Moduli and deformations
-
Klaus Altmann
Abstract
Moduli spaces are algebraic objects parametrizing the entire set of algebraic gadgets of a certain type (e.g. curves, vector bundles, singularities). However, to gain enumerative information by using a meaningful intersection theory, it is necessary to consider compactifications of these moduli spaces. Doing so, an essential point is to provide a modular interpretation of the new objects on the boundary - and this can be understood from two, rather opposite viewpoints: On the one hand, we will study degenerations of classical objects, and on the other, we might start with singular, sometimes rather combinatorial objects and look for deformations, e.g. smoothings. In the present survey we are going to explain these general approaches, and we present concrete results within the context of moduli of curves and abelian varieties, toric and spherical varieties.
Abstract
Moduli spaces are algebraic objects parametrizing the entire set of algebraic gadgets of a certain type (e.g. curves, vector bundles, singularities). However, to gain enumerative information by using a meaningful intersection theory, it is necessary to consider compactifications of these moduli spaces. Doing so, an essential point is to provide a modular interpretation of the new objects on the boundary - and this can be understood from two, rather opposite viewpoints: On the one hand, we will study degenerations of classical objects, and on the other, we might start with singular, sometimes rather combinatorial objects and look for deformations, e.g. smoothings. In the present survey we are going to explain these general approaches, and we present concrete results within the context of moduli of curves and abelian varieties, toric and spherical varieties.
Chapters in this book
- Frontmatter I
- Contents V
- Introduction VII
- Algebraic K-theory, assembly maps, controlled algebra, and trace methods 1
- Lorentzian manifolds with special holonomy – Constructions and global properties 51
- Contributions to the spectral geometry of locally homogeneous spaces 69
- On conformally covariant differential operators and spectral theory of the holographic Laplacian 90
- Moduli and deformations 116
- Vector bundles in algebraic geometry and mathematical physics 150
- Dyson–Schwinger equations: Fix-point equations for quantum fields 186
- Hidden structure in the form factors of N = 4 SYM 197
- On regulating the AdS superstring 221
- Constraints on CFT observables from the bootstrap program 245
- Simplifying amplitudes in Maxwell-Einstein and Yang-Mills-Einstein supergravities 266
- Yangian symmetry inmaximally supersymmetric Yang-Mills theory 288
- Wave and Dirac equations on manifolds 324
- Geometric analysis on singular spaces 349
- Singularities and long-time behavior in nonlinear evolution equations and general relativity 417
- Index 491
Chapters in this book
- Frontmatter I
- Contents V
- Introduction VII
- Algebraic K-theory, assembly maps, controlled algebra, and trace methods 1
- Lorentzian manifolds with special holonomy – Constructions and global properties 51
- Contributions to the spectral geometry of locally homogeneous spaces 69
- On conformally covariant differential operators and spectral theory of the holographic Laplacian 90
- Moduli and deformations 116
- Vector bundles in algebraic geometry and mathematical physics 150
- Dyson–Schwinger equations: Fix-point equations for quantum fields 186
- Hidden structure in the form factors of N = 4 SYM 197
- On regulating the AdS superstring 221
- Constraints on CFT observables from the bootstrap program 245
- Simplifying amplitudes in Maxwell-Einstein and Yang-Mills-Einstein supergravities 266
- Yangian symmetry inmaximally supersymmetric Yang-Mills theory 288
- Wave and Dirac equations on manifolds 324
- Geometric analysis on singular spaces 349
- Singularities and long-time behavior in nonlinear evolution equations and general relativity 417
- Index 491
