Osamu Fujino
We discuss the minimal model program for projective morphisms of complex analytic spaces. Roughly speaking, we show that the results obtained by Birkar--Cascini--Hacon--M\textsuperscript{c}Kernan hold true for projective morphisms between complex analytic spaces. We also treat some related topics.
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Infinifesimal Criterion for Flatness of Projective Morphism of Schemes
推广了著名的（希尔伯特多项式）准则平坦的射影态射的Noether计划的情况下，非约基的态射。
P1-bundles over projective manifolds of Picard number one each of which admit another smooth morphism of relative dimension one
本文给出了Picard数为1的射影流形上的P1-丛的一个完全分类，其中P1-丛允许另一个相对维数为1的光滑态射。
Finitude cohomologique des morphismes propres en géométrie algébrique : une preuve transcendante sans techniques projectives
本文利用代数簇的一个真态射，不使用Chow引理，也不使用任何射影方法，给出了相干层的高阶直象的相干性的一个超越证明.主要工具是与平凡值域上的（Berkovich）解析空间的相干上同调的比较，以及Kiehls关于刚性-解析空间之间的真态射定理。
In this note we describe the image of $\PP^2$ in $ Gr(2, \CC^{4})$ under a morphism given by a rank two vector bundle on $\PP^2$ with Chern classes $(2,2).$
Morphisms from Quintic Threefolds to Cubic Threefolds are Constant
证明了从四维射影空间中的5次超曲面到四维射影空间中的非奇异3次超曲面的每一个态射必然是常数。在这个过程中，我们还分类的态射从射影平面的非奇异三次三倍的3次多项式。
A simple remark on a flat projective morphism with a Calabi-Yau fiber
若K3曲面是复数域上连通Noether概型上平坦射影态射的纤维，则任何光滑连通纤维也是K3曲面.观察到这一点，Nam-Hoon Lee教授问，对于更高维度的Calabi-Yau纤维是否也是如此。我们将对他的问题给出明确的否定回答，并证明他最初的观察。
Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius morphism can be lifted modulo $p^2$, then $X$ admits, up to a finite étale Galois cover, a smooth morphism onto an ordinary abelian variety whose fibers are products of projective spaces.
Towards Characterizing Morphisms Between High Dimensional Hypersurfaces
本文证明了对于射影空间中任意一个一般型且维数至少为3的非奇异超曲面之间的态射f，存在一个处处定义的限制于f的射影空间的自同态F.作为推论，我们看到，如果X，Y是维数至少为3的一般型非奇异超曲面，使得存在从X到Y的非常数态射f，则degY整除degX的商为q，而且射影空间的自同态F由q次多项式给出.
On linear morphisms of product spaces
本文研究两个射影空间乘积到射影空间的线性态射，给出存在自同构与线性态射使 Segre 嵌入复合后等于该态射的充分条件。
Real-fibered morphisms of del Pezzo surfaces and conic bundles
本文基于 Ahlfors 关于实曲线的结果，利用拓扑环绕数刻画高维实代数簇的实纤维线性投影，并完整分类实 del Pezzo 曲面到射影平面的相关态射，同时探讨实圆锥丛。
The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms
经典的算术Grothendieck-Riemann-Roch定理只适用于在复数上光滑的射影态射。本文将算术Grothendieck-Riemann-Roch定理推广到正则算术簇之间的一般射影态射的情形。为此，我们依靠作者发展的广义解析挠理论。
We give a sufficient condition under which the moduli space of morphisms between logarithmic schemes is quasifinite under the moduli space of morphisms between the underlying schemes. This implies that the moduli space of stable maps from logarithmic curves to a target logarithmic scheme is finite over the moduli space of stable maps, and therefore that it has a projective coarse moduli space when the target is projective.
Frobenius morphism and vector bundles on cycles of projective lines
本文讨论了弗罗贝尼乌斯态射对射影线圈上不可分解的向量丛的作用。这给出了一个问题的答案保罗Monsky，这出现在他的研究希尔伯特-昆兹理论的平面三次曲线。
Rational curves on blowing-ups of projective spaces
我们证明了从一条射影线到一个射影空间的乘积的连续爆破的（自由）射影的模空间的不可约性（和有理连通性），如果提供了关于射影的一个适当的数值条件（并且爆破的所有中心都是有理连通的）.
