Bend and break lemma for higher dimension

 I. Introduction:

In this article, I would like to generate the work in bend and break lemma of professor Andreas Horing and professor Thomas Peternell in paper [HP16]. In the orginal paper, authors excellently resolved the problem of finding minimal model program for $\mathbb{Q}$- factorial Kahler threefold with terminal singularities and $K_X$ is not pseudo-effective. From [HP16] in general and especially in part five, I saw many proofs where the "threefold" assumption may not take a crucial role, and I try to generalize them in dimension four, at least for the case non-nef locus of $N(K_X)$ ("negative" part of Zariski decompostion of $K_X$) is not contained in any prime divisors which appear in Zariski decomposition of $K_X$. Since the closure of the union of curves' deformation could be surfaces, the restriction of $N(K_X)$ might not be pseudo-effective. So to reserve the pseudo-effectiveness, the assumption "non-nef locus of $N(K_X)$ is not contained in any prime divisors which appear in Zariski decomposition of $N(K_X)$" is required.   

Most of my results are just replicates of the proofs from lemmas, theorems or corollaries in [HP16]. There may have some changes, but not much. But there are somethings new in my article, that is the appearence of nef-curves belong to nef-cone $\overline{NM}$ in lemma 3 (which is the generalization of lemma 5.5) and lemma 5 (generalization of lemma 5.8) and non-rigid criterion in lemma 4. 

We know that in the proof of any version of "bend and break" lemma, the authors after finding out the curve has deformation of at least dimension two, will find a smooth surface contain that curve and work on that. For Mori, he used a smooth curve $D$ (which is obtained from the deformation ) and considered the surface $\mathbb{P}^{1} \times D$ and work on that. This is also true for professor Horing and professor Peternell, when they gave cycle structures for curves $C$ belong to a smooth projective uniruled surface $\widehat{S}$ such that $K_{\widehat{S}}.C < 0$. In the light of that method, I also search for a surface $S$ (in a smooth projective threefold $T$) containing curve $C$ with $K_T . C < 0$, and also satisfies the condition $S.C < 0$, in order to use the familiar technique on the pseudo-effective $K_S$, when the case $S$ is not uniruled occured, and the bend and break lemma when $S$ is uniruled. But there are also curves $C$ which have non-zero intersection for every surface over $T$, and hence these curves must be nef. Since we get a nef curve, we are not able to find a surface $S$ which contain it and $K_S.C < 0$, and our plan to restrict this curve to a surface and use some well-know bend and break results, are totally bankrupt. So using properties of nef cone is necessary here.

II. Main part:  

Lemma 1:  
 Let $X$ be a normal $\mathbb{Q}$-factorial compact Kähler fourfold with at most terminal singularities such that $K_X$ is pseudoeffective. 
a) Let $S$ be a surface which is different from the non-nef locus of $N(K_X)$ such that $K_X|_S$ is not pseudoeffective. Then $S$ is contain in a $T_j$ in the divisorial Zariski decomposition of $K_X$.
b) Let $T$  be a three fold which is not intersect non-nef locus of $N(K_X)$ such that $K_X|_T$ is not pseudoeffective. Then $T = T_j$ for some $j$ in the divisorial Zariski decomposition of $K_X$ and $T$ is a Moishezon, moreover any desingularisation $\widehat{T}$ is a uniruled projective threefold.

The proof of this lemma is just a repeat  of lemma 4.1 in [HP16].
For a), suppose $S$ is not contain in any $T_i$, then  we have:
$K_X|_S = \sum_{i=1}^{n} \lambda_{i} (T_j \cap S) + N(K_X)|_S$.
Since we assume that $S$ is has zero intersection with the non-nef locus, then $N(K_X)|_S$ is pseudoeffective, then right handside must pseudoeffective, but $K_X|_S$ is not, hence $S$ must be contained in a threefold.
For b), we just repeat the proof of lemma 4.1 .

