Ivan Smith
We show the cohomological monodromy for the universal family of smooth cubic threefolds does not factor through the genus five mapping class group. This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds.
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We study degree of irrationality of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces that have terminal singularities.
An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo $p$. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of $\mathbb F_3$. As a spin-off, it is shown that the 5-dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over $\mathbb F_q$ which attains Perret's and Weil's upper boun…
Irrationality of the first stabilization of a complex smooth cubic threefold
本文证明所有光滑复三次三维簇与射影直线乘积的非理性，并构造一个稳定有理但非理性的 Fano 簇，其高次幂与对称平方均为有理。
For $X$ a smooth cubic threefold we study the Plücker embedding of the Fano surface of lines $S$ of $X$. We prove that if $X$ is general then the minimal gonality of a covering family of curves of $S$ is four and that this happens for a unique family of curves. The analysis also shows that there is a unique pentagonal connecting family of curves, which leads to the fact that the connecting gonality of $S$ is five whereas the degree of irrationality, i.e.\ the minimal degree of a rational map from $S$ to $\mathbb{P}^2$, is six.
Let $S$ be a smooth projective surface on a smooth threefold $X$ such that $X$ has Picard rank 1 and NS$(S)$ is generated by the restriction of divisors from X. We show that if $X$ satisfies the Bogomolov-Gieseker type inequality for tilt semistable objects conjectured by Bayer-Macrì-Stellari, then the minimum degree of a dominant rational map $S\dashrightarrow\mathbb{P}^2$ is either relatively large or determined by a net of curves of low degree on $S$. As one application, we prove that the complete intersection of three very general quadrics in $\mathbb{P}^5$ has degree of irrationality 4.
We exhibit geometric conditions on a family of toric hypersurfaces under which the value of a canonical normal function at a point of maximal unipotent monodromy is irrational.
The cubic threefold is symplectically irrational
该研究证明光滑三次三维簇辛非有理，重现了Clemens-Griffiths经典非有理定理，其障碍源于量子联络在不规则奇点周围的形式单值性。
An atomic criterion for irrationality without quantum computations
该研究提出一种无需量子计算、基于原子理论的无理性判据，结合量子乘法单值群等变性与单值群表示不可约性，可判定三类特殊代数簇的无理性。
In this paper we are concerned with the monodromy of Picard-Fuch differential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the hypergeometric cases the matrix representations of monodromy relative to the Frobenius bases can be expressed in terms of the geometric invariants of the underlying Calabi-Yau threefolds. This phenomenon is also verified numerically for other families of Calabi-Yau threefolds in the paper. Furthermore, we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgrou…
Orphan Calabi-Yau threefold with arithmetic monodromy group
本文研究了孤立Calabi-Yau三流形单参数族Picard-Fuchs算子的单值群，为4阶双重八次孤立算子构造有理辛基，证明其单值群在Sp(4,Z)中稠密，并证明其中一个算子的单值群是算术的。
Recently J.C. Rohde constructed families of Calabi-Yau threefolds parametrised by Shimura varieties. The points corresponding to threefolds with CM are dense in the Shimura variety and, moreover, the families do not have boundary points with maximal unipotent monodromy. Both aspects are of interest for Mirror Symmetry. In this paper we discuss one of Rohde's examples in detail and we explicitly give the Picard-Fuchs equation for this one dimensional family.
Doran and Morgan have introduced certain rational basis for the monodromy group of the Picard-Fuchs operator of a hypergeometric family of Calabi-Yau threefolds. In this paper we compute numerically the transition matrix between a generalization of the Doran-Morgan basis and the Frobenius basis at a half-conifold point of a one-parameter family of double octic Calabi-Yau threefolds. We identify the entries of this matrix as rational functions in the special values $L(f,1)$ and $L(f,2)$ of the corresponding modular form $f$ and one constant. We also present related results concerning the rank o…
The Tannakian formalism allows to attach to any subvariety of an abelian variety an algebraic group in a natural way. The arising groups are closely related to moduli questions such as the Schottky problem, but their geometric interpretation is still mysterious. We show that for the theta divisor on the intermediate Jacobian of a cubic threefold, the Tannaka group is an exceptional group of type E_6. This is the first known exceptional case, and it suggests a connection with the monodromy of the Gauss map.