1-point Gromov-Witten invariants of the moduli spaces of sheaves over the projective plane
Gieseker-Uhlenbeck态射将光滑射影曲面上稳定秩为2的层的Gieseker-Uhlenbeck模空间映射到Uhlenbeck紧化，是Hilbert点格式的Hilbert-Chow态射的推广.当曲面是复射影平面时，我们确定了所有亏格为0的1点Gromov-Witten不变量关于Gieseker-Uhlenbeck态射的极值.主要思想是通过研究Gieseker模空间上的障碍层和利用亚纯2-形式来理解稳定映射的模空间的虚基本类。
Spaces of morphisms from a projective space to a toric variety
本文研究了从复射影空间到紧致光滑复曲面簇X的态射空间。证明了第一作者关于从CP^m到CP^n的有理映射空间的稳定性定理推广到了从CP^m到X的连续态射空间，并给出了同样的证明.在曲线的情况下，我们的结果提高了已知的界限的稳定尺寸。
We make a very detailed analysis of the numerical properties of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka's inequality on the number of canonical singularities on a projective normal surface with non-negative Kodaira dimension to the non minimal case, obtaining a slightly better result than known extensions by Megyesi and Langer.
Vanishing theorems for projective morphisms between complex analytic spaces
本文讨论复解析空间射影态射的消失定理，从最小模型理论观点建立了Kollár纲领的终极推广，对最小模型理论起关键作用。
Hypersurfaces in projective schemes and a moving lemma
设X/S是仿射基上的拟射影态射。本文发展了一种证明X/S的闭子概形H/S的存在性的技巧，它具有各种有利的性质.我们给出了这一技巧的几个应用，包括某些射影态射中有限拟截面的存在性，以及X/S中包含给定闭子概形C且与闭集F适当相交的超曲面的存在性。现在假设基S是环R的谱，使得对于任何有限态射Z -> S，Pic（Z）是挠群。如果R是数域上的整数环，或有限域上光滑仿射曲线的函数环，则满足这个条件。在这方面，我们证明了一个移动引理有关的水平1-圈的正规计划X准投射和平坦的S。当X/S的所有纤维都是固定维数d时，我们还证明了对任意在S上投射的概型X，存在到射影空间P_S^d的有限满射S-态射.
Three-dimensional toric morphisms with anti-nef canonical divisors
本文将Gorenstein环面3-折叠的环面双有理态射分类到具有较充分反典范除子的三维仿射空间上。
Let $X$ be a fixed projective scheme which is flat over a base scheme $S$. The association taking a quasi-projective $S$-scheme $Y$ to the scheme parametrizing $S$-morphisms from $X$ to $Y$ is functorial. We prove that this functor preserves limits, and both open and closed immersions. As an application, we determine a partition of schemes parametrizing rational curves on the blow-ups of projective spaces at finitely many points. We compute the dimensions of its components containing rational curves outside the exceptional divisor and the ones strictly contained in it. Furthermore, we provide …
Morphisms of Projective Varieties from the viewpoint of Minimal Model Theory
在这堂课的讲稿中，我们以一种足够独立的方式，介绍我们在极小模型理论领域的贡献和兴趣。我们从双正则和双有理的角度研究Fano-Mori空间。对于前者，我们回顾并发展了Kawamata的基点自由技术和Mori的一些变形论点。对于后者，我们依靠Sarkisov和#-极小模型程序。在写这些笔记时，我们想给出我们对这一研究领域的观点。我们并不打算对整个问题进行论述。这些笔记收集了我们分别在Wykno（Pl）（1999），Recife（Br）（2000）和Ferrara（It）（2000）举办的三个小型课程中提出的一些主题。
Isotriviality and the space of morphisms on projective varieties
设$K=k(C)$是无限域$k$上光滑射影曲线$C$的函数域，$X$是$k$上的射影簇.我们证明了两个结果。首先，我们证明了在一定条件下，一个次数至少为2的$K$ -态射$ϕ: X_K \to X_K$是等平凡的当且仅当$ϕ$在$K$的所有位置$v$都有潜在的好的约化。第二，让$(X,ϕ), (Y,ψ)$是动力系统，其中$X,Y$定义在$k$上，$g:X_{K} \to Y_{K}$是$K$ -态射，使得$g \circ ϕ= ψ\circ g$。我们证明了在一定条件下，如果$ϕ$是定义在$k$，那么$ψ$是定义在$k$。
Non-existence of separating morphisms of low degree
本文针对嵌入射影平面或Hirzebruch曲面的曲线，基于Manzaroli的策略改进分离亏度下界，研究低次数分离态射的不存在性。
