This helps make toric types a particularly appealing test case for many constructions in complex geometry. Samples of toric types involve complex projective spaces, and bundles above them.
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A single may perhaps similarly outline an affine complex algebraic variety being a subset X ⊆ C n displaystyle Xsubseteq mathbb C ^ n
, and that is a non-negative integer counting the quantity of holes from the given compact Riemann surface area.
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Additionally, the additional framework of complex geometry makes it possible for, particularly in the compact setting, for global analytic results for being demonstrated with fantastic results, such as Shing-Tung Yau's proof on the Calabi conjecture, the Hitchin–Kobayashi correspondence, the nonabelian Hodge correspondence, and existence benefits for Kähler–Einstein metrics and consistent scalar curvature Kähler metrics. These benefits usually feed back into complex algebraic geometry, and for example a short while ago the classification of Fano manifolds working with K-stability has benefited greatly both from strategies in Assessment As well as in pure birational geometry.
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