Robert Laterveer
We prove that certain Fano fourfolds of K3 type constructed by Fatighenti-Mongardi have a multiplicative Chow-Künneth decomposition. We present some consequences for the Chow ring of these fourfolds.
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We prove that Küchle fourfolds $X$ of type d3 have a multiplicative Chow-Künneth decomposition. We present some consequences for the Chow ring of $X$.
This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial $1$-cycles is generated by conics. We also show that Verra fourfolds admit a multiplicative Chow-Künneth decomposition, and draw some consequences for the intersection product in the Chow ring of Verra fourfolds.
This note concerns hyperkähler fourfolds $X$ having a non-symplectic involution $ι$. The Bloch-Beilinson conjectures predict the way $ι$ should act on certain pieces of the Chow groups of $X$. The main result is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has consequences for the Chow ring of the quotient $X/ι$.
Let $X$ be a hyperkähler variety. Voisin has conjectured that the classes of Lagrangian constant cycle subvarieties in the Chow ring of $X$ should lie in a subring injecting into cohomology. We study this conjecture for the Fano variety of lines on a very general cubic fourfold.
This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold $X$ in the family has an involution such that the induced involution on the Fano variety $F$ of lines in $X$ is symplectic and has a $K3$ surface $S$ in the fixed locus. The main result establishes a relation between $X$ and $S$ on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family.
This article is about hyperkähler fourfolds $X$ admitting a non-symplectic involution $ι$. The Bloch-Beilinson conjectures predict the way $ι$ should act on certain pieces of the Chow groups of $X$. The main result is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has some interesting consequences for the Chow ring of the quotient $X/ι$.
Cubic fourfolds, Kuznetsov components and Chow motives
本文证明了具有Fourier-Mukai导出等价Kuznetsov分支的两个光滑三次超曲面四维簇，其Chow动机作为Frobenius代数对象同构，并推得l-adic上同调Frobenius代数间存在Galois等变同构，同时讨论了与K3曲面导出等价的情形。
Let $X$ be a smooth complex cubic fourfold and let $F$ be the variety of lines of $X$. The variety $F$ is known to be a smooth projective hyperkaehler fourfold, which is moreover endowed with a self rational map $ϕ: F -\rightarrow F$ first constructed by C. Voisin. Here we define a filtration of Bloch--Beilinson type on the Chow group of zero-cycles $CH_0(F)$ which canonically splits under the action of $ϕ$, thereby answering in this case a question of A. Beauville. Moreover, we show that this filtration is of motivic origin, in the sense that it arises from a Chow--Kuenneth decomposition of t…
We construct new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form $Z\times A^1$, where $Z$ is a quasiprojective variety. The affine cones over such a fourfold admit effective $G_a$-actions. Similar constructions of cylindrical Fano threefolds and fourfolds were done previously in [KPZ11, KPZ14, PZ15].
We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.
We consider extremal contractions on smooth Fano fourfolds whose second Chern character is positive. We show that such contractions can neither be of fiber type nor contract a divisor to a point.
On locally factorial Fano fourfolds of Picard number two
本文分类了具有超曲面Cox环且允许三维环面有效作用的Picard数为2的局部析因Fano四重折叠。
In this paper, we classify Fano fourfolds whose the second exterior power of tangent bundles are numerically effective with Picard number greater than one.
We construct 4 di erent families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form Z x A1, where Z is a quasiprojective variety. The affi ne cones over such a fourfold admit eff ective Ga-actions. Similar constructions of cylindrical Fano threefolds were done previously in our papers jointly with Takashi Kishimoto.
In this paper we show that a general element of $|-K_X|$ on a four-dimensional Fano manifold has at most terminal singularities. We then determine an explicit local expression of these singular points.
