Gauduchon's form and compactness of the space of divisors

TL;DR

This paper investigates how the algebraic dimension of fibers in a holomorphic family of compact complex manifolds behaves under certain conditions, showing it is lower semi-continuous and preserving properties like being Moishezon.

Contribution

It establishes that in families where fibers satisfy the -lemma on a dense set, the algebraic dimension cannot decrease, extending understanding of complex manifold deformation.

Findings

Algebraic dimension is lower semi-continuous in the family.

If fibers are Moishezon on a dense set, all fibers are Moishezon.

The -lemma condition influences algebraic properties of fibers.

Abstract

We show that in a holomorphic family of compact complex connected manifolds parametrized by an irreducible complex space $S$, assuming that on a dense Zariski open set $S^{*}$ in $S$ the fibres satisfy the $\partial\bar\partial-$lemma, the algebraic dimension of each fibre in this family is at least equal to the minimal algebraic dimension of the fibres in $S^{*}$. For instance, if each fibre in $S^{*}$ are Moishezon, then all fibres are Moishezon.

Equations

$$a:=\inf \{alg(X_s),s\in D^{\ast }\}$$

$$\theta :\Gamma \to {\mathbb{R}}^{+}\mathrm{by}\gamma \mapsto \theta (\gamma ):={\int }_{\gamma }\omega .$$

$$T_{\pi }:T_{\mathcal{X}}\to {\pi }^{\ast }(T_D).$$

$$h^{\wedge q}\le C(V).\omega .$$

$$h_{|Z}^{\wedge q}\le C(V).{\omega }_{|Z}.$$

$${\int }_{\gamma }{h}^{\wedge q}\le C_L.{\int }_{\gamma }\omega .$$

$$H_{2q}({\pi }^{-1}({\Delta }_s),\mathbb{C})\to H_{2q}(\mathcal{X},\mathbb{C})$$

$$D\times X_0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\frac{\partial \omega }{\partial x}:=({\Phi }^{-1}{)}^{\ast }(\frac{\partial ({\Phi }^{\ast }(\omega )}{\partial x})\mathrm{and}\frac{\partial \omega }{\partial y}:=({\Phi }^{-1}{)}^{\ast }(\frac{\partial ({\Phi }^{\ast }(\omega )}{\partial y})$$

$$d\omega =\frac{\partial \omega }{\partial x}.{\pi }^{\ast }(dx)+\frac{\partial \omega }{\partial y}.{\pi }^{\ast }(dy)+d_{/\pi }\omega .$$

$$|{\nabla }_x\omega |\le C_K.\omega \mathrm{and}|{\nabla }_y\omega |\le C_K.\omega$$

$$d\omega\wedge\pi^{*}(\varphi)_{|\mathcal{H}}=\big{(}\nabla_{x}\,\omega\wedge\pi^{*}(\varphi_{y}.dx\wedge dy)-\nabla_{y}\,\omega\wedge\pi^{*}(\varphi_{x}.dx\wedge dy)\big{)}_{|\mathcal{H}}$$

$$D^{\prime \prime }\times U\subset D^{\prime \prime }\times V\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\frac{\partial \omega }{\partial x}.{\pi }^{\ast }({\phi }_y.dx\wedge dy)\mathrm{and}\frac{\partial \omega }{\partial y}.{\pi }^{\ast }({\phi }_x.dx\wedge dy)$$

$$|\big{(}d\varphi\wedge\pi^{*}(\varphi)\big{)}_{\mathcal{H}}|\leq C_{K}.(|\varphi_{x}|+|\varphi_{y}|).(\omega\wedge\pi^{*}(dx\wedge dy))_{|\mathcal{H}}.$$

$$|\frac{\partial \eta }{\partial x}|\le C.\eta \mathrm{and}|\frac{\partial \eta }{\partial y}|\le C.\eta \mathrm{in}U$$

$$-C\le \frac{{\eta }^{\prime }(x)}{\eta (x)}\le C$$

$$-C.|x-x_0|\le Log\frac{\eta (x)}{\eta (x_0)}\le C.|x-x_0|$$

$$\Phi :]a,b[\to Ux:=\phi (t),y:=\psi (t)\mathrm{with}{\phi }^{\prime }(t{)}^2+{\psi }^{\prime }(t{)}^2\equiv 1$$

$$\mathrm{and}\Phi (a+\epsilon )=s_0\mathrm{and}\Phi (b-\epsilon )=s\mathrm{with}0<\epsilon \ll 1.$$

$$\theta :\Gamma \to {\mathbb{R}}^{+\ast }\gamma \mapsto {\int }_{\gamma }\omega .$$

$$\Gamma \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\eta :D^{\ast }\to {\mathbb{R}}^{+\ast },s\mapsto \eta (s):=Trac{e}_{\tilde{\tau }}(\tilde{\theta })(s)$$

