Ziv Ran
We prove some lower bounds on certain twists of the canonical bundle of a codimension-2 subvariety of a generic hypersurface in projective space. In particular we prove that the generic sextic threefold contains no rational or elliptic curves and no nondegenerate curves of genus 2.
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This is the part II of our series of two papers, "Clemens' conjecture: part I", "Clemens' conjecture: part II". Continuing from part I, in this paper we turn our attention to general quintic threefolds. In a universal quintic threefold X, we construct a family of quasi-regular deformation B_b such that the generic member in this family is non-deviated, but some special member is deviated. By the result from part I, this is impossible unless there is no one parameter family of smooth rational curves in a generic quintic threefold.
This is a series of two papers in which we solve the Clemens conjecture: there are only finitely many smooth rational curves of each degree in a generic quintic threefold. In this first paper, we deal with a family of smooth Calabi-Yau threefolds f_εfor a small complex number ε. We give an geometric obstruction, deviated quasi-regular deformations B_b of c_ε, to a deformation of the rational curve c_εin a Calabi-Yau threefold f_ε.
The relative Clemens Conjectures for $\frac{1}{2}$-log Calabi-Yau threefolds
本文提出了1/2-log Calabi-Yau三维流形的相对Clemens猜想，并通过验证素Fano三维流形的框架，证明了相对Gromov-Witten理论的虚拟复杂性被抑制，从而实现了经典计数。
The Mirror Clemens-Schmid Sequence
本文提出联系光滑簇与补开集上同调的四项长正合序列，证明其兼容反常 Leray 滤过和混合 Hodge 结构，并通过 K3、Calabi–Yau 及五次三维簇退化验证其为 Clemens–Schmid 序列的镜像。
We introduce and study a likely condition that implies the following form of Clemens' conjecture in degrees $d$ between 10 and 24: given a general quintic threefold $F$ in complex $\IP^4$, the Hilbert scheme of rational, smooth and irreducible curves $C$ of degree $d$ on $F$ is finite, nonempty, and reduced; moreover, each $C$ is embedded in $F$ with balanced normal sheaf $Ø(-1)\oplusØ(-1)$, and in $\IP^4$ with maximal rank.
Infinitesimal invariants of mixed Hodge structures II: Log Clemens conjecture and log connectivity
本文研究混合Hodge结构的无穷小不变量，提出Log克莱门斯猜想并建立非消失性判据，推广了Calabi-Yau情形的对称性。
This is a sequel to our work in tropical Hodge theory. Our aim here is to prove a tropical analogue of the Clemens-Schmid exact sequence in asymptotic Hodge theory. As an application of this result, we prove the tropical Hodge conjecture for smooth projective tropical varieties which are rationally triangulable. This provides a partial answer to a question of Kontsevich who suggested the validity of the tropical Hodge conjecture could be used as a test for the validity of the Hodge conjecture.
In this paper we give an example to show Clemens' conjecture is not a first order deformation problem.
We prove the following statement, predicted by Clemens' conjecture: A generic quintic threefold contains only finitely many smooth rational curves of degree 12.
According to a conjecture of H. Clemens, the dimension of the space of rational curves on a general projective hypersurface should equal the number predicted by a naïve dimension count. In the case of a general hypersurface of degree 7 in $\mathbb{P}^5$, the conjecture predicts that the only rational curves should be lines. This has been verified by Hana and Johnsen for rational curves of degree at most 15. Here we extend their results to show that no rational curves of degree 16 lie on a general heptic fourfold.
Let $X$ be the product of two projective spaces and consider the general CICY threefold $Y$ in $X$ with configuration matrix $A$. We prove the finiteness part of the analogue of the Clemens' conjecture for such a CICY in low bidegrees. More precisely, we prove that the number of smooth rational curves on $Y$ with low bidegree and with nondegenerate birational projection is at most finite (even in cases in which positive dimensional families of degenerate rational curves are known).