Fano fourfolds of K3 type
本研究从旗流形乘积上齐次向量丛零轨迹构建的法诺四簇数据库中提取64个K3型族，探究其几何与K3结构起源，给出不变量、有理性及导出范畴半正交分解相关信息。
Prime Fano fourfolds in classical and generalized Grassmannians
本文在广义Grassmannian中完整分类了Picard秩1指数1的Fano四重簇，扩展了Küchle的结果，发现了三个新族，并计算了体积和Hodge数等不变量。
Higher Chow cycles, cyclic cubic fourfolds and Lagrangian subvarieties
本文研究了holomorphic李代数流形上的高维Chow循环，通过构造显式不可分解的(2,1)和(4,1)循环，首次在holomorphic李代数流形上提供了此类循环的例子，并探讨了循环在拉格朗日子变种上的限制现象。
Chow groups of surfaces of lines in cubic fourfolds
研究四次三次折叠中与固定直线相交的直线曲面的动机分解，定义Beauville-Voisin类的类似物，并利用Shen和Vial引入的Bloch-Beilinson滤过自然分裂研究到所有直线Fano簇的前推映射。
The affine cones over Fano-Mukai fourfold of genus $7$ are flexible
本文证明了亏格 $7$ 的任何光滑 Fano-Mukai 四重折叠的仿射锥具有柔性。
The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the $n$-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently.
On smooth Fano fourfolds of Picard number two
本文分类了具有一般超曲面Cox环且Picard数为2的光滑Fano四维簇。
We construct new families of smooth Fano fourfolds with Picard rank $1$ which contain open $\Bbb A^1$-cylinders, that is, Zariski open subsets of the form $Z \times \Bbb A^1$, where $Z$ is a quasiprojective variety. In particular, we show that every Mukai fourfold of genus $8$ is cylindrical and there exists a family of cylindrical Gushel-Mukai fourfolds.
We show that the affine cones over any Fano-Mukai fourfold of genus 10 are flexible; in particular, the automorphism group of such a cone acts highly transitively outside the vertex. Furthermore, any Fano-Mukai fourfold of genus 10, with one exception, admits a covering by open charts isomorphic to the affine four-space.
Derived categories of Gushel-Mukai surfaces and Fano fourfolds of K3 type
研究K3型Fano四维流形的半正交分解，证明非常一般的对偶Gushel-Mukai表面不相等，尽管导出和L等价。
Conjecturally, Fano varieties of K3 type admit a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. We prove this for many of the families of Fano varieties of K3 type constructed by Fatighenti-Mongardi. This has interesting consequences for the Chow ring of these varieties.
We describe the closed cone of moving curves of smooth Fano three- and fourfolds by giving finitely many equations that cut out the cone. The equations are induced by the exceptional divisors of divisorial contractions and by nef divisors on birational models obtained by sequences of flips.The results presented here are part of the author's Ph.D. thesis, written under the supervision of Stefan Kebekus.
We prove that the Fano variety of lines of a cuspidal cyclic cubic fourfold is a symplectic variety with transversal A2-singularities and we study the properties of the nonsymplectic order three automorphism induced by the covering automorphism on the irreducible holomorphic symplectic manifold obtained by blowing up the singular locus.
We show that certain Fano eightfolds (obtained as hyperplane sections of an orthogonal Grassmannian, and studied by Ito-Miura-Okawa-Ueda and by Fatighenti-Mongardi) have a multiplicative Chow-Künneth decomposition. As a corollary, the Chow ring of these eightfolds behaves like that of K3 surfaces.
Classification of Fano fourfolds with large anticanonical base locus
本文分类了反典范基轨迹为光滑曲面的光滑Fano四重，证明恰有22个形变族，均源于同一几何构造，并与具有$\mathbb P^1$-丛结构的Fano三重相关。
Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only ADE-singularities. This is shown as a corollary to the characterization of a singularity that is obtained as a $K3^{[2]}$-type contraction and has a unique symplectic resolution.
We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 surface. The moduli spaces of complexes and of sheaves are related by wall-crossing in the derived category of twisted sheaves on the corresponding K3 surface.
We exhibit several families of Fano threefolds with a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of these threefolds injects into cohomology. As a by-product of the argument, we observe that double covers of projective spaces admit a multiplicative Chow-Künneth decomposition.
In this text we prove that if a smooth cubic in $\PR^5$ has its Fano variety of lines birational to the Hilbert scheme of two points on a K3 surface, then there exists a smooth projective curve or a smooth projective surface embedded in the Fano variety, such that the kernel of the push-forward (at the level of zero cycles ) induced by the closed embedding is torsion.
