Deligne's conjecture and mirror symmetry

Wenzhe Yang

Published 2020-01-10Version 1

In this paper, we will study the connections between mirror symmetry and Deligne's conjecture on the values of the $L$-functions of critical pure motives. Using mirror symmetry, we will give an explicit method to compute Deligne's period for a smooth fiber in the deformation of a one-parameter Calabi-Yau threefold. We will give examples to show how this method works and express Deligne's period in terms of the classical periods of the threeform of Calabi-Yau threefolds and its derivative. We will also compute the Deligne's period of the Calabi-Yau threefolds studied in a recent paper by Candelas, de la Ossa, Elmi and van Straten, and verify that Deligne's conjecture is satisfied based on the numerical results in their paper.