Li Yi

Li Yi

I am a 3rd year PhD student majoring in birational geometry

Birational Geometry Notes: Around Moduli theory

2 minute read

Published:

In this part of the notes, we focus on the properties of moduli spaces. The guiding philosophy behind moduli theory is that

The moduli of a geometric object completely determine the object itself. In birational geometry, one of the central themes is that the geometry of a variety is largely governed by the curves and divisors it contains.

Part A. Basic Concepts in Moduli Theory

Note-A.1: A Brief Introduction to Stacks [upd 10.1]

Note-A.2: A Brief Introduction to Artin Stacks and Deligne–Mumford Stacks [upd 10.1]

Note-A.3: A Brief Introduction to Algebraic Spaces [upd 10.1]

Part B. Geometry of Moduli

Note-1: Dimension and Tangent Space of Moduli

Note-2: Boundedness of Moduli Spaces

Note-3: Properness and Separatedness of Moduli Spaces

Note-4: Projectivity and Positivity on Moduli Spaces [update 8.24]

Note-5: Irreducibility of Moduli Spaces

Note-6: K-stability of Moduli Spaces

Note-7: On Automorphism Groups

Part C. Singularities of Moduli

Note-1: Singularities on Moduli Spaces

Part D. Moduli of Curves and Surfaces

1. Construction of Teichmüller Space and Teichmüller Theorem

2. Moduli Space of Smooth Curves and Stable Curves [update 8.24]

3. Moduli Spaces for Surfaces of General Type

4. Moduli Spaces for Surfaces in $\mathbb{P}^3$

5. Semi-stable Reduction

Part E. Hodge Theory and Moduli

0. Gauss–Manin Connection, Griffiths Transversality, and Griffiths Curvature Formula

1. Local Torelli and Global Torelli

2. Betti Moduli, de Rham Moduli, and Dolbeault Moduli

3. Moduli Space of K3 Surfaces

4. Moduli Space of Abelian Varieties (ppav)

5. Calabi–Yau Moduli and Hyperbolicity

Part F. Theory of Hilbert Schemes and Chow Schemes

Part G. Moduli Techniques in Birational Geometry and Deformation Problems

1. Hilbert Scheme Techniques in Birational Geometry

2. Applications of Chow Varieties in Boundedness Problems

3. Applications of Douady Space/Barlet Chow Cycle Space in Analytic Deformation Problems

Part H. KSBA Moduli

1. Relative Canonical Sheaf, Kollár’s Package Theorem

2. Stable Families, Locally Stable Families, KSBA Morphisms

3. Base Change Properties, Flatness Criteria, Kollár’s Condition

4. Du Bois Singularities and CM Condition

5. Families of Divisors, Flatness Conditions, Mumford Divisors

6. Moduli of Stable Varieties

Part I. Theory of K-moduli of Fano Varieties

Part J. Moduli of Abelian Varieties and K3 Surfaces

In this part, we discuss further aspects of moduli of Abelian varieties and K3 surfaces.

Part K. Shafarevich Conjecture and Deformation Rigidity

Part J. Geometric Langlands and Mirror Symmetry

1. A Brief Introduction to D-modules

2. A Brief Introduction to Fourier–Mukai Transform

3. A Brief Introduction to Geometric Langlands

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The purpose of this series of notes is to discuss the recent paper Moishezon morphism by Professor Kollár in detail. Let us first briefly answer the question WHY Moishezon?

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