Unobstructedness of deformations of weak Fano manifolds
Taro Sano
证明了弱Fano流形具有无障碍变形。对于一般品种，我们调查的条件下，品种必然受阻。
We prove that a weak Fano manifold has unobstructed deformations. For a general variety, we investigate conditions under which a variety is necessarily obstructed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Unobstructedness of deformations of holomorphic maps onto Fano manifolds of Picard number 1
我们表明，满射态射到一个Fano流形皮卡德数1的变形是通畅的和刚性模的自同构的目标，如果各种最小有理切线的Fano流形是非线性或有限的。关于极小有理切线的多样性的条件对于Picard数为1的Fano流形的几乎所有已知例子都成立，除了射影空间。当最小有理切线的簇是非线性的时，证明是基于N. Mok和作者关于切映射的双有理性。当Fano流形的极小有理切线簇是有限的时，其关键思想是通过与极小有理曲线相关联的泛态射对给定的满射进行因式分解.
Unobstructedness of deformations for a d-semistable central corank one boundary point
针对复阿贝尔曲面的不可约II型退化，本文证明其局部平凡、平坦及对数光滑形变函子均光滑，确立了d-半稳定中心余秩一边界点形变的无阻碍性。
We generalize the Tian-Todorov Theorem in the case of Calabi-Yau varieties equipped with a line bundle.
Unobstructedness of filling secants and the Gruson-Peskine general projection theorem
我们证明了一个畅通无阻的结果变形的子品种限制交叉与另一个，固定的子品种。我们推导出光滑度和预期尺寸的结果为多点轨迹的通用投影，主要是从一个点或一条线，或纤维的嵌入维2或less. The当前版本是一个实质性的增强前一个，包括第一次从一条线的投影结果。
Unobstructedness of affine Gorenstein terminal toric fourfolds
该研究证明维数不超过4的仿射终态戈伦斯坦环面簇无阻碍，针对四维情形确定齐次阻碍次数，构造双参数形变消除潜在阻碍，完善了环面簇形变相关理论。
We prove that a weak $\mathbb{Q}$-Fano $3$-fold with terminal singularities has unobstructed deformations. By using this result and computing some invariants of a terminal singularity, we provide two results on global deformation of a weak $\mathbb{Q}$-Fano $3$-fold. We also treat a stacky proof of the unobstructedness of deformations of a $\mathbb{Q}$-Fano $3$-fold.
We consider compact Kählerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure $Π$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor $D(Π)$. We prove that $(X, Π)$ has unobsrtuced deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on $H^2$ of the open symplectic manifold $X\setminus D(Π)$, and in fact coincides with this $H^2$ provided the Hodge number $h^{2,0}_X=0$, and finally that the degeneracy locus $D(Π)$…
Lifting of cohomology and unobstructedness of certain holomorphic maps
设$f$是紧致复流形之间的全纯映射.我们给出了一个标准$f$有{ \it通畅的变形}，即局部模空间的$f$是光滑的：这说，粗略地说，这组无穷小变形的$f$，当被视为一个函子，本身满足一个自然提升性质的无穷小变形。这种提升性质是满意的，例如，每当组的问题承认一个“拓扑”或霍奇理论的解释，我们给出了一些例子，主要涉及卡-丘流形，在这种情况下。
A family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.
We determine obstructedness or unobstructedness of (holomorphic) Poisson deformations of ruled surfaces over an elliptic curve.
We give a new characterization of generalized Kähler structures in terms of their corresponding complex Dirac structures. We then give an alternative proof of Hitchin's partial unobstructedness for holomorphic Poisson structures. Our main application is to show that there is a corresponding unobstructedness result for arbitrary generalized Kähler structures. That is, we show that any generalized Kähler structure may be deformed in such a way that one of its underlying holomorphic Poisson structures remains fixed, while the other deforms via Hitchin's deformation. Finally, we indicate a close r…
Deformations of Q-Calabi-Yau 3-folds and Q-Fano 3-folds of Fano index 1
本文证明了任何只具有普通终端奇点的Q-Calabi-Yau 3-fold和任何只具有终端奇点的Fano指数为1的Q-Fano 3-fold都有Q-smoothing。
We calculate the Hodge numbers of quasismooth Fano 3-folds whose total anti-canonical embedding has small codimension, and relate these to the number of deformations.
Deformations of weak Fano 3-folds with only terminal singularities
本文证明了任何只有终端奇点的$\Bbb Q$ -阶乘弱Fano 3-重都是光滑的.
In this note we collect some results on the deformation theory of toric Fano varieties.
