Clemens' conjecture: part II

Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology class

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.

An example on deformation of a pair

In this paper we give an example to show Clemens' conjecture is not a first order deformation problem.

量子K不变量与Gopakumar-Vafa不变量II. 三阶Calabi-Yau流形在零亏格的情况

Quantum K-invariants and Gopakumar-Vafa invariants II. Calabi-Yau threefolds at genus zero

本文研究了Calabi-Yau三阶流形上量子K理论与Gopakumar-Vafa不变量之间的关系，证明了零亏格量子K不变量与GV不变量之间的转换关系，并验证了虚拟Clemens猜想。

Finiteness of rational curves of degree 12 on a general quintic threefold

We prove the following statement, predicted by Clemens' conjecture: A generic quintic threefold contains only finitely many smooth rational curves of degree 12.

Rational curves of degree 16 on a general heptic fourfold

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.

Rational curves of degree 10 on a general quintic threefold

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.

6维球面与Clemens三维簇上的纤维化

Fibrations on the 6-sphere and Clemens threefolds

该研究针对满足一阶、二阶Betti数为零且欧拉非零的3维复流形Z，证明其不存在到2维复空间的满全纯映射，仅可能纤维化到曲线，还为S⁶的假想复结构提供了新限制。

Periods of families of curves in threefolds

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.

Hilbert scheme of rational curves on a generic quintic threefold

Let $X_0$ be a generic quintic threefold in projective space $\mathbf P^4$ over the complex numbers. For a fixed natural number $d$, let $R_d(X_0)$ be the open sub-scheme of the Hilbert scheme, parameterizing irreducible rational curves of degree $d$ on $X_0$. In this paper, we show that (1) $R_d(X_0)$ is smooth and of expected dimension, \par (2) Combining the Calabi-Yau condition on $X_0$, we further show that it consists of immersed rational curves. (3) Parts (1) and (2) imply a statement of Clemens' conjecture: if $C_0\in R_d(X_0)$ and $c_0:\mathbf P^1\to C_0$ is the normalization, the \pa…

Rational curves of degree 11 on a general quintic threefold

We prove that the incidence scheme of rational curves of degree 11 on quintic threefolds is irreducible. This implies a strong form of the Clemens conjecture in degree 11. Namely, on a general quintic threefold $F$ in $\mathbb{P}^4$, there are only finitely many smooth rational curves of degree 11, and each curve $C$ is embedded in $F$ with normal bundle $\mathcal{O}(-1) \oplus \mathcal{O}(-1)$. Moreover, in degree 11, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components on $F$.

Rational curves of degree at most 9 on a general quintic threefold

We prove the following form of the Clemens conjecture in low degree. Let $d\le9$, and let $F$ be a general quintic threefold in $\IP^4$. Then (1)~the Hilbert scheme of rational, smooth and irreducible curves of degree $d$ on $F$ is finite, nonempty, and reduced; moreover, each curve is embedded in $F$ with normal bundle $Ø(-1)\oplusØ(-1)$, and in $\IP^4$ with maximal rank. (2)~On $F$, there are no rational, singular, reduced and irreducible curves of degree $d$, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3)~On $F$, there are no connected, reduced and reducible …

Shimura Varieties and the Mumford-Tate conjecture, part I

We prove the Mumford-Tate conjecture for those abelian varieties over number fields, whose simple factors of their adjoint Mumford-Tate groups have over $\dbR$ certain (products of) non-compact factors. In particular, we prove this conjecture for the orthogonal case ($B_n$ and $D_n^{\dbR}$ Shimura types). We construct embeddings of Shimura varieties of (whose adjoints are of prescribed abelian type) into unitary Shimura varieties. We also prove two general theorems involving Frobenius tori of abelian varieties over number fields.