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

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.

63 pages, 15 figures; an older version of these notes was published in the Proceedings of the Modave Summer School in Mathematical Physics 2005

References in corpus (8)

• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• An SU(5) Heterotic Standard Model
• String Theory on Calabi-Yau Manifolds
• Chern-Simons Theory and Topological Strings
• The Space of Kähler metrics (II)
• Tri-linear Couplings in an Heterotic Minimal Supersymmetric Standard Model
• Geometric Transitions
• Toric Geometry and String Theory

Cited by in corpus (24)

• Global F-theory Models: Instantons and Gauge Dynamics
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• Toric Construction of Global F-Theory GUTs
• Heterotic Models from Vector Bundles on Toric Calabi-Yau Manifolds
• Rank-two 5d SCFTs from M-theory at isolated toric singularities: a systematic study
• A First Course on Twistors, Integrability and Gluon Scattering Amplitudes
• Toric Resolutions of Heterotic Orbifolds
• A short overview of the "Topological recursion"
• Box Graphs and Resolutions II: From Coulomb Phases to Fiber Faces
• Toric geometry and local Calabi-Yau varieties: An introduction to toric geometry (for physicists)
• Typicality, Black Hole Microstates and Superconformal Field Theories
• Orientifolds of Warped Throats from Toric Calabi-Yau Singularities
• M2-brane Flows and the Chern-Simons Level
• Lectures on the Calabi-Yau Landscape
• Mass Deformations of Unoriented Quiver Theories
• F-theory and Heterotic Duality, Weierstrass Models from Wilson lines
• Toric Geometry and String Theory
• Topological strings, quiver varieties and Rogers-Ramanujan identities
• Quotients of the conifold in compact Calabi-Yau threefolds, and new topological transitions
• Lectures on Naturalness, String Landscape and Multiverse
• On perturbative field theory and twistor string theory
• Szasz Analytic Functions and Noncompact Kähler Toric Manifolds
• Combinatorics in N = 1 Heterotic Vacua
• Holomorphic Lagrangian fibrations of toric hyperkahler manifolds