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 will 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 theme is that the geometry of a variety is largely governed by the curves and divisors it contains.

Part A. Basic Conecpt in Moduli Theory

Note-1: A brief introduction to Stack [upd 10.1]

Note-2: A brief introduction to Artin Stack and Deligne-Mumford Stack [upd 10.1]

Note-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: Irreducible of moduli spaces

Note-6: K-stability of moduli spaces

Note-7: On automorphism group

Part C. Singularity of moduli

Note-1: Singularity on moduli spaces

Part D. Moduli of curves and surfaces

1. Construction of Teichmuller space and Teichmuller theorem

2. Moduli space of smooth curves, stable curves [update 8.24]

3. Moduli spaces for surfaces of general type

4. Moduli spaces for surfaces in P3

5. Semi-stable reduction

Part E. Hodge theory and moduli

0. Gauss Mannin connection, Griffith transversality and Griffith curvature formula

1. Local Torelli, Global Torelli

2. Betti Moduli, deRham 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 scheme and Chow schemes

Part G. Moduli techniques in birational geometry and deformation problems

1. Hilbert scheme technique in birational geometry

2. Applications of Chow variety in boundedness problems

3. Applications of Barlet-Chow cycle space in some analytic deformation problems

Part H. KSBA moduli

1. Relative canonical sheaf, Kollar’s package theorem

2. Stable family, locally stable family, KSB morphism

3. Base change properties, flatness criterion, Kollar’s condition

4. DuBois singularity and CM condition

5. Family of divisors, flatness condition, 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 will discuss more on moduli of Abelian varieties and K3 surfaces.

Part K. Shafarevich Conjecture and deformation rigidty

Part J. Geometric Langlands and Mirror symmetry

1. A brief introduction to D-module

2. A brief introduction to Fourier Mukai transform

3. A brief introduction to Geometric Langlands

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