Calabi-Yau geometry and mirror transforms of the Hodge conjecture

自20世纪90年代中期以来，S数学家证明了将复几何与Calabi-Yau流形对的辛几何联系起来的更强、更完整的“镜像定理”。镜像对称性允许一个流形上的复杂几何中的问题被其镜像上的辛几何中等价的、重新表述的问题所取代。通过这种方式，一方面棘手的问题可能会让另一方得到一个简单的答案。 霍奇猜想被描述为“将先验计算转化为代数计算的隐喻”。复代数几何中最重要的猜想远不止于此，但它只是涉及代数循环的一系列相关猜想中的一个。 我们最近在微观黑洞和L函数的特定值方面取得了惊人的结果，这些量是数论中一些最深的猜想的核心。这项工作建立了镜像变换对代数圈的算术结构的敏感性。在此基础上，以及模形式、K3曲面和环面几何的相关工作，我们将应用镜像对称来解决Hodge猜想的几个变体。通过“透过镜子”观察Calabi-Yau几何的最新结果，我们可以以全新的方式建立Hodge猜想的结果。

Since the mid 1990's mathematicians have proved stronger and more complete "mirror theorems" relating the complex geometry to the symplectic geometry of Calabi-Yau manifold pairs. Mirror symmetry permits a problem in complex geometry on one manifold to be replaced by equivalent, reformulated problems in symplectic geometry on its mirror. In this way intractable questions on one side may admit a simple answer on the other. The Hodge Conjecture has been described as "a metaphor for transforming transcendental computations into algebraic ones." The most important conjecture in complex algebraic geometry is much more than this, and yet it is merely one of a host of related conjectures involving algebraic cycles. We have recently obtained striking results relating microscopic black holes and special values of L-functions, quantities at the heart of some of the deepest conjectures in number theory. This work establishes the sensitivity of the mirror transform to the arithmetic structure of algebraic cycles. Building on this foundation, and also related work on modular forms, K3 surfaces, and toric geometry, we will apply mirror symmetry to tackle several variants of the Hodge Conjecture. By looking at state-of-the-art results on Calabi-Yau geometry "through the looking glass", we can establish consequences of the Hodge Conjecture in entirely new ways.