$$⟨d{\eta }_1,\phi .dy⟩=-⟨{\eta }_1,\frac{\partial \phi }{\partial x}.dx\wedge dy⟩=-{\int }_{\mathcal{Y}}\omega \wedge {\pi }^{\ast }(\frac{\partial \phi }{\partial x}.dx\wedge dy)$$

$$=-{\int }_{\mathcal{Y}}\omega \wedge {\pi }^{\ast }(d\phi \wedge dy)=-{\int }_{\mathcal{Y}}\omega \wedge d({\pi }^{\ast }(\phi .dy))\mathrm{then}$$

$$⟨d{\eta }_1,\phi .dy⟩={\int }_{\mathcal{Y}}\frac{\partial \omega }{\partial x}\wedge {\pi }^{\ast }(\phi .dx\wedge dy)={\int }_{\mathcal{Y}}{\nabla }_x\omega \wedge {\pi }^{\ast }(\phi .dx\wedge dy)$$

$$s\mapsto {\int }_{Y_s}{\nabla }_x\omega$$

$$|\frac{\partial {\eta }_1}{\partial x}|\le C.{\eta }_1$$

Full text

Gauduchon’s form and compactness of the space of divisors (second version)

Daniel Barlet111Institut Elie Cartan, Algèbre et Géomètrie,

Université de Lorraine, CNRS UMR 7502 and Institut Universitaire de France.

(16/05/17)

*En hommage à E. Bishop *

Abstract.

We show that in a holomorphic family of compact complex connected manifolds parametrized by an irreducible complex space $S$, assuming that on a dense Zariski open set $S^{*}$ in $S$ the fibres satisfy the $\partial\bar{\partial}-$lemma, the algebraic dimension of each fibre in this family is at least equal to the minimal algebraic dimension of the fibres in $S^{*}$. For instance, if each fibre in $S^{*}$ are Moishezon, then all fibres are Moishezon.

AMS Classification 2010.

32 G 10 - 32 J 18- 32 J 27- 32 Q 99- 32 G 05- 32 J 10.

Key words.

Family of compact complex manifods, Algebraic dimension, relative codimension 1 cycle-space, $\partial\bar{\partial}-$lemma, strongly Gauduchon manifolds.

1 Introduction

In this article we give a rather elementary proof of the following result which answers a “classical” question.

Theorem 1.0.1

Let $\pi:\mathcal{X}\to D$ be a smooth holomorphic family of compact complex connected manifolds parametrized by the unit disc $D$ in $\operatorname{\mathbb{C}}$. Assume that for each $s\in D^{*}$ the fiber $X_{s}$ of $\pi$ at $s$ satisfies the $\partial\bar{\partial}-$lemma. Let

[TABLE]

where $alg(X)$ denotes the algebraic dimension of the compact complex connected manifold $X$. Then we have $alg(X_{0})\geq a$.

As a special case (when $a=n:=dimX_{s}$) we obtain that, if for any $s\in D^{*}$ each $X_{s}$ is a Moishezon manifold, $X_{0}$ is also a Moishezon manifold.

The previous theorem gives easily the following corollary (see [B.15] for details).

Corollary 1.0.2

Let $\pi:\mathcal{X}\to S$ be a holomorphic family of compact complex connected manifolds parametrized by a reduced and irreducible complex space $S$. Let $S^{*}$ be a dense Zariski open set in $S$. Assume that for each $s\in S^{*}$ the fiber $X_{s}$ of $\pi$ at $s$ satisfies the $\partial\bar{\partial}-$lemma. Let $a:=\inf\{alg(X_{s}),s\in S^{*}\}$. Then for any $s\in S$ we have $alg(X_{s})\geq a$.$\hfill\blacksquare$

Note that the proof in [B.15] shows that the minimum of the algebraic dimension is obtained at the general point222this means on the complement of a countable union of closed nowhere dense analytic subsets in $S$ (see the proposition 3.3.1 below). in $S$.

In the first section we show that the existence of a Gauduchon metric on a compact complex connected manifold $X$ implies the compactness of the connected components of the space of divisors in $X$. This new proof of this classical result is the key of our proof for the theorem above which appears as a relative version of it.

I thank J-P. Demailly for some constructive comments which help to improve this article.

2 The absolute case

Let me begin by two simple lemmas.

Lemma 2.0.1

Let $M$ be a connected reduced complex space and let $T:M\to\mathbb{R}^{+}$ be a continuous function on $M$. Assume that $T$ is pluri-harmonic on the smooth part $M^{\prime}$ of $M$ and that the function $T$ achieves its minimum at a point $x_{0}\in M$. Then $T$ is constant on $M$.

Proof.