On threefolds with the smallest nontrivial monodromy group
本文利用伴随理论与SGA7命题，完整分类了光滑超平面截面上H^2单值群为Z/2Z的光滑三维流形，证实了Zak于1991年提出的分类可能性。
The quartic threefold is symplectically irrational
本文证明光滑四次三维流形是辛非有理的，即不能通过一系列辛爆破、爆缩和形变与射影空间相关联，从而恢复了Iskovskikh-Manin的经典代数非有理结果，证明中使用了Iritani的工作建立了沿辛爆破的量子上同调分解定理。
The weight-monodromy conjecture claims the coincidence of the weight filtration and the monodromy filtration, up to shift, on the $l$-adic étale cohomology of a proper smooth variety over a complete discrete valuation field. Although it has been proved in some cases, the case of dimension $\geq 3$ in mixed characteristic is still open so far. The aim of this paper is to give a proof of the weight-monodromy conjecture for a threefold which has a projective strictly semistable model such that, for each irreducible component of the special fiber, the Picard number is equal to the second $l$-adic …
Monodromy of Calabi-Yau threefold flops via grade restriction rule and their quantum Kahler moduli
本文基于等级限制规则和窗口类别，给出了Calabi-Yau三重域翻转相关Monodromy的精确表达式，并通过量子Kähler模空间的判别式进一步细化Monodromy形式，利用其与嵌套环链基本群的关系进行验证。
We analyze the link between the occurrence of massless B-type D-branes for specific values of moduli and monodromy around such points in the moduli space. This allows us to propose a classification of all massless B-type D-branes at any point in the moduli space of Calabi-Yau's. This classification then justifies a previous conjecture due to Horja for the general form of monodromy. Our analysis is based on using monodromies around points in moduli space where a single D-brane becomes massless to generate monodromies around points where an infinite number become massless. We discuss the various…
It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi-Yau threefolds coming from eight planes in $\mathbb{P}^3$ does {\em not} have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is natura…
In this paper we study properties of the locus of second type lines of a general cubic threefold and fourfold. By analysing the geometry of the Fano scheme of lines of the Fermat cubic fourfold and in particular giving an explicit description of the locus of second type lines, we deduce that the Voisin map is birational over the second type locus. For a general cubic threefold, by studying properties of the second type locus again, we compute that various natural geometric monodromy groups are the full symmetric group.
We show that some hypergeometric monodromy groups in Sp(4,Z) split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank 2. In particular, we show that the monodromy of the natural quotient of the Dwork family of quintic threefolds in P^{4} splits as Z*Z/5. As a consequence, for a smooth quintic threefold X we show that a certain group of autoequivalences of the bounded derived category of coherent sheaves is an Artin group of dihedral type.
A well-known theorem of Kulikov, Persson and Pinkham states that a degeneration of a family of K3-surfaces with trivial monodromy can be completed to a smooth family. We give a simple example that an analogous statement does not hold for Calabi-Yau threefolds.
The quintic-mirror family is a well-known one-parameter family of Calabi-Yau threefolds. A complete description of the global monodromy group of this family is not yet known. In this paper, we give a presentation of the global monodromy group in the general linear group of degree 4 over the ring of integers modulo 5.
It is well known since Noether that the gonality of a smooth plane curve of degree d>3 is d-1. Given a k-dimensional complex projective variety X, the most natural extension of gonality is probably the degree of irrationality, that is the minimum degree of a dominant rational map from X to $\mathbb{P}^k$. In this paper we are aimed at extending the assertion on plane curves to smooth hypersurfaces in $\mathbb{P}^n$ in terms of degree of irrationality. We prove that both surfaces in $\mathbb{P}^3$ and threefolds in $\mathbb{P}^4$ of sufficiently large degree d have degree of irrationality d-1, …
$E_6$-local systems from cubic threefolds
研究通过构造立方三项式三种fold模空间上的无穷多个局部系统，其代数单个群为例外群$E_6$，并基于阿贝尔 étale 覆盖的芬诺方案中位上同调得出主要贡献。
To a one-parameter family of Calabi-Yau threefolds, we can associate the extended period map by the log Hodge theory of Kato and Usui. In the present paper, we study the image of a maximally unipotent monodromy point under the extended period map. As an application, we prove the generic Torelli theorem for a large class of one-parameter families of Calabi-Yau threefolds.
