Abstract
Relations among fundamental invariants play an important role in algebraic geometry. It is known that an $n$-dimensional variety of general type whose image of its canonical map is of maximal dimension, satisfies $\textrm{Vol} \geq 2 (p_{g} - n)$. In this article, we investigate the very interesting extremal situation of varieties with $\textrm{Vol} = 2(p_{g} - n)$, which we call Horikawa varieties for they are natural higher dimensional analogues of Horikawa surfaces.
We obtain a structure theorem for Horikawa varieties and explore their pluriregularity. We use this to prove optimal results on projective normality of pluricanonical linear systems. We study the fundamental groups of Horikawa varieties, showing that they are simply connected. We prove results on deformations of Horikawa varieties, whose implications on the moduli space make them the higher dimensional analogue of curves of genus 2.
Even though there are infinitely many families of Horikawa varieties in any given dimension $n$, we show that when the image of the canonical map is singular, the geometric genus of the Horikawa varieties is bounded by $n + 4$.
Funding Statement
The first author was partially supported by GRF grant of the University of Kansas. The first author is also grateful for the hospitality of the Departamento de Álgebra, Geometría y Topología of the Universidad Complutense de Madrid and for the support of grant MTM2015-65968-P during his visit. The second author was partially supported by the National Center for Theoretical Sciences and the Ministry of Science and Technology of Taiwan. The third author was partially supported by grant MTM2015-65968-P and by UCM research group 910772.
Citation
Purnaprajna BANGERE. Jungkai A. CHEN. Francisco Javier GALLEGO. "On higher dimensional extremal varieties of general type." J. Math. Soc. Japan 75 (3) 761 - 783, July, 2023. https://doi.org/10.2969/jmsj/88668866
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