This is an elementary exercice.$\hfill\blacksquare$

Lemma 2.0.2

Let $X$ be a compact reduced complex space and let $\omega$ be a continuous real $(q,q)-$form on $X$ which is strictly positive in the Lelong sense (see section 3.1). Let $\Gamma$ be a connected component of the reduced complex space of compact $q-$cycles in $X$ and define

[TABLE]

Then the continuous function $\theta$ achieves its minimum on $\Gamma$.

Proof.

Note first that $\theta$ is continuous thanks to the prop. IV 3.2.1 of [B-M 1]. Let $\alpha:=\inf\{\theta(\gamma),\gamma\in\Gamma\}$. Then the subset $A:=\{\gamma\in\Gamma\ /\ \theta(\gamma)\leq\alpha+1\}$ is a compact subset in $\Gamma$ thanks to Bishop’s theorem [Bi.64], because it is a closed subset in $\Gamma$ such that each cycle in $A$ has bounded volume (relative to $\omega$) and support in the compact space $X$ (see th. IV 2.7.23 in [B-M 1]). Then $\theta$ achieves its minimum on $A$.$\hfill\blacksquare$

Proposition 2.0.3

Let $X$ be a compact reduced complex space and let $\omega$ be a continuous real $(q,q)-$form on $X$ which is strictly positive in the Lelong sense (see section 3.1) and satisfies $\partial\bar{\partial}\omega=0$ as a current on $X$. Then any connected component of the reduced complex space $\mathcal{C}_{q}(X)$ of compact $q-$cycles in $X$ is compact.

Proof.

The function $\gamma\to\theta(\gamma):=\int_{\gamma}\omega$ is continuous on $\mathcal{C}_{q}(X)$ and $\partial\bar{\partial}-$closed in the sense of currents; as its achieves its minimum on any connected component $\Gamma$ of $\mathcal{C}_{q}(X)$ thanks to the lemma 2.0.2 , we conclude that it is constant on each $\Gamma$, thanks to lemma 2.0.1, and then any such $\Gamma$ is compact (see [B-M 1] ch.IV th. 2.7.20).$\hfill\blacksquare$

Remark.

As any compact complex manifold $X$ of dimension $n$ admits a Gauduchon metric (see [G.77]), the proposition above gives a proof of the fact that the space of divisors of such a $X$ has always compact connected components, which is a well known classical result (see [C.82] and [Fu.82]).

3 The relative case

3.1 Relative strong positivity in the Lelong sense

Let $\pi:X\to S$ be a surjective holomorphic map between two irreducible complex spaces. Let $T_{\mathcal{X}/D}$ be the tangent Zariski linear space of $\pi$, so the kernel of the tangent map

[TABLE]

Let $1\leq q\leq\dim X-\dim S:=n$  and let $\omega$ a continuous relative $(q,q)-$differential form on $X$. Then $\omega$ induces a continuous hermitian form of the fibres on the linear space on $X$ associated to the coherent sheaf \Omega_{X/S}^{q}:=\Omega_{X}^{q}\big{/}\pi^{*}(\Omega^{1}_{S})\wedge\Omega_{X}^{q-1}.

Let $p:Gr_{q}(X/S)\to X$ be the grassmannian of $q-$planes in $T_{X/S}$, $U$ the universal $q-$vector bundle on $Gr_{q}(X/S)$ and $\theta:\Lambda^{q}(U)\to Gr_{q}(X/S)$ the line bundle which is the determinant of $U$. A $\pi-$relative continuous $(q,q)-$form on $X$ defines a continuous hermitian form on $\Lambda^{q}(U)$.

Definition 3.1.1

We shall say that a $\pi-$relative continuous $(q,q)-$form $\omega$ on $X$ is strongly Lelong positive at $x_{0}$ if the hermitian form on $\Lambda^{q}(U)$ defined by $\omega$ is a positive hermitian form at each point of $p^{-1}(x_{0})$ (so a continuous hermitian metric on this line bundle) .

In other word that means that for any $q-$plane $P$ in the Zariski tangent space $T_{X,x_{0}}$ which is vertical (i.e. contained in the kernel of $T_{\pi,x_{0}}$) then $\omega_{x_{0}}[v_{1}\wedge\dots\wedge v_{q}]>0$ when $v_{1},\dots,v_{q}$ is a basis of $P$.

As the map $p:Gr_{q}(X/S)\to X$ is proper, the condition above is open. Moreover, it is an easy exercice to prove the following properties:

Assume that the $\pi-$relative continuous $(q,q)-$form $\omega$ on $X$ is strongly Lelong positive at $x_{0}$. Then for any continuous hermitian metric $h$ on $X$ there exists a neighbourhood $V$ of $x_{0}$ in $X$ and a positive constant $C(V)$ such that on the open set $p^{-1}(V)$ the metric induced by $h^{\wedge q}$ and $\omega$ on the line bundle $\Lambda^{q}(U)$ satisfy

[TABLE] 2. 2.

