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Fano variety
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<p>In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a Fano variety, introduced by <a href="/facts/Gino_Fano/wMmxkVpM">Gino Fano</a> (<a href="/facts/Gino_Fano/wMmxkVpM">Fano</a> 1934, 1942), is an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a> that generalizes certain aspects of <a href="/facts/Complete_intersection/UoFKxRqC">complete intersections</a> of algebraic <a href="/facts/Hypersurface/cTvtLVsV">hypersurfaces</a> whose sum of degrees is at most the total dimension of the ambient <a href="/facts/Projective_space/7MSpA9cM">projective space</a>. Such complete intersections have important applications in geometry and <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, because they typically admit <a href="/facts/Rational_point/j5v4cxuE">rational points</a>, an elementary case of which is the <a href="/facts/Chevalley%25E2%2580%2593Warning_theorem/xjbRnJwq">Chevalley–Warning theorem</a>. Fano varieties provide an abstract generalization of these basic examples for which rationality questions are often still tractable. </p><p>Formally, a Fano variety is a <a href="/facts/Complete_algebraic_variety/4rhVhqAU">complete variety</a> <i>X</i> whose <a href="/facts/Anticanonical_bundle/nJKN8mpG">anticanonical bundle</a> <i>K</i>X* is <a href="/facts/Ample_line_bundle/XRQKvXGD">ample</a>. In this definition, one could assume that <i>X</i> is <a href="/facts/Smooth_scheme/Y76o7b9T">smooth</a> over a field, but the <a href="/facts/Minimal_model_program/f15RyS0J">minimal model program</a> has also led to the study of Fano varieties with various types of singularities, such as <a href="/facts/Canonical_singularity/cCbkz2NR">terminal</a> or <a href="/facts/Canonical_singularity/cCbkz2NR">klt</a> singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, and success has been found in constructing <a href="/facts/Moduli_space/9sqrdpwR">moduli spaces</a> of Fano varieties and proving the existence of <a href="/facts/K%25C3%25A4hler%25E2%2580%2593Einstein_metric/bHzK7qmU">Kähler–Einstein metrics</a> on them through the study of <a href="/facts/K-stability_of_Fano_varieties/P2y96Pu6">K-stability of Fano varieties</a>. </p>