The Hitchin morphism is a map from the moduli space of Higgs bundles $\mathscr{M}_X$ to the Hitchin base $\mathscr{B}_X$, where $X$ is a smooth projective variety. When $X$ has dimension at least two, this morphism is not surjective in general. Recently, Chen-Ngô introduced a closed subscheme $\mathscr{A}_X$ of $\mathscr{B}_X$, which is called the space of spectral data. They proved that the Hitchin morphism factors through $\mathscr{A}_X$ and conjectured that $\mathscr{A}_X$ is the image of the Hitchin morphism. We prove that when $X$ is a smooth projective surface, this conjecture is true fo…
Direct images of bundles under Frobenius morphisms
设$X$是代数闭域$k$上维数为$n$的光滑射影簇，其中${\rm char}(k)=p>0$和$F:X\to X_1$是相对Frobenius态射.对$X$上的任意向量丛$W$，证明了$F_*W$的不稳定性被$W\otimes{\rm T}^{\ell}(Ω^1_X)$（$0\le \ell\le n(p-1)$）的不稳定性所限制（推论\ref{cor3.8}）.当$X$是亏格为$g\ge 2$的光滑投影曲线时，它意味着只要$W$稳定，则$F_*W$稳定。
In this paper we define and study real fibered morphisms. Such morphisms arise in the study of real hyperbolic hypersurfaces in projective space and other hyperbolic varieties. We show that real fibered morphisms are intimately connected to Ulrich sheaves admitting positive definite symmetric bilinear forms.
A view on extending morphisms from ample divisors
“射影流形比它的任何光滑超平面截面都更特殊”的哲学是射影几何的经典原理之一。Lefschetz类型的结果和相关的消灭定理是其中典型使用的技术。我们将调查的大部分问题，结果和progratures在这方面，使用现代设置的充分因子，和（某些方面）森理论。
Over a field of positive characteristic, a semisimple algebraic group $G$ may have some nonreduced parabolic subgroup $P$. In this paper, we study the Schubert and Bott-Samelson-Demazure-Hansen (BSDH) varieties of $G/P$, with $P$ nonreduced, when the base field is perfect. It is shown that in general the Schubert and BSDH varieties of such a $G/P$ are not normal, and the projection of the BSDH variety onto the Schubert variety has nonreduced fibers at closed points. When the base field is finite, the generalized convolution morphisms between BSDH varieties are also studied. It is shown that th…
An affine local criterion for toric projective space bundles
该研究探讨等维环态射成为射影空间丛的条件，证明仿射刚性定理并将其应用于长极射线相关的环收缩，得到无需ℚ-因子性的射影空间丛定理。
Finite Morphisms to Projective Space and Capacity Theory
研究了交换环R上等价于以下条件的条件：当X是S = Spec（R）上纤维维数为\leq d的射影概形，其中d为整数d \geq 0时，存在从X到S上P^d_S的有限态射，使得坐标超平面的拉回给出X的指定子概型，只要这些子概型满足一定的自然条件.我们利用这些结果定义了整体域上射影概形X的广义子集的一种新的容量。这个容量可以用来推广Fekete-SzegIgn定理的逆部分。
Hitchin in [Duke Math. J. 54 (1), 91-114 (1987)] introduced a proper morphism from the moduli space of stable $G$-Higgs bundles ($G=\mathrm{GL}(n,\mathbb{C}),\mathrm{Sp}(2m,\mathbb{C})$ and $\mathrm{SO}(n,\mathbb{C})$) over a curve to a vector space of invariant polynomials and he described the generic fibers of that morphism. In this paper, we first describe the generic Hitchin fibers for the moduli space of stable parabolic projective symplectic/orthogonal Higgs bundles without fixing the determinant. We also describe the generic fibers when the determinant is trivial.