Lemma 2:

Let $X$ be a normal $\mathbb{Q}$-factorial compact Kähler threefold with at most terminal singularities such that $K_X$ is pseudoeffective. Let $C \subset X$ be a curve such that $(K_X.C) < 0$ and $dim_C chow(X) > 0$.
Then there exists a threefold from the divisorial Zariski decomposition such that $C$ and its deformations are contained in that threefold.
Moreover, if the deformation of  we have: $(K_T.C) < (K_X.C)$.

proof:
  
Let $(C_h)_{h \in H}$ be a deformation family of $C$. Since $C$ is integral, a general deformation $C_h$ is integral. Since $K_X.C_h < 0$ for all $h \in H$ we see that the locus $\overline{ \bigcup_{h \in H} C_h}$ has a property that the restriction of  $K_X$ to that locus is pseudoeffective. Since $H$ and $C_h$ are irreducible, then we can see that $\overline{\bigcup_{h \in H} C_h}$ is also irreducible. We assume that $\overline{\bigcup_{h \in H} C_h }=  S$ is a surface, for the threefold case, it just a consequence of lemma 5.4 in [HP16]. By lemma 1, $S$ is contained in some threefolds $T_j$ and we assume that it contained only in a threefold $T_j$. The deformation family $(C_h)_{h \in H}$ has no fixed component, in particular for every $k \in$ {$1, . . . , r$} such that $k \neq j$, there exists a $h \in H$ general such that $C_h$ is not in $T_k$. Moreover, the restriction $N(K_X)|_S$ is pseudoeffective since $S$ is different from the non-nef locus, then we have: $N(K_X).C=N(K_X)|_S.C=N(K_X)|_S.C_h \geq 0$.
Since: $ 0 > K_X.C= \sum_{i=1}^{r} \lambda_i T_j.C + N(K_X).C$,
Then $T_i.C < 0$ and hence $C \subset T_i$ and all of its deformations lie on $T$.
    
By lemma 2 we know that $C$ will contain in a $T_i$ which is a prime divisor of Zariski's decomposition. So now there is two case : $T_i$ is uniruled or not uniruled. If it is not a uniruled threefold, since $K_T = (K_X + T)|_T$ then $K_T.C = (K_X|_T.C+T|_T.C) < 0 $, hence there is an $S'$ in Zariski decomposition  contains $C$ by lemma 5.4 in [HP16] and lemma 5.5 will work. But for the case $T$ is not uniruled, we would need a small change.

Lemma 3: 

Let $T$ be a smooth projective 3-fold that is uniruled, and let $C' \subset T$ be an irreducible curve.:
a) Suppose that $K_{T}C < 0$ and there exist a surface such that $S.C < 0$. Then there exists an effective 1-cycle $\sum_{i} \alpha_{i} C_i$ with coefficients in $\mathbb{Q^+}$ such that $[\sum_{i} \alpha_{i} C_i] = [C'] $ and $C_1$ is a rational curve such that $K_{S'}.C_1 < 0$.
b) Suppose that $K_{S'}.C \leq -5$. Then there exists an effective 1-cycle $\sum_{i} \alpha_{i}C_{i}$ with $i \geq 2$ such that $[\sum_{i} \alpha_{i} C_i]=[C']$ such that $K_{S'}C_1 < 0$ and $K_{S'}.C_2 < 0$. 

proof:

a)  By adjunction formula: $K_{S}=(K_{T}+T)|_{S}$, we see that $S$ contain $C$, and $(S.C) < 0$, then $(T.C)|_{S} = (S.C)< 0$ by a theorem in [Mum-Oda], hence $K_{S}.C < 0$. Now we will a proof of this theorem by induction on degree $d = H.C$ with $H$ is a fix ample invertible sheaf in $S$. With $d=0$ it is trivial. suppose it is true for $d > 0$. Consider $\widehat{S}$ is a composition of normalisation and desingularisation. Hence $\widehat{S}$ is smooth uniruled surface with morphism $\pi : \widehat{S} \rightarrow S$. By the original theorem, $\widehat{S}.\widehat{C} < 0$ with $\widehat{C}$ a strict transform of $C'$ implies $[\widehat{C}]= [\sum_{i} \alpha_{i} \widehat{C_i}]$ with $\widehat{C_1}$ rational and $(\widehat{S}.\widehat{C_1}) < 0$. Hence $[C] = \sum_{i} \alpha_{i} \pi_{*}[\widehat{C_i}]$ = $\sum_{i} \alpha_{i} [C_i]$ with $C'_1$ is rational or $i \geq 2$. For the first case, for the second case, $H.C_i < H.C$ for some i and then applies the induction.

b) Since $K_T.C \leq -5$, we get $dim_C chow(T) \geq 5$ and so we have a five-dimensional family of deformations $(C_h)_{h \in H}$. Note that the family of deformations splits: arguing by contradiction, let $\sigma T \rightarrow T_0$ be a MMP to some Mori fibre space $T_0$ which has general fibres are dimension two Fano variety. Since $H$ has dimension at least five, the subfamily  $H_p$ parametrising the deformations through a general point $p \in T$ has dimension at least four. Thus $\sigma((C_h))_{h \in H}$ is positive-dimensional, and we can take a subvariety $H'$ of $H$ such that it does not split, has a fixed point $p \in S_0$ with $S_0$ is a Fano fibre, hence a ruled surface. This contradicts [Pet01, Lemma 3.3]. After this, we just use the induction. 


Remark 1a. There is another proof for part b) if we assume there exists an uniruled $S$ s.t $S.C <0$. For the same $\widehat{S}$ and $\pi$, we have $[\widehat{C'}]= \sum_{i} \alpha_{i} \widehat{C_i}]$ with $\widehat{C_1}$, $\widehat{C_2}$ rational and $(\widehat{S''}.\widehat{C_1}) < 0$, $(\widehat{S}.\widehat{C_2}) < 0$. Hence  $[C'] = \sum_{i} \alpha_{i} \pi_{*}[\widehat{C_i}]=[\sum_{i} \alpha_{i} C'_i]$ with $C'_1, C'_2$ are rational. By an arguement of the proof of the orginal lemma, $(K_{\widehat{S}}.\widehat{C'_1}) \leq (K_{\widehat{S}}.\widehat{C'_2})$ , hence $(K_{S}.C'_1) \leq (K_{S}.C'_2)$, if  $(K_{S}.C'_2) < 0$, we done, if not, we have $(H.C'_1) < (H.C')$, then applies induction. 

Remark 1b. After this lemma, we will see that if there is a uniruled surface $S$ such that $S.C < 0$ then C can deform just like lemma 5.5. But if the surface with $S.C < 0$ that is not uniruled, then $K_S$ must be pseudoeffective and by the formula of lemma 4.1 in [Hor15], we can lead to $C$ be a rational curve. If $C$ is rational, then both statement a) and b) of Lemma 3 are satisfied, especially the statement b) is follow from the original bend and break lemma of Mori. 

Remark 2. Now we see that there only curves such that $S.C \geq 0$ for all surface $S \subset T$ that do not have a deformating formula. For special, these curves are nef curves which belong to $\overline{NM_1}$ (for more detailed, c.f. [Arau10]). In this case, $C = z + \sum_{i=1}^{r} \mathbb{R}_{+} \Gamma_{i}$ with $z \in \overline{NE}_{K_T \geq 0}$ and $\Gamma_{i}$ are negative rational curves which contained in the covering family of $T$. If $C$ is a strict transform of a $C'$ in $T'$, with $T'$ a uniruled threefold in Zariski decomposition of $K_X$, then $C' = \pi_{*}z + \sum \pi_{*} \Gamma_{i}'$. Since  $z \in \overline{NE_{K_T \geq 0}}$, then there must be a sequence of positive cycles $[Z_j] \in \overline{NE_{K_T \geq 0}}$ converge to $z$, and thus $\pi_{*}[Z_j] \rightarrow \pi_{*}z$. Since $K_{T}.C \geq K_{T'}.C$ ($T$ is desingularisation of $T'$), then $\pi_{*}[Z_j]$ are all in $\overline{NE_{K_{T'} \geq 0}}$ and $\pi_{*}z$ must be in $\overline{NE_{K_{T'} \geq 0}}$, and hence $[C']= \pi_{*}z + \sum \pi_{*} \Gamma_{i}' \in \overline{NE_{K_T' \geq 0}}+\sum_{i}^{r} \mathbb{R}_{+} \Gamma_{i}'$.     