On deformations of Q-Fano threefolds
本文研究了只具有终端奇点的Q-Fano 3-fold的形变理论。首先，我们证明了Q-Fano 3-fold的Kuranishi空间是光滑的。其次，我们证明了每个只具有“普通”终端奇点的Q-Fano 3-fold是Q-smoothable的，也就是说，它可以变形为只具有商奇点的Q-Fano 3-fold。最后，我们证明了Q-光滑的Q-Fano 3-倍假设存在的DuVal anticanonical元素。作为应用，我们得到了具有Du Val反正则元的准素Q-Fano 3-折叠的亏格界.
Deformations of holomorphic Poisson manifolds
证明了紧致全纯Poisson流形变形的一个无障碍性定理，并应用于一类例子。这些包括某些合理的表面和希尔伯特计划的点泊松表面。我们特别研究了射影平面的Hilbert方案，并表明一般变形由两个参数决定--椭圆曲线和椭圆曲线上的平移。
Deformations and obstructions of pairs (X,D)
研究了特征为0的代数闭域上定义的对（X，D）的变形，其中X是光滑的射影簇，D是光滑的或正规交叉因子。利用微分分次李代数理论和Cartan同伦构造，我们能够用完全代数的方法证明在许多情况下（X，D）的变形是无障碍的，例如，当（X，D）是一个对数Calabi-Yau对时，在光滑因子D在Calabi-Yau簇X中的情况下，当D是一个光滑因子在|-m K_X|，对于某个正整数m.
Deformations of canonical pairs and Fano varieties
本文致力于研究对数对变形的各个方面，特别是与奇异点不变性和对数多属性有关的问题。特别是，使用最近的结果从极小模型计划，我们得到了一个扩展定理的伴随因子的精神，萧和Kawamata和最近的作品Hacon和McKernan。然而，我们的主要动机来自于研究变形的法诺品种。我们的第一个应用程序的行为Mori室分解的Fano品种的家庭：我们证明，在温和的奇异性的情况下，这种分解是刚性变形时的尺寸是小的。然后，我们转而分析复曲面Fano品种的变形特性，并证明，每一个单纯复曲面Fano品种与最终端的奇异性是刚性变形下（特别是不光滑，如果奇异）。
Let X be a connected family of complex Fano manifolds. We show that if some fiber is the product of two manifolds of lower dimensions, then so is every fiber. Combining with previous work of Hwang and Mok, this implies immediately that if a fiber is (possibly reducible) Hermitian symmetric space of compact type, then all fibers are isomorphic to the same variety.
Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric $3$-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a consequence, we prove that the number of local …
In this article, we investigate deformations of a Calabi-Yau manifold $Z$ in a toric variety $F$, possibly not smooth. In particular, we prove that the forgetful morphism from the Hilbert functor $H^F_Z$ of infinitesimal deformations of $Z$ in $F$ to the functor of infinitesimal deformations of $Z$ is smooth. This implies the smoothness of $H^F_Z $ at the corresponding point in the Hilbert scheme. Moreover, we give some examples and include some computations on the Hodge numbers of Calabi-Yau manifolds in Fano toric varieties.
We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3-fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3-fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3-folds by building degenerations to K-polystable toric Fano 3-folds.
