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

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 $I=1$.