Other methods of treating moduli problems. Artin’s method of algebraic stacks. Griffiths’s method of period maps

1977 ◽  
pp. 93-116
Author(s):  
Herbert Popp
Keyword(s):  
Algebraic Stacks ◽  
Moduli Problems

Representability of Moduli Problems

2013 ◽  
Author(s):  
Kai-Wen Lan
Keyword(s):  
Invariant Theory ◽  
Deformation Theory ◽  
Geometric Invariant Theory ◽  
Geometric Invariant ◽  
Algebraic Stacks ◽  
Formal Smoothness ◽  
Abelian Schemes ◽  
Moduli Problems

This chapter elaborates on the representability of the moduli problems defined in the previous chapter. The treatment here is biased towards the prorepresentability of local moduli and Artin's criterion of algebraic stacks. The geometric invariant theory or the theory of Barsotti–Tate groups has been set aside: the argument is very elementary and might be considered outdated by the experts in this area. The chapter, however, discusses the Kodaira–Spencer morphisms of abelian schemes with PEL structures, which are best understood via the study of deformation theory. It also considers the proof of the formal smoothness of local moduli functors, illustrating how the linear algebraic assumptions are used.

scholarly journals STEENROD OPERATIONS ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC STACKS

2021 ◽  
pp. 1-48
Author(s):  
Federico Scavia
Keyword(s):  
Finite Groups ◽  
Reductive Groups ◽  
Orthogonal Groups ◽  
De Rham Cohomology ◽  
Algebraic Stacks ◽  
Steenrod Operations ◽  
De Rham

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .

Algebraic stacks

2001 ◽  
Vol 111 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Tomás L Gómez
Keyword(s):  
Algebraic Stacks

The pro-étale topology for algebraic stacks

2020 ◽  
Vol 48 (9) ◽  
pp. 3761-3770
Author(s):  
Chang-Yeon Chough
Keyword(s):  
Algebraic Stacks

scholarly journals Geometricity for derived categories of algebraic stacks

2016 ◽  
Vol 22 (4) ◽  
pp. 2535-2568 ◽  
Author(s):  
Daniel Bergh ◽  
Valery A. Lunts ◽  
Olaf M. Schnürer
Keyword(s):  
Derived Categories ◽  
Algebraic Stacks

scholarly journals Moduli of Bridgeland semistable objects on 3-folds and Donaldson–Thomas invariants

2019 ◽  
Vol 2019 (747) ◽  
pp. 175-219 ◽  
Author(s):  
Dulip Piyaratne ◽  
Yukinobu Toda
Keyword(s):  
Finite Type ◽  
Equivalent Form ◽  
Stability Conditions ◽  
Weak Stability ◽  
Algebraic Stacks ◽  
Moduli Stacks ◽  
Étale Quotients

Abstract In this paper we show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are quasi-proper algebraic stacks of finite type if they satisfy the Bogomolov–Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson–Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi–Yau 3-folds satisfying the BG inequality conjecture, for example on étale quotients of abelian 3-folds.

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