Editing Moduli space (section)

===Moduli of varieties===
In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. For instance, the higher-dimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties, such as the [[Siegel modular variety]]. This is the problem underlying [[Siegel modular form]] theory. See also [[Shimura variety]].

Using techniques arising out of the minimal model program, moduli spaces of varieties of general type were constructed by [[János Kollár]] and [[Nicholas Shepherd-Barron]], now known as KSB moduli spaces.<ref>J. Kollar. Moduli of varieties of general type, Handbook of moduli. Vol. II, 2013, pp. 131–157.</ref>

Using techniques arising out of differential geometry and birational geometry simultaneously, the construction of moduli spaces of [[Fano varieties]] has been achieved by restricting to a special class of [[K-stability of Fano varieties|K-stable]] varieties. In this setting important results about boundedness of Fano varieties proven by [[Caucher Birkar]] are used, for which he was awarded the 2018 [[Fields Medal|Fields medal]]. 

The construction of moduli spaces of Calabi-Yau varieties is an important open problem, and only special cases such as moduli spaces of [[K3 surface|K3 surfaces]] or [[Abelian varieties]] are understood.<ref>Huybrechts, D., 2016. ''Lectures on K3 surfaces'' (Vol. 158). Cambridge University Press.</ref>