Rigidity of wonderful group compactifications under Fano deformations
本文证明了De Concini和Procesi构造的精彩群紧化在Fano形变下具有刚性，即族中一个纤维同构于该紧化则所有纤维均同构，从而回答了Bien和Brion提出的问题。
We study the question whether rational homogeneous spaces are rigid under Fano deformation. In other words, given any smooth connected family f:X -> Zof Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space S, whether is f an S-fibration? The cases of Picard number one were studied in a series of papers by J.-M. Hwang and N. Mok. For higher Picard number cases, we notice that the Picard number of a rational homogeneous space G/P is less or equal to the rank of G. Recently A. Weber and J. A. Wisniewski proved that rational homogeneous spaces G/P with Picard numbers equal…
Deformations of the Fano scheme of a cubic
本文研究有限奇点 d 维三次超曲面的直线 Fano 概形形变，通过相对 Fano 概形构造局部模函子态射，并证明 d≥5 时其在一阶形变上为同构。
On deformations of Q-Fano threefolds II
本文研究了$3$维终端奇点的余边界映射，这在研究奇异$3$-褶皱的形变中具有重要意义.我们证明了这个地图消失，只有商奇点和$A_{1,2}/4$-奇点，即终端奇点解析同构的$\mathbb{Z}_4$-商的奇点$ (x^2+y^2 +z^3+u^2=0)$。作为应用，我们证明了具有终端奇点的$\mathbb{Q}$-Fano $3$-折叠可以变形为只具有商奇点和$A_{1,2}/4$-奇点的折叠.我们也处理$\mathbb{Q}$-光滑性问题的$\mathbb{Q}$-Calabi-Yau $3$-折叠。
Deformations, local freeness, and base change for higher Du Bois singularities
该研究证明严格高阶Du Bois奇点在小形变下不变，针对严格高阶Du Bois纤维的相对Du Bois复形建立基变换定理，展示1-Du Bois纤维的相关失效并推广局部自由性定理、证明Hodge数不变性。
Smoothings of Fano schemes with normal crossing singularities of dimension at most three
研究了具有维数不超过3的正规交叉奇点的Fano簇X的形变理论。我们得到了X的一阶变形层T^1（X）在X的适当对数分解及其奇异轨迹Z中的一个公式，并得到了X光滑存在的显式判据。
Jumping of the nef cone for Fano varieties
我们构建Q-阶乘终端Fano品种，从4维开始，其nef锥跳跃时，品种变形。因此，de Fernex和Hacon关于三维Fanos变形的结果是最优的。这些例子是基于存在的高维翻转变形同构，推广Mukai翻转。我们还推广了de Fernex和Hacon关于Fano簇的变形的结果。复曲面Fano簇在余维2上是光滑的，在余维3上是Q-阶乘的，它们是刚性的。除子类群是变形不变的klt Fanos，这是光滑的余维2和Q-阶乘余维3。当余维3中Q阶乘的终端Fano变形时，Cox环在平坦族中变形。一个侧面的结果，这似乎是新的是，除子类群的klt法诺品种映射同构到普通的同源性。
Deformations of Toric Varieties via Minkowski Sum Decompositions of Polyhedral Complexes
推广了K的仿射复曲面簇的变形的构造。Altmann和我们以前通过引入多面体复形的Minkowski和分解的概念，构造了弱Fano复曲面簇到任意复曲面簇的变形。我们的建设嵌入到一个高维的复曲面品种的图像是由一个素二项式完全相交理想在考克斯齐次坐标的原始复曲面品种。变形是通过完全相交的族来实现的。对于最坏情况下Gorenstein终端奇异的紧致单纯复曲面簇，我们证明了我们的变形跨越变形的无限小空间Kodaira-Spencer映射。对于Fano复曲面品种，我们表明，它们的变形可以在高维Fano复曲面品种相关的Batyrev-Borisov镜像对称建设。
105 families of Fano threefolds in positive characteristic
在正特征代数闭域上，证明光滑Fano三维簇共有105个不可约形变族。
Weak Fano threefolds with del Pezzo fibration
本文讨论了具有D型极值射线的光滑弱Fano 3-折叠。将具有除6次外的D型极值射线的光滑弱Fano 3-褶皱划分为47种变形类型。
We study deformations of a pair of a $\mathbb{Q}$-Fano $3$-fold $X$ with only terminal singularities and its elephant $D \in |{-}K_X|$. We prove that, if there exists $D \in |{-}K_X|$ with only isolated singularities, the pair $(X,D)$ can be deformed to a pair of a $\mathbb{Q}$-Fano $3$-fold with only quotient singularities and a Du Val elephant. When there are only non-normal elephants, we reduce the existence problem of such a deformation to a local problem around the singular locus of the elephant. We also give several examples of $\mathbb{Q}$-Fano $3$-folds without Du Val elephants.
We construct a deformation family for each of the 34 Hilbert series of index 2 Fano 3-folds. In 18 cases we construct two different families, distinguished by the topology of their general members.
