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

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

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