Some inequalities between \(K^ 3_ X\) and \(\chi({\mathcal O}_ X)\) for a threefold of general type
type: Item
instance of: scholarly article
title: Some inequalities between \(K^ 3_ X\) and \(\chi({\mathcal O}_ X)\) for a threefold of general type
zbMATH Open document ID: 0871.14032
DOI: 10.1007/PL00004595
author: Dong-Kwan Shin
published in: Mathematische Zeitschrift
publication date: 1997-07-03T00:00:00Z
review text: On a smooth minimal threefold of general type, there are inequalities between \(K_X^3\) and \(\chi ({\mathcal O}_X)\) as we have on a minimal surface of general type, namely \[ -{7\over 6} K_X^3-1 \leq\chi ({\mathcal O}_X) \leq- {1\over 72} K_X^3. \] The right inequality comes from Miyaoka's inequality. The left inequality comes from the inequality between some multiple of \(K_X^3\) and the dimension of the cohomology \(H^0 (X, {\mathcal O}_X (nK_X))\). These inequalities give some restrictions on the existence of smooth minimal threefolds of general type. It means that there are inequalities between \(c_1 \cdot c_2\) and \(c^3_1\).
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14F45
zbMATH DE Number: 1028180
zbMATH Keywords: canonical divisor
zbMATH Keywords: Euler characteristic
zbMATH Keywords: Chern class
zbMATH Keywords: minimal threefold of general type
zbMATH Keywords: Miyaoka's inequality
MaRDI profile type: Publication
full work available at URL: pl00004595
OpenAlex ID: W2012089389
description: scientific article; zbMATH DE number 1028180