Mirror Symmetry and Rational Curves on Quintic Threefolds: a Guide for Mathematicians
نویسنده
چکیده
We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new q-expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the “mirror symmetry” phenomenon recently observed by string theorists.
منابع مشابه
Calabi-Yau Manifolds Over Finite Fields, II
We study ζ-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The ζ-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of these factors is intriguing since we have been unable to ‘see’ these curves in the geometry of the quin...
متن کامل2 5 A ug 1 99 9 Another way to enumerate rational curves with torus actions
and Parkes used the " string-theoretic " principle of mirror symmetry to predict the numbers of rational curves of any degree on a quintic threefold [7]. This prediction took mathematicians by surprise, as the best results at the time only counted rational curves of degree three or less. Since then, exciting new developments have led to mathematical proofs of these predictions and many other re...
متن کاملFrobenius Map for Quintic Threefolds
We calculate the matrix of the Frobenius map on the middle dimensional cohomology of the one parameter family that is related by mirror symmetry to the family of all quintic threefolds.
متن کاملCalabi-Yau Manifolds Over Finite Fields, I
We study Calabi–Yau manifolds defined over finite fields. These manifolds have parameters, which now also take values in the field and we compute the number of rational points of the manifold as a function of the parameters. The intriguing result is that it is possible to give explicit expressions for the number of rational points in terms of the periods of the holomorphic three-form. We show a...
متن کاملClemens’s Conjecture: Part Ii
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 deformations Bb such that the generic member in this family is non-deviated, but some special member is deviated. By the ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1993