Abundance theorem for log surfaces
We establish the abundance theorem for log surfaces without assuming $\mathbb{Q}$-factoriality or log canonicity.
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arXiv subjects
Nao Moriyama
Publications and source records attributed to Nao Moriyama.
Non-projective complete log canonical surfaces
We construct non-projective complete log canonical algebraic surfaces whose canonical divisors are semi-ample over an algebraically closed field of any characteristic other than the algebraic closure of a finite field. We provide a unified framework to construct such surfaces for any given non-negative Kodaira dimension, namely, zero, one, or two. Furthermore, we show that any complete log canonical algebraic surface with Kodaira dimension minus infinity is automatically projective. This projectivity result confirms that our construction covers all possible values for the Kodaira dimension of non-projective complete log canonical surfaces.
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A note on $\mathbb{Q}$-Gorenstein surfaces
We construct a normal projective $\mathbb{Q}$-Gorenstein surface over an algebraically closed field whose canonical ring is not finitely generated. Moreover, we provide a counterexample to the minimal model program for $\mathbb{Q}$-Gorenstein surfaces, which was previously unknown in positive characteristic.
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Remarks on the minimal model theory for log surfaces in the analytic setting
We discuss the relative log minimal model theory for log surfaces in the analytic setting. More precisely, we show that the minimal model program, the abundance theorem, and the finite generation of log canonical rings hold for log pairs of complex surfaces which are projective over complex analytic varieties.
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On a vanishing theorem for surfaces
We propose a new formulation of a vanishing theorem for surfaces. Although this vanishing theorem follows easily from the well-known Kawamata--Viehweg vanishing theorem, it turns out to be remarkably useful. In particular, it is sufficient for the minimal model theory of log surfaces, and it allows one to carry out both the minimal model program and the abundance theorem for log surfaces without invoking any of the deeper vanishing theorems.
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