An Introduction to Calabi-Yau Manifolds

TL;DR

The paper surveys the geometric foundations of Calabi–Yau manifolds, emphasizing the condition $c_1(M)=0$, Ricci-flat metrics guaranteed by the Calabi–Yau theorem, and the holonomy constraint $\mathrm{Hol}^0(g) \subset SU(m)$. It then details the construction and classification of complete intersection Calabi–Yau manifolds (CICYs), including large-scale enumerations of CY3 types and their Hodge diamonds computed via Lefschetz theory. The discussion of Hodge theory and mirror symmetry highlights the structure of $h^{p,q}$ for CY3 and the canonical swap $h^{1,1} \leftrightarrow h^{2,1}$ between mirror pairs, illustrated by the quintic and its mirror. Collectively, the work underscores the role of CY geometry in string theory compactifications and the rich interplay between curvature, topology, and complex geometry in this class of manifolds.

Abstract

The goal of this paper is to develop the theory of Courant algebroids with integrable para-Hermitian vector bundle structures by invoking the theory of Lie bialgebroids. We consider the case where the underlying manifold has an almost para-complex structure, and use this to define a notion of para-holomorphic algebroid. We investigate connections on para-holomorphic algebroids and determine an appropriate sense in which they can be para-complex. Finally, we show through a series of examples how the theory of exact para-holomorphic algebroids with a para-complex connection is a generalization of both para-Kähler geometry and the theory of Poisson-Lie groups.

Figures (1)

• Figure 1: A cross section of the Calabi–Yau quintic in $\mathbb{CP}^4$.

Theorems & Definitions (14)

• Definition 1.1
• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 3.1
• Definition 4.1
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3