Now, we come to another important lemma and theorem for bend and break. Here we need to change a part of theorem 4.5 [HP16] a little bit. Since for fourfolds, the inequality of $dim_C Chow(X)$ is bad because of the existing of $\chi(\tilde{C})$ with $\tilde{C}$ is the normalisation of $C$, so we need an integer number $a$ such that if $K_X.C < -1 - a$, then $C$ is not very rigid. To see this, we need to look at the proof of lemma 4.5. Let $\widehat{C}$, $\tilde{C}$, $e$, $d$, $D$,... just like theorem 4.5 since all of it can generate in higher dimension, now we will revise the claim.

Remark 2. Since $K_{\widehat{X}}.C$ is an integer multiple of $e$ , our assumption implies $K_{\widehat{X}}.C \leq -1 + a - \frac{1}{e}$. Since $D.C = 1$ we obtain from this :
                            $K_{\widehat{C}}.C + (1 - \frac{1}{e})D.C \leq a - \frac{2}{e}$ 
The order of ramification for every point $p_0$ over $D$ is equal to $e$, so we have $deg(\overline{C} / C} \geq e$. Thus by the ramification formula and the preceding
inequality gives: 
                            $K_{\overline{C}}.\overline{C} = deg (\overline{C} / C)(K_{\widehat{C}}.C+(1-\frac{1}{e})D.C \leq -2 + e.a$
 We know that the deformation inequality is: 
                            $dim_{\overline{C}} chow (\overline{X}) \geq -K_{\overline{X}}.\overline{C} + \chi (\overline{C}) \geq 2 - e.a +\chi (\overline{C}) $
From this, we can take $a = min(\chi (\overline{C}),0)$, then the last right handside always larger than $1$. Hence, $\overline{C}$ deforms and so is $C$.  

Lemma 4: Let $X$ be a normal $\mathbb{Q}$-factorial compact Kähler fourfold with at most terminal singularities such that $K_X$ is pseudoeffective. Let $T_1, T_2, ..., T_r$ be the threefolds appearing in the divisorial Zariski decomposition. Set $b=max \lbrace 1-a, -K_X.Z | Z \in T_{j,sing} or Z \in T_j \cap T_i \rbrace$ with $a = min (\chi (\overline{Z}),0)$ with $\overline{Z}$ was defined in theorem 4.5 [HP16] . If $C \in X$ is a curve such that $-K_X.C > b$ then we have $dim_C Chow(X) > 0$.

proof:

Since $b \geq 1 - a$, the curve $C$ is not very rigid . Let $m \in \mathbb{N}$ be minimal such that $dim_{mC} chow(X) > 0$, and let $(C_h)_{h \in H}$ be a family of deformations. We can assume that union of these deformations are irreducible . The family $(C_h)_{h \in H}$ has no fixed component, i.e. there does not exist a curve  $B$ that is contained in the support of every $C_h$. Indeed, since $mC$ is a member of the family, we would have $B = C$, but then $dim_{(m-1)C}chow(X) > 0$, contradicting the minimality of  $m$. Thus if $C_h$ is general and $C_h = \sum_l \alpha_l C_{h,l}$ is its decomposition, we have $dim_{C_{h,l}}chow C > 0 $ for all $l$. Up to renumbering we can suppose that $K_X.C_{h,l} < 0$. By Lemma  applied to $C_{1,l}$ there exists a threefold $T_j$ from the decomposition such that 
                                         $\overline{\bigcup_{h \in H} C_{1,h}} = S \subset T_j$
Since the locus covered by the family $(C_h)_{h \in H} is irreducible, we see that $C \in T_j$.
By the definition of  $b$, we have $C \notin S_i$ for every $i \neq j$, thus $T_i.C \geq 0$ for $i \neq j$.The restriction $N(K_X)|_S$ is pseudoeffective, and for $h \in H$ general all the curves $C_{1,h}$ are movable in $S$, so we get 
                                        $N(K_X).C=N(K_X)|_S.C=N(K_X)|_S.C_h = \sum_l \alpha_l (N(K_X)|_S.C_{h,l} \geq 0$
Since $K_X.C < 0$, the equality from lemma 1 implies $T_j.C < 0$. Thus we have $K_{T_j}.C < K_X .C < -b $.By definition of $b$, the curve $C$ is not contained in the singular locus of $T_j$. Let : $\pi_j: \widehat{T_j} \rightarrow T_j$ be the composition of normalisation and minimal resolution. The strict transform $\widehat{C}$ of $C$ is well-defined and we have $K_{\widehat{T_j}}.\widehat{C} < K_{T_j}.C < b$ .Since $b \geq 0$ we obtain by deformation theory that $dim_{\widehat{C}} chow (\widehat{T_j}) > 0$, so $\widehat{C}$ deforms. Thus its push-forward $\pi_{*} \widehat{C}=C$ deforms.

Corollary 1: Let $X$ be a normal $\mathbb{Q}$-factorial compact Kähler fourfold with at most terminal singularities such that $K_X$ is pseudoeffective. Let $b$ be the constant from lemma 4 and set $d = max (b,4)$. If $C \subset X$ is a curve such that $-K_X.C > d$, then we have $[C] = [C_1]+[C_2]$ with $C_1$ and $C_2$ effective 1-cycles (with integer coefficients) on $X$.

The proof of this is just follow by corollary 5.7 in [HP16].

We also have another lemma, which is from lemma 5.8 in [HP16].

Lemma 5: Let X be a normal Q-factorial compact Kähler threefold with at most terminal singularities such that $K_X$ is pseudoeffective. Let $\mathbb{R}_{+}[\Gamma_i]$ be a $K_X$-negative extremal ray in $\overline{NE}$ where $\Gamma_i$ is a curve that is not very
rigid. Then the following holds:
a) There exists a curve $C \in X$ such that $[C] \in \mathbb{R}_{+}[\Gamma_i]$ and $dim_C Chow(X) > 0$.
b) There exists a rational curve $C \subset X$ such that $[C] \in \mathbb{R}_{+}[\Gamma_i]$.

proof:

The proof this lemma is just a repeat the proof of lemma 5.8, but there is a special case for curves which have strict transform are nef curves. By remark 2, we have $C = Z + \sum_{i = 1}^{r} \mathbb{R}_{+}[\Gamma'_i]$ with $\Gamma'_i$ are multiple of rational curves. Because $\mathbb{R}_{+}[\Gamma_i]$ is extremal ray, hence $\Gamma'_i \in \mathbb{R}_{+}[\Gamma_i]$.

Conferences:

[HP16] Andreas Höring and Thomas Peternell. Minimal models for Kähler threefolds. Invent.
Math., 203(1):217–264, 2016.
[Ara10] Carolina Araujo. The cone of pseudo-effective divisors of log varieties after Batyrev. Math. Z., 264(1):179 - 193, 2010. 
[Pet01] Thomas Peternell. Towards a Mori theory on compact Kahler threefolds. III. Bull. Soc. Math, France, 129(3): 339-356, 2001.