An easy consequence of the above estimate is the fact that on any closed (complex) analytic subset $Z$ of pure dimension $q$ in a fiber of $\pi$ over $V$ we have the inequality of volume forms on $Z$

[TABLE] 3. 3.

Assume now that $\omega$ is strongly Lelong positive on an open set $V$ in $X$ containing a compact set $L$. Then there exists a positive constant $C_{L}$ such that for any compact relative $q-$cycle $\gamma$ contained in $L$ we have the inequality

[TABLE]

This means that the integral over compact relative $q-$cycles of a continuous $\pi-$relative $(q,q)-$form which is strongly Lelong positive on $X$ controls the volume of these cycles contained in a given compact set $L\subset X$ for any given hermitian metric on $X$.

The following result follows easily of the descrition of compact subsets in the space of compact cycles which is a consequence of E. Bishop’s theorem.

Theorem 3.1.2

Let $\pi:\mathcal{X}\to S$ a proper surjective map between irreducible complex spaces. Assume that there exists a smooth $d-$closed $2q-$differential form $\tilde{\omega}$ on $\mathcal{X}$ such that its $(q,q)-$part induces a relative strongly Lelong positive form on $X$. Then any connected component of $\mathcal{C}_{q}(\pi)$, the space of relative compact $n-$cycles of $\pi$, is proper over $S$.

proof.

Take such a connected component $\Gamma$. Then the continuous function given by $\gamma\mapsto\int_{\gamma}\tilde{\omega}$ is constant on $\Gamma$. This implies that for any compact subset $K$ in $S$ the volume of cycles in $\Gamma\cap\pi^{-1}(K)$ for any continuous metric is bounded. But then, as $\pi^{-1}(K)$ is compact in $X$ this implies that $\Gamma\cap\pi^{-1}(K)$ is a compact subset in $\mathcal{C}_{q}(\pi)\subset\mathcal{C}_{q}(X)$ (see [B-M 1] IV th.2.7.20), concluding the proof.$\hfill\blacksquare$

Remark.

In the case of smooth family of compact connected manifolds on a complex disc (or polydisc) $\pi:\mathcal{X}\to D$ each of them satisfying the $\partial\bar{\partial}-$lemma, the existence of a smooth $(q,q)-$relative differential form $\omega$ such that $\partial_{/\pi}\bar{\partial}_{/\pi}\omega=0$ on $\mathcal{X}$ which is strongly Lelong positive on the fibres of $\pi$ allows to prove, in the same way than in the case of relative divisors, the properness over $D$ of the connected component of the relative cycle’s space $\mathcal{C}_{q}(\pi)$ :

the first step is to produce for any given $s\in D$, using the $\partial\bar{\partial}-$lemma on $X_{s}:=\pi^{-1}(s)$ smooth forms $\alpha$ and $\beta$ on $X_{s}$ of type $(q,q-1)$ and $(q-1,q)$ respectively such that the form $\omega-\partial\alpha-\bar{\partial}\beta$ is $d-$closed on $X_{s}$, but with $(q,q)-$ part strongly Lelong positive on $X_{s}$. Then, using the local $\mathscr{C}^{\infty}$ triviality of $\pi$ around $s$ to produce on $\pi^{-1}(\Delta_{s})$, where $\Delta_{s}$ is a small open (poly-)disc around $s$ in $D$, a smooth $d-$closed $2q-$form $\Omega_{s}$ inducing $\omega-\partial\alpha-\bar{\partial}\beta$ on $X_{s}$. Then, the $(q,q)-$part of $\Omega_{s}$ is $\omega_{|X_{s}}$ so it is strongly Lelong positive on $X_{s}$ and then also on $\pi^{-1}(\Delta_{s})$, if $\Delta_{s}$ is small enough around $s$, because the strong Lelong positivity is an open property (and in our $\mathscr{C}^{\infty}-$trivialisation used to construct $\Omega_{s}$ the complex structure varies smoothly). The theorem above gives then the properness on $\Delta_{s}$ of the connected components of the space $\mathcal{C}_{q}(\pi_{|\Delta_{s}})$.

To conclude, consider now a connected component $\Gamma$ of $\mathcal{C}_{q}(\pi)$ and remark that two $q-$cycles in $\Gamma$ which are contained in $\pi^{-1}(\Delta_{s})$ have not only the same image in $H_{2q}(\mathcal{X},\mathbb{C})$ but also in $H_{2q}(\pi^{-1}(\Delta_{s}),\mathbb{C})$ because the natural map

[TABLE]

is injective (in fact bijective). This implies that the integral of $\Omega_{s}$ on any cycle in $\Gamma\cap\mathcal{C}_{q}(\pi^{-1}(\Delta_{s}))$ is constant, so $\Gamma\cap\mathcal{C}_{q}(\pi^{-1}(\Delta_{s}))$ has only finitely many connected components, each of them being proper on $\Delta_{s}$.$\hfill\blacksquare$

Terminology.