We provide a new construction of rationality for cubic fourfolds via Mori's theory and the minimal model program. As an application, we present the solution of the Kuznetsov's conjecture for $d=42$ (the first open case). Our methods also show an explicit connection between the rationality of cubic fourfolds belonging to the first four admissible families $\mathcal C_d$, with $d=14,26,38$, and $42$ and some birational models of minimal K3 surfaces of degree $d$ contained in well known rational Fano fourfolds.
Terminalizations of quotients of Fano varieties of lines on cubic fourfolds
本文研究了立方四维流形上线的Fano簇商的终端化，计算了相关拓扑不变量，并识别出新的四维不可约解析辛流形变形类。
Cubic fourfolds with birational Fano varieties of lines
本文给出了几对非同构三次四重体例子，证明其 Fano 直线簇是有理等价的，为 Fourier-Mukai 伙伴猜想提供新证据，并提出三次四重体与其 Fano 直线簇有理等价性相关的猜想。
This paper constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth $4$-dimensional toric Fano varieties. The tilting bundles lead to a large class of explicit Calabi-Yau-$5$ algebras, obtained as the corresponding rolled-up helix algebra. A database of the full strong exceptional collections can be found in the package QuiversToricVarieties for the computer algebra system Macaulay2.
On the Chow Ring of Fano Fourfolds of K3 type
本文证明了一类新构造的 K3 型 Fano 簇具有乘性 Chow-Künneth 分解，使其 Chow 环表现如 K3 曲面，并给出了爆破射影簇 Franchetta 性质的判据。
Let X be a complex Fano manifold of dimension n. Let s(X) be the sum of l(R)-1 for all the extremal rays of X, the edges of the cone NE(X) of curves of X, where l(R) denotes the minimum of (-K_X \cdot C) for all rational curves C whose class [C] belongs to R. We show that s(X)\leq n if n\leq 4. And for n\leq 4, we completely classify the case the equality holds. This is a refinement of the Mukai conjecture on Fano fourfolds.
Fatighenti and Mongardi have defined Fano varieties of type S6 as zero loci of a certain vector bundle on the Grassmannian $\hbox{Gr}(2,10)$. These varieties have 3 Hodge structures of K3 type in their cohomology. We show that the Chow ring of these varieties also displays "K3 type" behaviour.
On the geometry of some quiver zero loci Fano fourfolds (with an appendix by E. Kalashnikov and F. Tufo)
本文对 Kalashnikov 提出的 170 族箭图旗零轨迹 Fano 四维流形进行研究，将其视为齐性向量丛截面的零轨迹，并给出全部 170 族的双有理与双正则刻画。
The Fano variety of lines on singular cyclic cubic fourfolds
本文研究奇异循环三次四重簇上直线Fano簇的辛解消，特别关注沿具有$A_i$型孤立奇点的三次三重簇分支的情形，并发现覆盖自同构诱导的三阶非辛自同构。
These are notes of lectures given at the school `Birational Geometry of Hypersurfaces' in Gargnano in March 2018. The main goal was to discuss the Hodge structures that come naturally associated with a cubic fourfold. The emphasis is on the Hodge and lattice theoretic aspects with many technical details worked out explicitly. More geometric or derived results are only hinted at.
