Positivity of Δ-genera for connected polarized demi-normal schemes

Abstract

In this paper, we show that the Δ-genus $\Delta (X,\mathcal{L})\ge 0$ for any connected polarized demi-normal scheme $(X,\mathcal{L})$ over an algebraically closed field k. As an application, we obtain $\Delta (X,I(K_X+\Lambda ))\ge 0$ for any KSBA stable log scheme $(X,\Lambda )$, where I is the Cartier index of $K_X+\Lambda$. We also construct examples of KSBA stable log schemes with I = 1 and $\Delta (X,K_X+\Lambda )=0$, which shows the inequality is sharp when I = 1.

KEYWORDS:

• Δ-genus
• Gorenstein KSBA stable log schemes
• inequality of Noether type
• KSBA stable schemes
• polarized demi-normal schemes

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 14C20
• 14J10

Acknowledgments

We would like to express our sincere gratitude to Prof. Jinxing Cai and Prof. Wenfei Liu, for their invaluable guidance and instruction. Additionally, we would like to extend our thanks to the anonymous reviewers for their insightful comments and suggestions. Their insightful comments and suggestions have been instrumental in improving this paper.