Cone and contraction theorem for projective morphisms between complex analytic spaces
本文在复解析空间中建立正规对的锥与收缩定理，作为复解析对数典范对极小模型纲领的起点，并重点处理奇点比川又对数终端更差的情形。
Existence of the Shafarevich morphism for semisimple local systems on quasi-projective varieties
本文证明了在极大性假设下，拟射影簇上半单局部系统关联覆盖空间存在连通纤维的正规满全纯映射，确立了 Shafarevich 态射的存在性。
Analytic and Algebraic Oka-1 Approximation for Smooth Projective Morphisms with Rationally Connected Fibers
本文证明光滑射影态射（纤维有理连通）在开黎曼曲面上的连续提升可被全纯或代数逼近并插值，从而建立代数 Oka-1 性质与有理连通性的等价。
Generalized Albanese morphisms
我们定义了一个射影簇X的Albanese簇的推广。广义Albanese态射X --> Alb_r（X）收缩X中的那些曲线C，对于这些曲线C，诱导态射Hom（π_1（X），U（r））--> Hom（π_1（C），U（r））具有有限像。因此，它们可以被解释为U（r）-版本的Shafarevich态射。
Finite Morphisms between Projective Varieties and Skeleta
本文利用不可约投射簇之间诱导的态射来研究不可约投射簇之间的有限态射。主要结果的背景如下。设$V'$和$V$是代数闭的非阿基米德值域$k$上的不可约投射簇，$ϕ$是有限态射$ϕ: V' \to V$ .设$x \in V^{an}(L)$，其中$L/k$是代数闭的完全非阿基米德值域扩张.我们将$x$与位于$x$上的纤维上的空间$(V \times_k L)^{an}$的一个$L$ -点正则地关联起来，并将这个点记为$x_L$。将$V$嵌入到某个$n$维射影空间中，以自然的方式定义了$x_L$的$(V \times_k L)^{an}$中的一族开邻域$\mathcal{O}_{x_L}$.该族的每个元素由量化其大小的$(n + 1)^2$元组参数化。我们特别感兴趣的是集合$\mathcal{O}_{x_L}$中的元素$O$，其态射$(ϕ\times id_L)^{an}$的原像通过$(ϕ\times id_L)^{an}$分解为$O$的同胚副本的不交并。令$\mathcal{G}_{x_L} \subset \mathcal{O}_{x_L}$表示该形式的元素的子集合。定理1.3表明，存在空间$V$到有限单纯复形上的变形收缩，使得沿着收缩的纤维，属于$\mathcal{G}_{x_L}$的最大元素的大小是恒定的。
Toroidalization of birational morphisms of 3-folds
我们证明了在特征为零的域上，通过对域和目标上的点和非奇异曲线进行一系列的爆破，可以使射影3-折叠的双有理态射成为环面。
The Space of Morphisms on Projective Space
P^n上态射的模理论推广了P ^1上有理映射的研究。本文证明了P^n上d > 1次态射空间的三个结果，并利用PGL（n+1）的共轭作用证明了它的商.首先，我们证明了这个商是几何的，并计算了态射空间的稳定和半稳定完备化。这加强了Silverman，以及Petsche，Szpiro和Tepper以前的结果。其次，我们仅用n和d来限制每个态射的PGL（n+1）中稳定子群的大小。第三，我们专门研究了n = 1的情形，证明了商空间对所有d > 1都是有理的;这部分地推广了Silverman关于d = 2情形的一个结果。
Vector Bundles over Normal Varieties Trivialized by Finite Morphisms
设$Y$是代数闭域$k$上的正规投射簇，$V$是$Y$上的向量丛.证明了如果存在$k$ -方案$X$和使$V$平凡化的有限满射$g:X\to Y$，则$V$本质有限.