In the case of a compact complex connected manifold $X$ of dimension $n$ a strongly Gauduchon form in the sense of [B.15] is a smooth $d-$closed $(2n-2)-$form $\omega$ on $X$ such that its $(n-1,n-1)-$part is strongly positive in the sense of Lelong (in this case it is the “usual” sense). Assuming that $X$ satisfies the $\partial\bar{\partial}-$lemma, any Gauduchon form333so a $(n-1,n-1)$ smooth differential form positive definite and $\partial\bar{\partial}-$closed. gives rise to a strongly Gauduchon form via the method described above. And, thanks to [G.77] a Gauduchon form always exists on a compact complex connected manifold.

3.2 Estimation of volumes

Let now consider a holomorphic family $\pi:\mathcal{X}\to D$ of compact complex connected manifolds of dimension $n$ parametrized by the unit disc in $\operatorname{\mathbb{C}}$. So $\mathcal{X}$ is a smooth complex manifold of dimension $n+1$, and we fix a smooth relative Gauduchon form $\omega$ on $\mathcal{X}$. Then $\omega$ is a smooth $\pi-$relative differential form on $\mathcal{X}$ on type $(n-1,n-1)$ which is positive definite in the fibres and satisfies $\partial_{/\pi}\bar{\partial}_{/\pi}\omega\equiv 0$ on $\mathcal{X}$.

Using Ehresmann’s theorem we may assume that we have a $\mathscr{C}^{\infty}$ trivialization

[TABLE]

of the map $\pi$ where $X_{0}:=\pi^{-1}(0)$. Then the form $\Phi^{*}(\omega)$ is a smooth differential form on $X_{0}$ of degree $2n-2$ depending in a smooth way of the real variables $x$ and $y$ in $D$, and we then identify $\omega$ with the smooth (absolute) $(2n-2)-$differential form of $(\Phi^{-1})^{*}(\Phi^{*}(\omega))$ on $\mathcal{X}$.

We shall define

[TABLE]

on $\mathcal{X}$. Then $\frac{\partial\omega}{\partial x}$ and $\frac{\partial\omega}{\partial y}$ are smooth $(2n-2)-$differential forms on $\mathcal{X}$ and satisfy

[TABLE]

where $d_{/\pi}$ is the $\pi-$relative differential.

Now define the $\pi-$relative smooth differential forms $\nabla_{x}\,\omega$ and $\nabla_{y}\,\omega$ on $\mathcal{X}$ as the $(n-1,n-1)-$parts of the restrictions of $\frac{\partial\omega}{\partial x}$ and $\frac{\partial\omega}{\partial y}$ to each fibre of $\pi$. We may also consider $\nabla_{x}\,\omega$ and $\nabla_{y}\,\omega$ as smooth hermitian forms on the holomorphic vector bundle $\Lambda^{n-1}(T_{\mathcal{X}/D})$ on $\mathcal{X}$. Then the following lemma is given by an elementary compactness argument, because $\omega$ is a smooth positive definite hermitian form on this vector bundle.

Lemma 3.2.1

For any compact set $K$ in $D$ there exists a constant $C_{K}>0$ such on $\pi^{-1}(K)$ we have the following inequalities

[TABLE]

between hermitain forms on the vector bundle $\Lambda^{n-1}(T_{\mathcal{X}/D})$ on $\pi^{-1}(K)$.$\hfill\blacksquare$

The next lemma will allow us to prove our main estimate of the integral of $\omega$ over an analytic family of relative divisors in $\mathcal{X}$.

Lemma 3.2.2

Consider over an open disc $D^{\prime}\subset D$ a closed and reduced complex hypersurface $\mathcal{H}$ in $\pi^{-1}(D^{\prime})$ which is $(n-1)-$equidimensional on $D^{\prime}$. Then, for any smooth differential $1-$form $\varphi:=\varphi_{x}.dx+\varphi_{y}.dy$ on $D^{\prime}$ we have at the generic point of $\mathcal{H}$ the equality

[TABLE]

proof.

Near a generic point in $\mathcal{H}$ we may assume that we have a local holomorphic trivialisation

[TABLE]

where $D^{\prime\prime}$ is a small open disc in $D$ and $V=U\times\Delta$ is the product of an open polydisc in $\operatorname{\mathbb{C}}^{n-1}$ by an open disc $\Delta$ in $\operatorname{\mathbb{C}}$. On a fibre of the restriction of $\pi$ to $\mathcal{H}$ the form $d\omega\wedge\pi^{*}(\varphi)$ is of pure type $(n,n)$.

