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Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
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Consequence of the failure of Nagata's conjecture
A modern version of the Nagata's conjecture says that
$$
L_{N,t}:=f_{N}^{*}(-K_{\mathbb{P}^{2}})-t\sum_{j=1}^{N}E_{j}
$$
is Ample for any $t<\frac{3}{\sqrt{N}}$, where $f_{N}:Y_{N}\to \mathbb{P}^{2}$ …