Let $S$ be a finite dimensional noetherian scheme. For any proper morphism between smooth $S$-schemes, we prove a Riemann-Roch formula relating higher algebraic $K$-theory and motivic cohomology, thus with no projective hypothesis neither on the schemes nor on the morphism. We also prove, without projective assumptions, an arithmetic Riemann-Roch theorem involving Arakelov's higher $K$-theory and motivic cohomology as well as an analogue result for the relative cohomology of a morphism. These results are obtained as corollaries of a motivic statement that is valid for morphisms between oriente…
Let $X$ be a projective variety (possibly singular) over an algebraically closed field of any characteristic and $\mathcal{F}$ be a coherent sheaf. In this article, we define the determinant of $\mathcal{F}$ such that it agrees with the classical definition of determinant in the case when $X$ is non-singular. We study how the Hilbert polynomial of the determinant varies in families of singular varieties. Consider a singular family such that every fiber is a normal, projective variety. Unlike in the case when the family is smooth, the Hilbert polynomial of the determinant does not remain consta…
We give sufficient conditions to ensure that the ideal $Φ(\mathcal E)$ of $\mathcal E$-phantom maps in a locally $λ$-presentable exact category $(\mathcal{A}, \mathcal{E})$ is (special) (pre)covering ideal, where $\mathcal E$ is an exact substructure of $(\mathcal{A}, \mathcal{E})$. As a byproduct, we infer the existence of various covering ideals in categories of sheaves which have a meaningful geometrical motivation. In particular we deal with a Zariski-local notion of phantom maps in categories of sheaves. We would like to point out that our approach is necessarily different from [FGHT13], …
On morphisms onto quadrics
证明了从Picard数为1的光滑射影n重射到光滑n二次曲面的态射的次数是有界的（当然，前提是n至少为3）。事实上，它在几年前就被证明了，但我从来没有写过证明，直到J。M.黄禹锡最近建议我这么做。
Galois subspaces for projective varieties
该研究针对射影簇到射影空间的嵌入场景，探究满足“与嵌入复合后形成同维射影空间间Galois态射”的线性投影空间的结构，属于代数几何领域的理论研究。
We show a few basic results about moduli spaces of semistable modules over Lie algebroids. The first result shows that such moduli spaces exist for relative projective morphisms of noetherian schemes, removing some earlier constraints. The second result proves general separatedness Langton type theorem for such moduli spaces. More precisely, we prove S-completness of some moduli stacks of semistable modules. In some special cases this result identifies closed points of the moduli space of Gieseker semistable sheaves on a projective scheme and of the Donaldson--Uhlenbeck compactification of the…
Making the motivic group structure on the endomorphisms of the projective line explicit
本文在 Jouanolou 装置到射影直线的带点朴素同伦类上构造显式群结构，并证明 Cazanave 幺半群可经自然态射群完备化到由简单态射生成的子群。
The Hitchin morphism for certain surfaces fibered over a curve
本文证明Chen-Ngô猜想对直纹曲面及非各向同性椭圆纤维化（的爆破）上的任何约化群成立，并进一步对经典群证明了限制在半调和Higgs丛模空间上的Hitchin态射的满射性。
The Hilbert-Chow morphism and the incidence divisor
对于光滑射影簇$P$，我们构造了一个支持在$\mathscr{C}_a (P) \times \mathscr{C}_{\dim(P)-a-1}(P)$中关联轨迹上的Cartier因子.有一个自然的定义，相应的线丛上的产品的希尔伯特计划，我们表明这个丛下降到周品种。这回答了Mazur提出的一个问题。
Hermitian structures on the derived category of coherent sheaves
本文的主要目的是建立理论基础和所需的语言，以处理问题的直接图像的埃尔米特向量丛的投射不一定光滑态射。为此，我们首先定义厄米结构的对象上的有界导出范畴的连贯层光滑的复杂品种。其次，我们将Bott-Chern类的理论推广到这些厄米结构。最后，我们引入范畴$\oSm_{\ast/\CC}$，它的态射是相对切复形上具有厄米结构的射影态射。
Let $k$ be a field, let ${\sf C}$ be a $k$-linear abelian category, let $\underline{\mathcal{L}}:=\{\mathcal{L}_{i}\}_{i \in \mathbb{Z}}$ be a sequence of objects in ${\sf C}$, and let $B_{\underline{\mathcal{L}}}$ be the associated orbit algebra. We describe sufficient conditions on $\underline{\mathcal{L}}$ such that there is a canonical morphism from the noncommutative space ${\sf Proj }B_{\underline{\mathcal{L}}}$ to a noncommutative projective line in the sense of \cite{abstractp1}, generalizing the usual construction of a map from a scheme $X$ to $\mathbb{P}^{1}$ defined by an invertible…
Morphisms from Azumaya prestable curves with a fundamental module to a projective variety: Topological D-strings as a master object for curves
这是我们从Grothendieck的观点研究D-膜的基础的一个延续，在相关的Wilson理论的区域中，“膜”仍然是膜。在这项工作中，我们专注于D-字符串和构造的模栈的态射Azumaya预稳定曲线$C^{Az}$与一个基本模块${\cal E}$到一个固定的目标$Y$的一个给定的组合类型。这样的态射给出了$Y$上的B型威克旋转D弦的原型，它遵循了波尔钦斯基-格罗滕迪克模型，而这一堆可以作为拓扑D弦世界片瞬子的数学理论的基础。
Singularities appearing on generic fibers of morphisms between smooth schemes
该研究针对正特征下光滑概形态射的几何一般纤维的奇点问题，给出涉及局部基本群、雅可比理想等的多种判别准则，并确定了出现在这类纤维上的有理二重点类型。
We show that every smooth projective curve over a finite field k admits a finite tame morphism to the projective line over k. Furthermore, we construct a curve with no such map when k is an infinite perfect field of characteristic two. Our work leads to a refinement of the tame Belyi theorem in positive characteristic, building on results of Saïdi, Sugiyama-Yasuda, and Anbar-Tutdere.