We prove the "strong form" of the Clemens conjecture in degree 10. Namely, on a general quintic threefold F in P^4, there are only finitely many smooth rational curves of degree 10, and each curve is embedded in F with normal bundle O(-1)^2. Moreover, in degree 10, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components in F.
Clemens' conjecture states that the the number of rational curve in a generic quintic threefold is finite. If it is false we prove that certain periods of rational curves in such a quintic threefold must vanish. Our method is based on a generalization of a proof of Max Noether's theorem using infinitesimal variation of Hodge structures and its reformulation in terms of integrals and Gauss-Manin connection.
Generalised Cone Conjecture, I: Beyond Calabi--Yau
该研究为系列论文的第一篇，旨在证明曲面上的广义锥猜想，属于数学代数几何领域的相关工作。
We survey three results on syzygies of curves beyond Green's conjecture, with a particular emphasis on drawing connections between the study of syzygies and other topics in moduli theory.
We prove an analogue for holonomic DQ-modules of the codimension-three conjecture for microdifferential modules recently proved by Kashiwara and Vilonen. Our result states that any holonomic DQ-module having a lattice extends uniquely beyond an analytic subset of codimension equal to or larger than three in a Lagrangian subvariety containing the support of the DQ-module.
Singularities on Fano fibrations and beyond
本文研究Fano型等纤维化的奇点问题，证明了McKernan、Shokurov及作者提出的相关猜想，推导出有理连通簇猜想的变体，给出多个代数几何问题的应用，并证明了控制纤维化纤维重数的关键一般性结果。
Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves II: Beyond Riemann Hypothesis
本文提出不依赖黎曼假设的新方法，利用更强渐近界建立椭圆曲线高阶Zeta函数的Sato-Tate定律并表述Murmuration现象。
Existence of global Néron models beyond semi-abelian varieties
本文证明了完美剩余域情形下非半阿贝尔代数群的Bosch-Lütkebohmert-Raynaud猜想，给出了非完美剩余域的反例，并分类了完美剩余域上的单有理有界幂幺群。
We introduce an analogue of the Novikov Conjecture on higher signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain "higher Todd genera" are birational invariants. This implies birational invariance of certain extra combinations of Chern classes (beyond just the classical Todd genus) in the case of varieties with large fundamental group (in the topological sense). We prove the conjecture under the assumption of the "strong Novikov Conjecture" for the fundamental group, which is known to be correct for many groups …
Zilber-Pink in $Y(1)^n$: Beyond multiplicative degeneration
在Zilber-Pink框架下，通过假设曲线与边界的相交条件，证明了Y(1)^n中曲线的不可能交点的大伽罗瓦轨道猜想，并得到Zilber-Pink猜想的新情形。
We discuss conjectures following from the attractor mechanism in type II string theory about the possible Chern classes of stable holomorphic vector bundles on Calabi-Yau threefolds. In particular, we give sufficient conditions for Chern classes to correspond to stable bundles.
We prove that if $X \to Y$ is a (geometrically) regular morphism of Noetherian schemes, then from a Nisnevich-local perspective, the Gersten complex for Quillen $K$-theory on $X$ becomes acyclic in degrees beyond the Krull dimension of $Y$. Using our methods, we also reduce the general Gersten conjecture for regular, unramified local rings to the case of a discrete valuation ring which is essentially smooth over $\mathbb{Z}$. We apply our results to the the theory of algebraic cycles --- globally to obtain relative versions of Bloch's Formula and locally to address the Claborn-Fossum Conjectur…
Consider a simplicial complex that allows for an embedding into $\mathbb{R}^d$. How many faces of dimension $\frac{d}{2}$ or higher can it have? How dense can they be? This basic question goes back to Descartes' "Lost Theorem" and Euler's work on polyhedra. Using it and other fundamental combinatorial problems, we introduce a version of the Kähler package beyond positivity, allowing us to prove the hard Lefschetz theorem for toric varieties (and beyond) even when the ample cone is empty. A particular focus lies on replacing the Hodge-Riemann relations by a non-degeneracy relation at torus-inva…
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