A Superficial Working Guide to Deformations and Moduli

Abstract

This is the first part of a guide to deformations and moduli, especially viewed from the perspective of algebraic surfaces (the simplest higher dimensional varieties). It contains also new results, regarding the question of local homeomorphism between Kuranishi and Teichmueller space, and a survey of new results with Ingrid Bauer, concerning the discrepancy between the deformation of the action of a group G on a minimal models S, respectively the deformation of the action of G on the canonical model X. Here Def(S) maps properly onto Def(X), but the same does not hold for pairs: Def(S,G) does not map properly onto Def(X,G). Indeed the connected components of Def(S), in the case of tertiary Burniat surfaces, only map to locally closed sets. The last section contains anew result on some surfaces whise Albanese map has generic degree equal to 2.

56 pages, revision to appear in the Handbook of Moduli, in honour of David Mumford, to be published by International press, editors Gavril Farkas and Ian Morrison. The former theorem 29 on moduli spaces for minimal surfaces has been corrected

References in corpus (7)

• A simply connected surface of general type with p_g=0 and K^2=2
• Geometry of nilmanifolds with left-invariant complex structure and deformations in the large
• Lie-algebra Dolbeault cohomology and small deformations of nilmanifolds
• Burniat surfaces II: secondary Burniat surfaces form three connected components of the moduli space
• Symplectic structures of algebraic surfaces and deformation
• Surfaces with p_g=q=2, K^2=6 and Albanese map of degree 2
• Teichmueller space via Kuranishi families

Cited by in corpus (20)

• Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratum
• Surfaces with p_g=q=2, K^2=6 and Albanese map of degree 2
• On the existence of universal families of marked irreducible holomorphic symplectic manifolds
• Contraction centers in families of hyperkahler manifolds
• Teichmuller space for hyperkahler and symplectic structures
• Kobayashi pseudometric on hyperkahler manifolds
• A new family of surfaces with $p_g=q=2$ and $K^2=6$ whose Albanese map has degree $4$
• Deformations of Dolbeault cohomology classes
• Families of Lagrangian fibrations on hyperkaehler manifolds
• Ergodic complex structures on hyperkahler manifolds
• Teichmuller spaces, ergodic theory and global Torelli theorem
• Automorphisms of surfaces of general type with q>=2 acting trivially in cohomology
• Deformations of symplectic groupoids
• Uniformization of compact complex manifolds by Anosov representations
• On the classification of surfaces of general type with $p_g=q=2$
• On topologically trivial automorphisms of compact Kähler manifolds and algebraic surfaces
• A note on the automorphism group of a compact complex manifold
• Complex structures, moment maps, and the Ricci form (Extended Version)
• Cyclic covers of Stable curves and their moduli spaces
• On the obstructions of deforming vector forms