On the birational isotriviality of the Albanese morphism of a log Calabi-Yau pair with a torus action
本文研究了带有环面作用的log Calabi-Yau对的Albanese映射的birational isotrivial性，证明了在特定条件下，Albanese映射的两个一般纤维是birationally等价的。
For a smooth projective complex variety whose Albanese morphism is finite, we show that every Bridgeland stability condition on its bounded derived category of coherent sheaves is geometric, in the sense that all skyscraper sheaves are stable with the same phase. Furthermore, we describe the stability manifolds of irregular surfaces and abelian threefolds with Picard rank one, and show that they are connected and contractible.
On the Hitchin morphism in positive characteristic
设X是特征p>0的域上的光滑射影曲线。我们表明，希钦态射，关联到一个希格斯丛的特征多项式，有一个非平凡的变形的仿射直线。这种变形是通过考虑X上向量丛上的t-联络的模堆叠和p-曲率的模拟，并通过观察相关的特征多项式在适当的意义上是p次幂来构造的。
An infinitesimal condition to deform a finite morphism to an embedding
在本文中，我们给出了从光滑簇$X$到有限到光滑图像上的射影空间的态射$φ$变形为嵌入的充分条件。这一结果使一些关于曲线曲面态射变形的定理，如$K3$和一般型的定理，在一个新的，更概念化的光中，得到了特殊的方法。本文结果的主要意义之一是应用于射影空间中具有给定不变量的光滑簇的构造。我们说明了这一点，使用我们的结果来构造正则嵌入曲面与$c_1^2=3p_g-7$，并得出一些有趣的性质，其模空间。我们的结果的另一个有趣的应用是光滑的绳子。得到了嵌入射影空间的绳可光滑的一个充分条件。因此，我们证明了正则嵌入地毯满足一定的条件，可以平滑。对$1$维绳和$K3$维地毯的光滑化定理也给出了简单、统一的证明。我们的变形$φ$嵌入的条件，可以非常透明的$φ$的一个适当的一阶无穷小变形的上同调类。它在一个非常一般的设置（任何$X$的任意尺寸和任何$φ$畅通无阻的代数形式的半通用变形）。例如，当我们将其专门用于曲线的情况时，可以看到结果的简单性。
A mathematical theory of D-string world-sheet instantons, II: Moduli stack of $Z$-(semi)stable morphisms from Azumaya nodal curves with a fundamental module to a projective Calabi-Yau 3-fold
在D（10）的第二部分D（10.2）中，我们以D（10.1）（arXiv：1302.2054 [math.AG]）为基础，定义了来自一般Azumaya节曲线的$Z$ -半稳定态射的概念，属$\ge 2$，其基本模为投射的Calabi-Yau 3-fold，并证明了固定类型的这种$Z$ -半稳定态射的模栈是紧的。这给了我们一个D弦的反模栈，就像Gromov-Witten理论中稳定映射到基本弦的模栈一样。它为我们提供并准备了一个新的卡-丘三重不变量的基础，这个不变量在超弦理论中捕捉了软D弦世界片的瞬子数。本说明与D（10.1）一起手写，并与同上一起阅读。
Let $π: X \to Y$ be a morphism of projective varieties and suppose that $α$ is a pseudo-effective numerical cycle class satisfying $π_*α= 0$. A conjecture of Debarre, Jiang, and Voisin predicts that $α$ is a limit of classes of effective cycles contracted by $π$. We establish new cases of the conjecture for higher codimension cycles. In particular we prove a strong version when $X$ is a fourfold and $π$ has relative dimension one.
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