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

TL;DR

We address the problem of extending Fujita's nonnegativity of the Δ-genus to connected polarized demi-normal schemes and derive a corresponding Noether-type inequality for KSBA stable log schemes. The approach develops a gluing framework via normalization and connecting subschemes, using Mayer-Vietoris-type arguments to establish $Δ(X,L) ≥ 0$ and to classify extremal cases as trees of components glued along hyperplanes. Key results include $Δ(X,L) ≥ 0$ for connected demi-normal $(X,L)$, $Δ(X, I(K_X+Λ)) ≥ 0$ for KSBA stable log schemes, and $Δ(X, K_X+Λ) ≥ 0$ when $I=1$, with a sharp equality case demonstrated by explicit constructions. These findings enhance the understanding of polarizations on singular varieties and have implications for the compactification of moduli spaces of varieties of general type.

Abstract

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 $I=1$.

Figures (1)

• Figure 1: $n=2$ case.

Theorems & Definitions (37)

• Theorem 1.1: see Theorem \ref{['mainthm']}
• Definition 2.1
• Remark 2.2
• Definition 3.1
• Remark 3.2
• Definition 3.3
• Remark 3.4
• Proposition 3.5
• Definition 3.6
• Definition 3.7