Lectures on complex geometry, Calabi-Yau manifolds and toric geometry

TL;DR

This paper provides an introductory overview of complex geometry, Calabi-Yau manifolds, and toric geometry, focusing on their definitions, properties, and methods for constructing Calabi-Yau manifolds within toric varieties.

Contribution

It offers a comprehensive introduction to key concepts and construction techniques in complex and toric geometry relevant to Calabi-Yau manifolds.

Findings

Clarifies definitions of Calabi-Yau manifolds

Explains construction of Calabi-Yau as hypersurfaces in toric varieties

Introduces local toric Calabi-Yau threefolds

Abstract

These are introductory lecture notes on complex geometry, Calabi-Yau manifolds and toric geometry. We first define basic concepts of complex and Kahler geometry. We then proceed with an analysis of various definitions of Calabi-Yau manifolds. The last section provides a short introduction to toric geometry, aimed at constructing Calabi-Yau manifolds in two different ways; as hypersurfaces in toric varieties and as local toric Calabi-Yau threefolds. These lecture notes supplement a mini-course that was given by the author at the Modave Summer School in Mathematical Physics 2005, and at CERN in 2007.