$ in $S_\epsilon$, where $-C < f\restrict{\6 S_\epsilon}$. The new $\zeta$ is positive, closed, and equal to $dd^c(\phi)$ on $S_\epsilon$. \newpage {\bf\blue Finite correspondences and K\"ahlerness (2)} \theorem Let $M_1$ be a complex surface, admitting a birationally finite correspondence to $M_2$. Then {\bf \red $M_1$ is K\"ahler if and only if $M_2$ is K\"ahler.} {\bf \green Step 3:} Let $S_\epsilon$ be an epsilon-neighbourhood of $\sing(\zeta)$. Using the local $dd^c$-lemma in $S_\epsilon$, we could write $\zeta = dd^c f$. Since a maximum of plurisubharmonic functions is plurisubharmonic, we can replace $\zeta$ by a current which is equal to $\zeta$ outside $S_\epsilon$ and equal to $dd^c\max(f, -C)$ in $S_\epsilon$, where $-C < f\restrict{\6 S_\epsilon}$. The new $\zeta$ is positive, closed, and equal to $dd^c(\phi)$ on $S_\epsilon$. {\bf \green Step 4:} Consider a smooth convex function $\max_\epsilon:\; \R^2 \arrow \R$ equal to $\max(x,y)$ for $|x-y|>\epsilon$ and monotonous in each argument. Using $\max_\epsilon$ instead of $\max$ in Step 3, we may assume that $\zeta$ is actually smooth. {\bf \green Step 5:} Suppose that $M_1$ is non-K\"ahler. Then Harvey-Lawson bring an exact current $\xi=d\alpha$ with positive (1,1)-part. The current $\zeta\wedge \xi$ is positive; on the other hand, it is exact, giving $0=\int_M \zeta\wedge \xi>0$ -- a contradiction. \endproof \remark {\bf \red In dimension $>2$, this theorem is false;} it should be clear where the argument fails. \newpage {\bf\blue K\"ahler cone and Nakai-Moishezon-Demailly-Paun theorem} \theorem {\bf \blue (Nakai-Moishezon-Demailly-Paun)}\\ Let $M$ be a K\"ahler surface, and \[K:=\{\alpha\in H^{1,1}(M)\ \ |\ \ \alpha^2 > 0, \ \ \alpha\restrict{\text{all complex curves}}>0\}. \] {\bf \red Then $K$ is a K\"ahler cone of $M$ unless $M$ has no complex curves.} In the latter case, K\"ahler cone of $M$ is one of two connected components of $K$. \endproof \theorem Let $M_1$ be a complex surface, admitting a birationally finite correspondence to $M_2$. Then {\bf \red $M_1$ is projective if and only if $M_2$ is projective.} {\bf \green Proof. Step 1:} Using the Harvey-Lawson argument such as above, we prove that {\bf \purple $M_1$ is K\"ahler.} Let $\omega$ be a rational K\"ahler form on $M_2$, and $\zeta:= \pi_1{}_*\pi_2^*\omega$. {\bf \purple Then $\zeta^2>0$} by the same argument (with removal of isolated singularities) as used above. {\bf \green Step 2:} For any curve $C\subset M_1$, let $C_Z$ be its proper preimage in $Z$. Then $\langle \xi, C\rangle= \int_{C_Z}\pi_2^*\omega>0$. Then by Nakai-Moishezon-Demailly-Paun, {\bf \purple $[\xi]$ is a K\"ahler class.} {\bf \green Step 3:} This class is rational by construction, hence {\bf \purple $M_1$ is projective by Kodaira.} \endproof \newpage {\bf\blue Non-projective surfaces and the intersection form} \theorem Let $M$ be a complex surface, and $NS(M)$ the group of all integer cohomology classes represented by closed $(1,1)$-currents. {\bf \red Then $M$ is non-projective if and only if the intersection form on $NS(M)$ is non-positive.