Remark that $d_{/\pi}\omega\wedge\pi^{*}(dx)$ and $d_{/\pi}\omega\wedge\pi^{*}(dy)$ vanish on $\mathcal{H}\simeq D^{\prime\prime}\times U$ because such forms present only the types $(n,n-1)$ and $(n-1,n)$ in the variables $t_{1},\dots,t_{n-1}$ in $U\subset\operatorname{\mathbb{C}}^{n-1}$. Then, using $(0)$ and the fact that the restrictions to $\mathcal{H}$ of the forms

[TABLE]

come only from the $(n-1,n-1)-$parts of the restriction to the fibre of $\pi$ of $\frac{\partial\omega}{\partial x}$ and $\frac{\partial\omega}{\partial y}$ respectively because the restriction to $\mathcal{H}$ of $\pi^{*}(dx\wedge dy)$ is of type $(1,1)$, the conclusion follows, by definition of $\nabla_{x}\,\omega$ and $\nabla_{y}\,\omega$.$\hfill\blacksquare$

The previous lemma gives the following simple estimate on $\mathcal{H}$.

Corollary 3.2.3

We keep the notations of the previous lemma. Assuming that the $1-$form $\varphi$ has a compact support in $D^{\prime}$ contained in the compact set $K$ in $D$ we have on $\mathcal{H}$ the inequality

[TABLE]

*between (smooth) real $2n-$differential forms on $\mathcal{H}$ where $C_{K}$ is the constant introduced in the lemma 3.2.1.$\hfill\blacksquare$

The following classical lemma will be useful to conlude our estimation of the volume.

Lemma 3.2.4

Let $U$ a bounded domain in $\operatorname{\mathbb{C}}$. Assume that there is a point $s_{0}\in U$ such that any point in $U$ can be joined to $s_{0}$ by a $\mathscr{C}^{1}$ path in $U$ with length uniformly bounded by a number $L$. For instance, we can restrict ourself to $U:=D^{*}\setminus A$ where $D^{*}$ is a punctured open disc in $\operatorname{\mathbb{C}}$ and $A$ a closed discrete subset in $D^{*}$. Let $\eta:U\to\mathbb{R}^{+*}$ be a $\mathscr{C}^{1}$ function on $U$ such that

[TABLE]

where $C>0$ is a given constant. Then this implies that the function $\eta$ is uniformly bounded on $U$ by $\eta(s_{0}).exp(2C.L)$.

Proof.

Firstly consider the case of an open interval $]a,b[$ in $\mathbb{R}$ and let $L:=b-a$. If $\eta:]a,b[\to\mathbb{R}^{+*}$ is a $\mathscr{C}^{1}$ function such that $|\eta^{\prime}(x)|\leq C.\eta(x)\quad\forall x\in]a,b[$; we have

[TABLE]

which gives, after integration on $[x_{0},x]\subset]a,b[$

[TABLE]

and then, for any choice of $x_{0}\in]a,b[$ the estimate $\eta(x)\leq\eta(x_{0}).\exp(C.L)$ is valid.

In the case of the bounded domain $U\subset\operatorname{\mathbb{C}}$, choose for any $s\in U$ a $\mathscr{C}^{1}$ path of length strictly less than $L$ and consider a parametrization of this path extended a little around $s_{0}$ and $s$ :

[TABLE]

Then apply the one dimensional case to the function $F(t):=\eta(\Phi(t))$ which satisfies $|F^{\prime}(t)|\leq 2C.F(t)$. This gives $\eta(s)\leq\eta(s_{0}).exp(2C.L)\quad\forall s\in U$.$\hfill\blacksquare$

3.3 proof of the theorem 1.0.1

First step.

For each $s\in D^{*}$ we know that $X_{s}$ admits a strongly Gauduchon form and then there exists a strongly Gauduchon form for the family over a small disc around $s$ (see section 3.1). So we have a smooth $d-$closed $2(n-1)-$form such that its $(n-1,n-1)$ part is positive definite on each fibre. This implies that each connected component of the relative cycle space $\mathcal{C}_{n-1}(\pi^{*})$ is proper on $D^{*}$, where $\pi^{*}$ is the restriction of $\pi$ to $\pi^{-1}(D^{*})$:

in a connected component $\Gamma$ of the relative cycle space $\mathcal{C}_{n-1}(\pi^{*})$, for any disc $\Delta\subset D^{*}$ all cycles in $\pi^{-1}(\Delta)\cap\Gamma$ are homologuous in $\pi^{-1}(\Delta)$ because we have a $\mathscr{C}^{\infty}-$ relative isomorphism $\mathcal{X}\simeq D\times X_{0}$. So if we dispose of a $d-$closed $2(n-1)-$ smooth form $\Omega$ on $\pi^{-1}(\Delta)$ with a positive definite $(n-1,n-1)$ part on fibres we see that the volume for $(n-1)-$cycles in $\Gamma\cap\pi^{-1}(\Delta)$ relative to the “volume” defined by $\Omega$ has to be constant, proving the properness on $\Delta$ of $\Gamma\cap\pi^{-1}(\Delta)$.

Second step.

