Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Vincent Bouchard

TL;DR

These lectures provide a concise bridge between differential and algebraic approaches to complex geometry, Calabi–Yau manifolds, and toric geometry, with emphasis on concrete computations and string-theory applications. They develop a bundle-centric view of complex geometry, establish key results on Kähler and Calabi–Yau structures (including Yau’s solution to the Calabi conjecture), and then deploy toric methods to construct and analyze Calabi–Yau manifolds both as hypersurfaces in toric varieties and as local noncompact geometries. The notes culminate with explicit examples—the quintic in ${\mathbb{C}P^4}$ and the Tian–Yau manifold—alongside toric-diagram and reflexive-polytope techniques, and they connect these mathematical structures to mirror symmetry and topological string considerations. Overall, the material offers a compact, application-oriented primer for mathematical physics, illuminating how topological invariants, holonomy, and polyhedral data drive the geometry underlying string compactifications.

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.

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

TL;DR

These lectures provide a concise bridge between differential and algebraic approaches to complex geometry, Calabi–Yau manifolds, and toric geometry, with emphasis on concrete computations and string-theory applications. They develop a bundle-centric view of complex geometry, establish key results on Kähler and Calabi–Yau structures (including Yau’s solution to the Calabi conjecture), and then deploy toric methods to construct and analyze Calabi–Yau manifolds both as hypersurfaces in toric varieties and as local noncompact geometries. The notes culminate with explicit examples—the quintic in and the Tian–Yau manifold—alongside toric-diagram and reflexive-polytope techniques, and they connect these mathematical structures to mirror symmetry and topological string considerations. Overall, the material offers a compact, application-oriented primer for mathematical physics, illuminating how topological invariants, holonomy, and polyhedral data drive the geometry underlying string compactifications.

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.

Paper Structure

This paper contains 45 sections, 6 theorems, 107 equations, 15 figures.

Table of Contents

  1. Complex geometry
  2. Complex manifolds
  3. Tensors on complex manifolds
  4. Exterior forms on complex manifolds
  5. Cohomology
  6. Chern classes
  7. Properties of Chern classes --- and a digression on exact sequences
  8. Chern character
  9. Holomorphic vector bundles
  10. Holomorphic line bundles
  11. Refined Dolbeault cohomology groups
  12. Divisors and line bundles
  13. Kähler geometry
  14. Kähler manifolds
  15. Forms on Kähler manifolds
  16. ...and 30 more sections

Key Result

Proposition 2.8

Let $M$ be a complex manifold of real dimension $2m$, with a Kähler metric $g$. The holonomy group of $M$ is contained in $U(m)$.

Figures (15)

  • Figure 1: Pictorial representation of a complex manifold, according to definition \ref{['d:compl1']}.
  • Figure 2: Schematic visualization of exact sequences. The vertical lines represent the vectors spaces, the dots represent the zeroes, and the dashed lines represent the maps. This pictorial representation helped me visualize the condition that ${\rm Im} (\alpha_k) = {\rm Ker} (\alpha_{k+1})$ for all $k$. And from this picture it is easy to see that giving a short exact sequence, as in the second picture, is equivalent to saying that $A \subseteq B$ and $C = B/A$.
  • Figure 3: Pictorial representation of a holomorphic vector bundle.
  • Figure 4: Pictorial representation of a holomorphic map of complex manifolds.
  • Figure 5: Representation of the normal bundle of a one-dimensional hypersurface $X$ in ${\mathbb C} {\mathbb P}^2$, which can be understood as the quotient $N_X = \frac{T^{(1,0)}{\mathbb C} {\mathbb P}^2|_X}{T^{(1,0)}X}$.
  • ...and 10 more figures

Theorems & Definitions (50)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Example 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 40 more