We prove that for every smooth prime Fano $3$-fold $V$, the Hilbert scheme $\operatorname{Hilb}^{sc} V$ of smooth connected curves on $V$ contains a generically non-reduced irreducible component of Mumford type. We also study the deformations of degenerate curves $C$ in $V$, i.e., curves $C$ contained in a smooth anti-canonical member $S \in |-K_V|$ of $V$. We give a sufficient condition for $C$ to be stably degenerate, i.e., every small (and global) deformation of $C$ in $V$ is contained in a deformation of $S$ in $V$. As a result, by using the Hilbert-flag scheme of $V$, we determine the dim…
Deformations of generalized complex branes
研究了广义复流形上（秩1）膜的形式变形理论。例如，这推广了固定复流形中复子流形的变形理论。对于GC流形$(X,\mathbb{J})$上的每个GC膜$\mathcal{B}$，我们构造了一个形式（点）广群$\textbf{Def}^{\mathcal{B}}(X,\mathbb{J})$（定义在某类实Artin代数上），它编码了$\mathcal{B}$的形式变形。我们研究了几何内容的广群在一些不同的情况。利用（双）半单微分分次李代数（DGLA）理论，我们对每个膜$\mathcal{B}$构造了一个DGLA $L_{\mathcal{B}}$，它控制$\mathcal{B}$的“局部可平凡化”变形.作为该构造的一个具体应用，我们证明了一个无障碍结果。
On Fano manifolds of Picard number one
K{ü}chle将Grassmannian上齐次向量丛的整体瓣的零轨迹所得到的Fano四重分类。令人惊讶的是，他的分类展示了两个家庭的四倍相同的离散不变量。库兹涅佐夫问这两种类型的四重是否是形变等价的。我们表明，答案是肯定的，在一个非常强的意义上，因为这两个家庭实际上是相同的！这种现象在更高的维度也会发生。
Existence and deformations of Kahler-Einstein metrics on smoothable Q-Fano varieties
我们证明了Q-Gorenstein光滑，K-polystable Q-Fano品种的Kahler-Einstein度量的存在，我们展示了这些度量的行为，在Gromov-Hausdorff意义上，Q-Gorenstein光滑。
One-dimensional components in the K-moduli of smooth Fano 3-folds
本文通过在 4 个特定形变族中识别 K-多稳定极限，完成了可光滑化 Fano 3-folds 的 K-模空间中一维分支的分类。
Smoothings of Fano varieties with normal crossing singularities
本文给出了定义在特征为零的代数闭域上的具有正规交叉奇点的Fano簇X是光滑的判别准则。与原始版本的不同之处在于，本文利用对数结构和变形理论证明了X可被一个光滑簇光滑的充要条件是T^1（X）=O_D，其中D是X的奇异轨迹。
Seshadri constants, introduced by Demailly, measure the local positivity of a nef divisor at a point. In this paper, we compute the Seshadri constants of the anticanonical divisors of Fano manifolds with coindex at most $3$ at a very general point. As a consequence, if $X$ is a nonsingular Fano threefold which is very general in its deformation family, then $\varepsilon(X,-K_X;x)\leq 1$ for all points $x\in X$ if and only if $\vert-K_X\vert$ is not base point free.
Unprojection and deformations of tertiary Burniat surfaces
构造了一类4维一般型曲面族，其中p_g=0，K^2=3，基本群Z/2xQ_8，其中Q_8是四元数群.所构造的族包含K^2=3的Burniat曲面。此外，我们还构造了曲面族中曲面的泛覆盖作为（\PP^1）^4上的完全交，并给出了Z/2xQ_8在（\PP^1）^4上的作用，从而提升了曲面上的自然作用。战略如下。我们考虑一个p_g=0，K^2=3的曲面的一个三角形（Z/2）^3-覆盖T，并假设它可以嵌入一个Fano 3-重V中。由于V是一个Enriques-Fano 3-fold，考虑到它的Fano覆盖，得到了上面泛覆盖的简单描述。
A Conditional Reduction of the Rational Hodge Conjecture for Threefolds and Deformation-Theoretic Verifications in Several Families
本文提出几何逼近假设 Hypothesis~BB，证明其蕴含三流形的有理 Hodge 猜想，并利用形变理论判据在 Calabi--Yau、Fano 及一般五次三流形族中验证了该假设，将其约化为可计算条件。
Geometric Manin's Conjecture for Fano 3-Folds
本文通过分类光滑 Fano 三维流形上的自由有理曲线族，证明了表示固定数值曲线类的非常自由有理曲线族不可约或为空，从而在三维情形下证明了几何 Manin 猜想，并给出了模空间分量个数的显式计数。
Some properties of Fano manifolds that are zeros of sections in homogenous vector bundles over Grassmannians
设$X$是Grassmannian上不可约齐次向量丛中一般整体截面$s$的零概型的Fano流形.我们证明了Plücker嵌入的限制嵌入$X$射影正常，和每一个小的变形$X$来自变形的部分$s$。这些结果在Fano 4-folds的情况下得到了加强。
On Global Deformations of Quartic Double Solids
证明了四次二重体的光滑整体变形，即沿光滑四次体分支的$\mathbb P^3$的双重覆盖，在不假设整体变形的射影性的情况下，仍然是四次二重体.对于$\mathbb P^ 5$中的两个二次曲面的光滑相交也给出了类似的结果，但这要容易得多。在一个弱形式下，这推广了J.Kollár和I. Nakamura关于Moishezon流形同胚于某些Fano三重流形的结果，并进一步证明了Picard秩$1$的Fano流形的整体变形是否是Fano流形本身.