Applying the theorem 1.0.2 of [B.15] we obtain that the minimum of the algebraic dimension of the fibres over $D^{*}$ is obtained on a dense (in fact the complement of a countable subset) subset of $D^{*}$. This implies that it is enough, under the hypothesis of the theorem 1.0.1, to prove that the conclusion holds on a small open disc $D_{0}$ with center [math] in $D$ because this “weak” version of this theorem may be apply then to a small disc centered at any point point in $D^{*}$ and this gives the theorem 1.0.1.

Thanks to the second step, we may now assume that there exists a positive constant $C$ such that the inequalities of the the lemma 3.2.1 holds on $\mathcal{X}$.

Now, using the proposition 3.3.1 (recalled from [B.15]) it is enough to show that each irreducible component $\Gamma$ of the space $\mathcal{C}_{n-1}(\pi)$ of relative $(n-1)-$cycles in $\mathcal{X}$ is proper over $D$.

Step 3.

Let $\Gamma$ be an irreducible component of $\mathcal{C}_{n-1}(\pi^{*})$ with reduced generic cycle and such that the projection $p:\Gamma\to D^{*}$ is surjective. As the map $p$ is proper, let $\tau:\pi^{-1}(D^{*})\to S$ be a Stein factorization (we may assumed that $S$ is normal) of $p$ and let $\tilde{\tau}:S\to D^{*}$ the corresponding proper finite surjective map.

Now let $\omega$ be a smooth $\pi-$relative Gauduchon form on $\mathcal{X}$ and define on $\Gamma$ the function

[TABLE]

As the function $\theta$ is continuous, pluri-harmonic along the fibres of $p$ as a distribution444meaning that it is $\partial\bar{\partial}-$closed as a distribution along the fibres of $p:\Gamma\to D^{*}$. and proper, it is constant in the fibres of $\tau$ which are the connected components of fibres of $p$ over $D^{*}$. So it defines a continuous function $\tilde{\theta}:S\to\mathbb{R}^{+*}$. This situation corresponds to the following diagram:

[TABLE]

Step 4.

Let $\tilde{A}_{0}\subset S$ be the set of points $\sigma\in S$ where the fibre $\tau^{-1}(\sigma)$ is not contained in the union of the singular set of $\Gamma$ with the critical set of $p:\Gamma\to D^{*}$. Then $\tilde{A}_{0}$ is a closed analytic subset in $S$ with no interior point. So $A_{0}:=\tilde{\tau}(A_{0})$ is a closed discrete subset in $D^{*}$. Let $A_{1}\subset D^{*}$ be the ramification set of $\tilde{\tau}$. It is also closed and discrete in $D^{*}$ and finally put $A:=A_{0}\cup A_{1}$.

We shall show now that the continuous function

[TABLE]

satisfies the following properties :

1. i)

The partial derivatives $\frac{\partial\eta}{\partial x}$ and $\frac{\bar{\partial}\eta}{\partial y}$ are continuous on $D^{*}\setminus A$ (with $s:=x+i.y$). 2. ii)

They satisfy the inequalities $|\frac{\partial\eta}{\partial x}|\leq C.\eta$ and $|\frac{\partial\eta}{\bar{\partial}y}|\leq C.\eta$ on $D^{*}\setminus A$, where $C$ is the constant introduced above (see lemma 3.2.1 and the end of step 2).

Note that it is enough to prove the properties i) and ii) near each point in $D^{*}\setminus A$.

Choose a point $s_{0}\in D^{*}\setminus A$ and fix an open disc $\Delta_{0}\subset D^{*}\setminus A$ with center $s_{0}$. Choose now in $p^{-1}(s_{0})$ generic point $\gamma_{0}$ and, as $\gamma_{0}$ is a smooth point of $\Gamma$, such that $p$ has rank $1$ at this point, we can find a smooth locally closed curve $\Sigma$ through $\gamma_{0}$ in $\Gamma$ such that the restriction of $p$ to $\Sigma$ induces an isomorphism of $\Sigma$ onto $\Delta_{1}$ where $\Delta_{1}\subset\Delta_{0}$ is an open disc with center $s_{0}$.

Define, for $s\in\Delta_{1}$ the cycle $Y_{s}$ as the cycle corresponding to the point $p_{|\Sigma}^{-1}(s)$ in $\Sigma\subset\Gamma$. This defines an analytic family of $\pi-$relative $(n-1)-$cycles. So the function $\eta_{1}$ which is given by $s\mapsto\int_{Y_{s}}\ \omega$ is continuous on $\Delta_{1}$ (see the proposition IV 2.3.1 in [B-M 1]) and coincides with $\frac{1}{\delta}.\eta$ where $\delta$ is the degree of the map $\tilde{\tau}$. We shall compute now the partial derivatives in the distribution sense of this function $\eta_{1}$.

