This short note investigates a special case of the Miyaoka-Yau inequality, specifically <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$-c_2(X)\cdot c_1(X)\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>·</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, for an elliptic threefold X under the condition that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$K_X^{\otimes m}=f^*L$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>K</mi> <mi>X</mi> <mrow> <mo>⊗</mo> <mi>m</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mi>f</mi> <mo>∗</mo> </msup> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation> for some positive line bundle L on the base surface Y. We provide an explicit formula connecting this Chern number to the Weil-Petersson form <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\omega _{WP}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mrow> <mi mathvariant="italic">WP</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>: <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$-c_2(X)\cdot c_1(X)=\frac{6 [\omega _{WP}]\cdot L}{\pi m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>·</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>6</mn> <mo stretchy="false">[</mo> <msub> <mi>ω</mi> <mrow> <mi mathvariant="italic">WP</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>·</mo> <mi>L</mi> </mrow> <mrow> <mi>π</mi> <mi>m</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. Consequently, equality <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$-c_2(X)\cdot c_1(X)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>·</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> holds if and only if the j-invariant of the fibers is constant.

A short note on Miyaoka-Yau inequality of elliptic threefold

期刊论文
发表日期：

2026年5月28日

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Zhenqu Wang

摘要

This short note investigates a special case of the Miyaoka-Yau inequality, specifically $-c_2(X)\cdot c_1(X)\ge 0$ $- c_{2} (X) \cdot c_{1} (X) \geq 0$ , for an elliptic threefold X under the condition that $K_X^{\otimes m}=f^*L$ $K_{X}^{\otimes m} = f^{*} L$ for some positive line bundle L on the base surface Y. We provide an explicit formula connecting this Chern number to the Weil-Petersson form $\omega _{WP}$ $ω_{WP}$ : $-c_2(X)\cdot c_1(X)=\frac{6 [\omega _{WP}]\cdot L}{\pi m}$ $- c_{2} (X) \cdot c_{1} (X) = \frac{6 [ω_{WP}] \cdot L}{π m}$ . Consequently, equality $-c_2(X)\cdot c_1(X)=0$ $- c_{2} (X) \cdot c_{1} (X) = 0$ holds if and only if the j-invariant of the fibers is constant.