Let $X$ be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that $X$ admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if $X$ has toroidal singularities, if $X$ has finite quotient singularities, and if the second cohomology group of its tangent sheaf vanishes.
Deformation and Unobstructedness of Determinantal Schemes
本文研究 Hilbert 概形中行列式概形轨迹闭包的维数、一般光滑性、正规模深度及外幂形变，将标准行列式情形推广到一般 $r$。
We extend the notion of essential deformations from the case of the Iwasawa manifold, for which they were introduced recently by the first-named author, to the general case of page-$1$-$\partial\bar\partial$-manifolds that were jointly introduced very recently by all three authors. We go on to obtain an analogue of the unobstructedness theorem of Bogomolov, Tian and Todorov for page-$1$-$\partial\bar\partial$-manifolds. As applications of this discussion, we study the small deformations of certain Nakamura solvmanifolds and reinterpret the cases of the Iwasawa manifold and its $5$-dimensional …
K-stability of Fano 3-folds in the World of Null-A
本文研究了不满足条件 (A) 的光滑 Fano 三维流形的 K-稳定性，证明除八个例外形变族外，此类流形均为 K-多重稳定的。
Deformations of Compact Calabi--Yau and Fano Varieties with Isolated Singularities
该研究针对带孤立奇点的紧对数典范凯勒簇，推广了Tenie与Namikawa的相关结果，证明其半通用形变的一般纤维满足特定Du Bois与链环不变量条件。
Hodge numbers of a Fano eightfold of K3 type
研究 K3 型法诺八重体的霍奇数，通过构造显式半稳定退化来推导，表明其皮卡秩为一，还给出了亏格为八的 K3 曲面的希尔伯特平方的射影模型描述。
In this article we study the deformations of hyperelliptic polarized varieties $(X,L)$ of dimension $m$ and sectional genus $g$ such that the image $Y$ of the morphism $φ$ induced by $|L|$ is smooth. If $L^m < 2g-2$, it is known that, by adjunction and the Clifford's theorem, any deformation of $(X,L)$ is hyperelliptic. Thus, we focus on when $L^m=2g-2$ or $L^m=2g$. We prove that, if $(X,L)$ is Fano-K3, then, except when $Y$ is a hyperquadric, all deformations of $(X,L)$ are again hyperelliptic (if $Y$ is a hyperquadric, the general deformation of $φ$ is an embedding). This contrasts with the …
Kuznetsov's Fano threefold conjecture via K3 categories and enhanced group actions
本文解决了 Kuznetsov 关于 Fano 三重形导出范畴猜想的最后开放情形，通过范畴 Enriques-K3 对应与 Hodge 理论证明四次双三维体和 Gushel-Mukai 三重形的 Kuznetsov 分量不等价，并利用增强群作用的障碍证明其形变等价。
Deformations of singular Fano and Calabi-Yau varieties
本文推广了奇异Fano与Calabi-Yau三维簇的形变理论，将末端奇点条件放宽至典范奇点，并在更高维数下借助更高Du Bois与有理奇点获得广义Fano及Calabi-Yau簇的光滑化结果。
Deformations of Smooth Toric Surfaces
对于完备光滑复曲面簇Y，我们描述分次向量空间T_Y^1。此外，我们表明，光滑复曲面是通畅的，光滑复曲面是刚性的当且仅当它是法诺。然后，对于给定的复曲面，我们通过多面体细分的闵可夫斯基分解来构造齐次变形，计算它们在Kodaira-Spencer映射下的像，并证明它们跨越T_Y ^[1]。
Deformations of Fano manifolds with weighted solitons
本文研究Fano流形上加权孤立子的形变问题，给出了加权孤立子可沿Kuranishi族小形变延拓的等价条件，推广了Cao-Sun-Yau-Zhang关于Kaehler-Einstein度量的相关结果。
Combinatorial and Algebraic Mutations of Toric Fano 3-folds and Mass Deformations of 2d (0,2) Quiver Gauge Theories
本文证明Fano 3-fold的多面体变异可等同于2d (0,2)膜砖模型的质量形变，且相关模型的介子模空间生成元数目、味荷多面体对偶性及Hilbert级数在特定细化下保持一致。
We present a new method to solve certain $\bar{\partial}$-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a $\bar{\partial}$-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at $E_1$-level, as well as certain injectivity theorem on compact Kahler manifolds. Furthermore, for a family of logarithmic deformatio…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。