Let $\mathcal{Y}\subset\Delta_{1}\times\mathcal{X}$ be the graph of the analytic family $(Y_{s})_{s\in\Delta_{1}}$ and let $\varphi$ be in $\mathscr{C}_{c}^{\infty}(\Delta_{1})$. Then we have

[TABLE]

by Stokes formula and using the lemma 3.2.2.

We conclude that the distribution $\frac{\partial\eta_{1}}{\partial x}$ is equal to the continuous function

[TABLE]

with the estimate

[TABLE]

deduced from the inequality of the lemma 3.2.1.

We have the analogous result for $\frac{\partial\eta_{1}}{\partial y}$.

So the lemma 3.2.4 applies and the function $\theta$ on $\Gamma$ is uniformly bounded. Then the closure of $\Gamma$ in $\mathcal{C}_{n-1}(\mathcal{\pi})$ is proper over $D$, using again the characterization of compact subsets in $\mathcal{C}_{n-1}(\mathcal{X})$ (see [B-M 1] IV th.2.7.20) .

Final step.

Consider now any irreducible component $\tilde{\Gamma}$ of $\mathcal{C}_{n-1}(\pi)$. Remark that to prove the properness of $\tilde{\Gamma}$ on $D$ it is enough to consider the case where the generic cycle of $\tilde{\Gamma}$ is reduced.

Then either $\tilde{\Gamma}$ is contained in some $\mathcal{C}_{n-1}(X_{s})$ for some $s\in D$, and then it is a connected closed subset in $\mathcal{C}_{n-1}(X_{s})$ and then it is compact, or $\tilde{\Gamma}\setminus p^{-1}(0)$ is a closed irreducible subset in $\mathcal{C}_{n-1}(\pi^{*})$ which is proper and surjective on $D^{*}$ because we may apply the previous result. In this cases we conclude also that $\tilde{\Gamma}$ (which is the closure of $\tilde{\Gamma}\setminus p^{-1}(0)$) is proper over $D$.

As any irreducible component of $\mathcal{C}_{n-1}(\pi)$ is proper over $D$, the conclusion follows.$\hfill\blacksquare$

For the convenience of the reader let me recall the proposition 4.0.8 of [B.15] which is an easy generalization of an old result of F. Campana (see [C.81]).

Proposition 3.3.1

Let $\pi:\mathcal{X}\to S$ be a proper surjective holomorphic $n-$equidimensional map between two irreducible complex spaces. Assume that any irreducible component of the complex space $\mathcal{C}_{n-1}(\pi)$ is proper over $S$. Then there exists a countable union $\Sigma$ of closed irreducible analytic subsets with no interior points in $S$ and a non negative integer $a$ such that:

For any $s\in S\setminus\Sigma$ the algebraic dimension of $X_{s}$ is equal to $a$; 2. 2.

For all $s\in S$ the algebraic dimension of $X_{s}$ is at least equal to $a$.

We conclude by noticing that there exists an analytic family of smooth complex compact surfaces of the class VII (not Kähler) parametrized by a disc such that the central fibre has algebraic dimension 0 and all other fibres have algebraic dimension equal to $1$, see [F-P.09].

This shows that in our theorem 1.0.1 some “Kähler type” assumption on the general fibre $X_{s}$ cannot be avoided in order that the general algebraic dimension gives a lower bound for the algebraic dimensions of all fibres.

Note that our assumption that the generic fibres satisfy the $\partial\bar{\partial}-$lemma (in fact for the type $(n,n-1)$) is a rather weak such assumption.

4 Bibliography

[B.15] Barlet, D. Two semi-continuity results for the algebraic dimension of compact complex manifolds, J. Math. Sci. Univ. Tokyo 22 (2015), no. 1, pp.39-54. 2. 2.

[B-M 1] Barlet, D. et Magnússon, J. Cycles analytiques complexes. I. Théorèmes de préparation des cycles, Cours Spécialisés 22., Société Mathématique de France, Paris, 2014. 525 pp. 3. 3.

[Bi.64] Bishop, E. Conditions for the analyticity of certain sets, Mich. Math. Journal (1964) p.289-304. 4. 4.

[C.82] Campana, F., *Sur les diviseurs non-polaires d’un espace analytique

compact*, J. Reine Angew. Math. 332 (1982), 126-133. 5. 5.

[Fu.82] Fujiki, A., Projectivity of the space of divisors of a normal compact complex space, Publ. Res. Inst. Math. Sci. 18 (1982), no. 3, 1163-1173. 6. 6.

[F-P.09] Fujiki, A. and Pontecorvo, M. Non-upper continuity of algebraic dimension for families of compact complex manifolds

arXiv: 0903.4232v2 [math. AG] . 7. 7.

[G.77] Gauduchon, P., Le théorème de l’excentricité nulle, C. R. Acad. Sci. Paris, S er. A 285 (1977), 387-390.