Mukai proved that most prime Fano fourfolds of degree 10 and index 2 are contained in a Grassmannian G(2,5). They are all unirational and some are rational, as remarked by Roth in 1949. We show that their middle cohomology is of K3 type and that their period map is dominant, with smooth 4-dimensional fibers, onto a 20-dimensional bounded symmetric period domain of type IV. Following Hassett, we say that such a fourfold is special if it contains a surface whose cohomology class does not come from the Grassmannian G(2,5). Special fourfolds correspond to a countable union of hypersurfaces in the …
An irregular smooth Fano fourfold in positive characteristic
在特征二的代数闭域上构造了一个光滑射影 Fano 四维簇，具有非零的 $H^1$ 且无自由有理曲线，从而不是可分有理连通的，该例子借助 ChatGPT 构建。
Birational geometry of Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls
研究包含三次滚动面对的三次四重线上Fano簇（超Kähler四重）的双有理几何，识别所有双有理模型并给出显式映射，同时刻画双有理自同构群的生成元和关系。
The automorphism groups of the Fano-Mukai fourfold of genus 10 were studied in our previous paper [arXiv:1706.04926]. In particular, we found in [arXiv:1706.04926] the neutral components of these groups. In the present paper we finish the description of the discrete parts. Up to isomorphism, there are two special Fano--Mukai fourfold of genus 10 with the automorphism groups $GL_2(k)\rtimes\mathbb{Z}/2\mathbb{Z}$ and $(\mathbb{G}_a\times\mathbb{G}m)\rtimes\mathbb{Z}/2\mathbb{Z}$, respectively. For any other Fano-Mukai fourfold $V$ of genus 10 one has $\mathrm{Aut}(V)=\mathbb{G}_m^2\rtimes \math…
Constant cycle surfaces on Fano varieties of cubic fourfolds
本文研究光滑四次三维簇的线Fano簇上常循环曲面的有限性，证明了固定阶的形如F(Y)的常循环曲面至多有限个。
Central Weyl involutions on Fano-Mukai fourfolds of genus 10
本文研究了亏格10的Fano-Mukai四重流形上源自G2型Weyl群中心的唯一对合τ，详细刻画了其不动点集合及由τ不变直线扫出的直纹面。
Using a codimension-$1$ algebraic cycle obtained from the Poincaré line bundle, Beauville defined the Fourier transform on the Chow groups of an abelian variety $A$ and showed that the Fourier transform induces a decomposition of the Chow ring $CH^*(A)$. By using a codimension-$2$ algebraic cycle representing the Beauville--Bogomolov class, we give evidence for the existence of a similar decomposition for the Chow ring of hyperKähler varieties deformation equivalent to the Hilbert scheme of length-$2$ subschemes on a K3 surface. We indeed establish the existence of such a decomposition for the…
Fano fourfolds with large anticanonical base locus
针对法诺流形反典范除子的奇异性问题，本文对比三元体与四元体的差异，证明当四维法诺流形反典范系统基底轨迹为余维二的正规曲面时，所有反典范除子均奇异。
Lines on cubic threefolds and fourfolds containing a plane
该研究刻画了特征非2域上含平面的一般三次三维簇的直线Fano概形，以此刻画非闭域上这类三次三维簇的有理性，并显式构造含平面三次四维簇直线Fano簇的拉格朗日纤维化。
On a numerical criterion for Fano fourfolds
本文证明了Campana--Peternell猜想在四维情形的一个特例，即满足特定数值条件且反典范除子严格数值有效的射影光滑四折叠必为Fano四折叠，并完全解决了相关非消失猜想。
Let $Y$ be a smooth dimensionally transverse intersection of the Grassmannian $\hbox{Gr}(2,n)$ with 3 Plücker hyperplanes. We show that $Y$ admits a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.
On the Chow ring of double EPW quartics
本研究通过双EPW四次曲线的几何特性，探讨了超kähler流形上代数循环的一般猜想。
K{ü}chle classified the Fano fourfolds that can be obtained as zero loci of global sections of homogeneous vector bundles on Grassmannians. Surprisingly, his classification exhibits two families of fourfolds with the same discrete invariants. Kuznetsov asked whether these two types of fourfolds are deformation equivalent. We show that the answer is positive in a very strong sense, since the two families are in fact the same! This phenomenon happens in higher dimension as well.
Cohomological rigidity of smooth toric Fano fourfolds
该研究证明整上同调环分次同构的光滑环面法诺四维簇必微分同胚，确立其微分同胚类由上同调环决定的刚性性质。
Given a smooth projective variety, a Chow-Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow-Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow-Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Kü…
Families of smooth Fano fourfolds of Picard rank 1 without Bott vanishing
本文证明当前已知的 Picard 秩 1 指数 1 的光滑 Fano 四重族满足 χ(X,T_X)<0，结合指数>1 情形，表明除射影空间外所有已知 Picard 秩 1 的光滑 Fano 四重均不满足 Bott 消失，从而任何具有大于 1 次自同态的此类簇必为 P^4。
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