A geometric Jacquet-Langlands correspondence for paramodular Siegel threefolds
本文研究仿模层级西格尔模三重fold的 Picard-Lefschetz 公式，证明权-单值猜想，并借助 Rapoport-Zink 一致化构造 GSp₄ 与定内形式间的几何 Jacquet-Langlands 对应，同时得到整数值结果与降层级定理。
We prove that the space of pairs $(X,l)$ formed by a real non-singular cubic hypersurface $X\subset P^4$ with a real line $l\subset X$ has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surface $F_\mathbb{R}(X)$ formed by real lines on $X$. For another interpretation we associate with each of the 18 components a well defined real deformation class of real non-singular plane quintic curves and show that this deformation class together wit…
Mumford defined a natural isomorphism between the intermediate jacobian of a conic-bundle over $P^2$ and the Prym variety of a naturally defined étale double cover of the discrminant curve of the conic-bundle. Clemens and Griffiths used this isomorphism to give a proof of the irrationality of a smooth cubic threefold and Beauville later generalized the isomorphism to intermediate jacobians of odd-dimensional quadric-bundles over $P^2$. We further generalize the isomorphism to the primitive cohomology of a smooth cubic hypersurface in $P^n$.
The F-theory vacuum constructed from an elliptic Calabi-Yau threefold with section yields an effective six-dimensional theory. The Lie algebra of the gauge sector of this theory and its representation on the space of massless hypermultiplets are shown to be determined by the intersection theory of the homology of the Calabi-Yau threefold. (Similar statements hold for M-theory and the type IIA string compactified on the threefold, where there is also a dependence on the expectation values of the Ramond-Ramond fields.) We describe general rules for computing the hypermultiplet spectrum of any F-…
Compact Complex Threefolds Fibred by Two-Dimensional Complex Tori
本文构造了两族以二维复环面为纤维的紧致复三维流形，计算了其拓扑不变量，并证明每个光滑六维流形上存在不可数多个非双全纯的复结构。
Extension of the Topological Abel-Jacobi Map for Cubic Threefolds
本文研究三次超曲面上偏斜线对定义的消没闭链，通过 Stein 引理构造了 72 重覆盖空间的紧致化，将拓扑 Abel-Jacobi 映射延拓至该紧致化，并证明其在基本群上的诱导映射为满射。
Consider the family of smooth cubic surfaces which can be realized as threefold-branched covers of $\mathbb{P}^{2}$, with branch locus equal to a smooth cubic curve. This family is parametrized by the space $\mathcal{U}_{3}$ of smooth cubic curves in $\mathbb{P}^{2}$ and each surface is equipped with a $\mathbb{Z}/3\mathbb{Z}$ deck group action. We compute the image of the monodromy map $ρ$ induced by the action of $π_{1}\left(\mathcal{U}_{3}\right)$ on the $27$ lines contained on the cubic surfaces of this family. Due to a classical result, this image is contained in the Weyl group $W\left(E_…
We study a certain family of determinantal quintic hypersurfaces in $\mathbb{P}^{4}$ whose singularities are similar to the well-studied Barth-Nieto quintic. Smooth Calabi-Yau threefolds with Hodge numbers $(h^{1,1},h^{2,1})=(52,2)$ are obtained by taking crepant resolutions of the singularities. It turns out that these smooth Calabi-Yau threefolds are in a two dimensional mirror family to the complete intersection Calabi-Yau threefolds in $\mathbb{P}^{4}\times\mathbb{P}^{4}$ which have appeared in our previous study of Reye congruences in dimension three. We compactify the two dimensional fam…
Matroids and the integral Hodge conjecture for abelian varieties
研究显示非常一般的极极化阿贝尔簇上曲线的上同调类是基本类的偶数倍，推翻了阿贝尔簇的整数霍奇猜想，并证明非常一般的立方三次曲面不稳理性。
For a projective variety $Z$ and for any integer $p$, define the $p$-th Néron-Severi group $NS_p(Z)$ of $Z$ as the image of the cycle map $A_{p}(Z)\to H_{2p}(Z; \mathbb{C})$. Now let $X\subset \Ps^{2m+1}$ ($m\geq 1$) be a projective variety of dimension $2m-1$, with isolated singularities, complete intersection of a smooth hypersurface of degree $k$, with a hypersurface of degree $n>max\{k, 2m+1\}$, and let $F$ be a general hypersurface of degree $n$ containing $X$. We prove that the natural map $NS_m(X)\to NS_m(F)$ is surjective, and that if $dim NS_m(F)=1$ then $dim NS_m(X)=1$. In particular…
Geometry of the mirror models dual to the complete intersection of two cubics
本文研究了双立方完全交集的镜模型几何，构造了自然的crepant解析，并分析了奇异纤维的Picard-Fuchs方程和极限混合Hodge结构。
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