论文标题
在de rham-betti猜想周围
Around the de Rham-Betti conjecture
论文作者
论文摘要
在代数扩展的平滑投影品种$ x $上的de rham-betti类是理性数字的$ k $，是$ x $分析的betti共同班级的一个理性类，它可以通过$ x $ $ x $的$ x $比较时期的同学同位素化。 Grothendieck的时期猜想意味着De Rham-Betti类应该是代数。我们证明，椭圆曲线产物上的任何de rham-betti类都是代数。这是通过证明与De Rham-Betti对象相关的Tannakian Torsor并利用Wüstholz的分析亚组定理来实现的。 In the case of products of non-CM elliptic curves, we prove the stronger result that $\overline{\mathds{Q}}$-de Rham-Betti classes are $\overline \mathds{Q}$-linear combinations of algebraic classes by showing that the period comparison isomorphism generates the torsor of motivic periods.一个关键步骤包括建立具有$ \ overline \ mathds {q} $系数的分析子组定理的版本。最后，基于Deligne和André关于KUGA-SATAKE对应的结果，我们进一步表明，Hodge的第二个Hyper-Kähler品种的第二个共同体学组之间的任何De Rham-Betti等轴测图都是Hodge。作为两种应用，我们表明，关于已知变形类型的Hyper-Kähler品种的Codimension-2 de Rham-Betti类是Hodge，我们在$ \ overline \ mathds {q} $上获得了用于K3表面的全局de rham-betti torelli定理。
A de Rham-Betti class on a smooth projective variety $X$ over an algebraic extension $K$ of the rational numbers is a rational class in the Betti cohomology of the analytification of$X$ that descends to a class in the algebraic de Rham cohomology of $X$ via the period comparison isomorphism. The period conjecture of Grothendieck implies that de Rham-Betti classes should be algebraic. We prove that any de Rham-Betti class on a product of elliptic curves is algebraic. This is achieved by showing that the Tannakian torsor associated to a de Rham-Betti object is connected, and by exploiting the analytic subgroup theorem of Wüstholz. In the case of products of non-CM elliptic curves, we prove the stronger result that $\overline{\mathds{Q}}$-de Rham-Betti classes are $\overline \mathds{Q}$-linear combinations of algebraic classes by showing that the period comparison isomorphism generates the torsor of motivic periods. A key step consists in establishing a version of the analytic subgroup theorem with $\overline \mathds{Q}$-coefficients. Finally, building on results of Deligne and André regarding the Kuga-Satake correspondence, we further show that any de Rham-Betti isometry between the second cohomology groups of hyper-Kähler varieties, with second Betti number not 3, is Hodge. As two applications we show that codimension-2 de Rham-Betti classes on hyper-Kähler varieties of known deformation type are Hodge and we obtain a global de Rham-Betti Torelli theorem for K3 surfaces over $\overline \mathds{Q}$.