} {\bf \green Proof. Step 1:} Suppose that $[\xi]\in NS(M)$ is an integer class with $[\xi]^2=0$, and $L$ the line bundle with its curvature equal to $\xi$ (it exists by $dd^c$-lemma). Then either $h^0(L^k)$ or $h^0(L^k \otimes K_M)$ grows quadratically with $k$ by Riemann-Roch. {\bf \green Step 2:} Proposition 1 implies that $M$ admits a birationally finite correspondence with a projective manifold, hence $M$ is projective as shown above. {\bf \green Step 3:} Conversely, if $M$ is projective, it admits a curve with positive self-intersection, namely, the hyperplane section. \endproof \definition {\bf \blue Elliptic surface} is a complex surface $M$ equipped with a holomorphic map $M\arrow S$, with generic fiber a curve of genus 1. \newpage {\bf\blue Curves on non-projective surfaces} \theorem Let $M$ be a non-projective surface. Then {\bf \red either all curves on $M$ are isolated, or $M$ admits an elliptic fibration $\pi:\; M\arrow S$,} with all curves belonging to the fibers of $\pi$. {\bf \green Step 1:} Since the intersection form on $NS(M)$ is non-positive, for any two distinct irreducible curves $C, C'$ in the same cohomology class, the intersection $C\cap C'$ is empty. Therefore, all non-isolated curves $C$ satisfy $C\cdot C=0$. {\bf \green Step 2:} Given a family $C_t$ of non-intersecting curves parametrized by $t\in D$, consider the corresponding fibration from a subset $M_0:=\bigcup_{t\in D} C_t\subset M$ to $D$. Since $NC_t= \pi^* TD\restrict{C_t}$, the normal bundle to each $C_t$ is trivial. {\bf \green Step 3:} Adjunction formula gives $TC_t \otimes NC_t= \Lambda^2 TM\restrict{C_t}$, hence $\Omega^1C_t = K_M \restrict{C_t}$, where $K_M$ is a canonical bundle. If $(K_M, C_t)=l\neq 0$, one would have $(K_M+ tC_t,K_M+ tC_t)= (K_M,K_M)+ 2tl>0$ for appropriate choice of $t$. Then $M$ would be projective. Therefore, $(K_M, C_t)= 0 = \deg \Omega^1C_t$. This implies that all smooth fibers of $\pi$ are elliptic curves. {\bf \green Step 5:} Existence of elliptic fibration $M\arrow B$ would follow if we show that the deformation space of $C_t$ is compact for any curve on a complex surface (see the next slide). \newpage {\bf\blue Gromov's compactness theorem} {\bf \green Step 6:} The same argument as in Step 4 is used to show that $(C_t, v)=0$ for any $v\in NS(M)$, hence any irreducible curve belongs to a fiber of $\pi$. \endproof \theorem {\bf \blue (Gromov's compactness theorem)}\\ Let $(M,I, \omega)$ be a compact (almost) complex Hermitian manifold, ${\goth D}$ the space of all (pseudo-) holomorphic curves on $M$, with topology induced by the Hausdorff metric, $p>0$ a real number, and ${\goth D}_p\subset {\goth D}$ the space of all curves $S$ with $\Vol(S):=\int_S \omega\leq p$. Then ${\goth D}_p$ is compact. \newpage {\bf\blue Moduli of curves on complex surfaces} \theorem Let $M$ be a compact complex surface, $C\subset M$ a curve, and $B$ a connected component of its Douady space. {\bf \red Then $B$ is compact.