Positivity of $\Delta$-genera for connected polarized demi-normal schemes

TL;DR

This paper proves the non-negativity of the $ riangle$-genus for connected polarized demi-normal schemes and applies this to KSBA stable log schemes, demonstrating the sharpness of the inequality with explicit examples.

Contribution

It establishes the positivity of the $ riangle$-genus for a broad class of schemes and constructs examples showing the inequality's sharpness.

Findings

$ riangle$-genus $ riangle(X,rak L)\ge 0$ for connected polarized demi-normal schemes

$ riangle(X,I(K_X+rak ext{Lambda}))\ge 0$ for KSBA stable log schemes

Constructed examples with $I=1$ and $ riangle=0$ showing sharpness

Abstract

In this paper, we show that the $\Delta$-genus $\Delta(X,\mathcal{L})\ge 0$ for any connected polarized demi-normal scheme $(X,\mathcal{L})$. 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 $I=1$.