Kawamata–Miyaoka type inequality for ℚ-Fano varieties with canonical singularities

Abstract Let X be an n -dimensional normal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℚ</m:mi> </m:math> {\mathbb{Q}} -factorial projective variety with canonical singularities and Picard number one such that X is smooth in codimension two, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi>K</m:mi> <m:mi>X</m:mi> </m:msub> </m:mrow> </m:math> {-K_{X}} is ample and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {n\geq 2} . We prove that X satisfies the following Kawamata–Miyaoka type inequality: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>c</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>n</m:mi> </m:msup> </m:mrow> <m:mo>&lt;</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:msub> <m:mi>c</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>⋅</m:mo> <m:msub> <m:mi>c</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> c_{1}(X)^{n}&lt;4c_{2}(X)\cdot c_{1}(X)^{n-2}. If additionally X is a threefold with terminal singularities, then a stronger inequality is also obtained.