} {\bf \green Proof. Step 1:} Fix a Hermitian metric $\omega$ on $M$. By Gromov's theorem, the space of curves of bounded volume is compact. {\bf \purple It remains to show that the volume stays bounded on each connected component of the Douady space,} for appropriate choice of $\omega$. Notice that this is vacuously true when $\omega$ is K\"ahler, because then the volume is a cohomological invariant. {\bf \green Step 2:} Consider the incidence variety $Z\subset M\times B$ consisting of pairs $C\in B, x\in C\subset M$, and let $\pi_1:\; Z \arrow M$, $\pi_2:\; Z \arrow B$ be the standard projections. Denote by $\Vol:\; B \arrow \R^{>0}$ the volume function, $\Vol(C)= \int_C \omega$. Then $\Vol= \pi_2{}_* \pi_1^* \omega$. {\bf \green Step 3:} Choose now $\omega$ Gauduchon. Then $dd^c\Vol= \pi_2{}* \pi_1^* dd^c\omega=0$. Therefore, $\Vol$ has no local minima or local maxima on $B$. However, $\Vol$ is a proper function by Gromov's theorem, hence it has to reach minimum somewhere. {\bf \purple Therefore, $\Vol=\const$.} Now, $B$ is compact by Gromov's theorem. \endproof \newpage {\bf\blue Class VII surfaces (a survey)} \definition Define the {\bf \blue Kodaira dimension} of a manifold $M$ as \\ $\kappa(M):=\lim\sup_n \frac {\log h^0(K^n_M)}{n}$. \definition A {\bf \blue Kodaira class VII surface} is a surface with $\kappa(M)=-\infty$ and $b_1(M)=1$. \example All Hopf surfaces and all Inoue surfaces are Kodaira class VII. \definition {\bf \blue Kato surface} is a surface $M$ which contains a 3-dimensional sphere $S^3$ such that $M\backslash S^3$ is connected, and a neighbourhood of $S^3$ is biholomorphic to a neighbourhood of standard $S^3$ in $\C^2$ (``global spherical shell''). \theorem All Kato surfaces are class VII. \newpage {\bf\blue The rest of classification theorems (a survey)} \theorem Let $M$ be a non-projective K\"ahler minimal surface. Then $M$ is elliptic, or isomorphic to a K3 or a torus. \theorem Let $M$ be a non-K\"ahler elliptic minimal surface. Then $M$ is isotrivial (all fibers are isomorphic) and Vaisman. \theorem Let $M$ be a non-K\"ahler non-elliptic surface. Then $M$ is class VII. \theorem (Bogomolov) All class VII surfaces with $b_2=0$ are Inoue or Hopf. \theorem (Andrei Teleman) All class VII surfaces with $b_2=1$ are Kato. \conjecture (GSS conjecture) All minimal class VII surfaces with $b_2>0$ are Kato. \theorem (Brunella) All Kato surfaces are LCK. %{\bf \green Step 2:} Since $\xi$ is non-negative on closure of %$A$, it is non-negative on all (2,0) and (0,2)-forms. Then %$\xi$ is of type (n-1, n-1). %{\bf \green Step 3:} Since $(\xi, dd^cf)=0$ for each $f\in C^\infty M$, %this gives $dd^c\xi =0$. Define {\bf \blue Aeppli cohomology} %as %\[ % H^{p,q}_{AE}(M):= \frac {\ker dd^c\restrict % \Lambda^{p,q}(M)} {\im d+ \im d^c}. %\] %The natural pairing $H^{p,q}_{AE}(M)\times %H^{n-p,n-q}_{BC}(M)\arrow \C$ is non-degenerate %{\bf \red (prove this)!} This gives a commutative diagram %with horizontal arrows induced by the de Rham %differential, and vertical arrows by duality. %\[\begin{CD} %\bigoplus_{p+q=m} H^{p,q}_{AE}(M)@>d>> %\bigoplus_{p+q=m+1} H^{p,q}_{BC}(M) \\ %@VVV @VVV\\ %\bigoplus_{p+q=2n-m} H^{p,q}_{AE}(M)^*@>{d^*}>> %\bigoplus_{p+q=2n-m-1} H^{p,q}_{BC}(M). %\end{CD} %\] %From this diagram, we obtain that $\ker d^*\subset H^*_{AE}(M)^*$ is %identified with $\im d\subset H^*_{AE}(M)$. %Since $\langle \xi, \ker d^*\rangle=0$, %this diagram gives $\xi\in d(H^{2n-3}_{AE}(M))$, %hence $\xi \end{document}