On syzygies of Calabi-Yau varieties and varieties of general type
type: Item
instance of: scholarly article
title: On syzygies of Calabi-Yau varieties and varieties of general type
zbMATH Open document ID: 1407.14036
author: Wenbo Niu
published in: Advances in Mathematics
publication date: 2019-01-18T00:00:00Z
review text: In the paper under review, the author studies projective normality, \(N_{p}\) properties, and effective very ampleness for line bundles on a Calabi-Yau variety, as well as for pluricanonical bundles of a variety of general type. Let us recall that for \(X\) a nonsingular projective complex variety of dimension \(n \geq 1\), Fujita's conjecture predicts that the adjoint line bundle \(L_{d} = K_{X}+ dB\) is globally generated for \(d \geq n + 1\) and is very ample for \(d \geq n + 2\), where \(K_{X}\) is the canonical line bundle of \(X\) and \(B\) is an ample line bundle. It was suggested through years that that the conjecture could be extended to higher syzygies through \(N_{p}\) property. Specifically, suppose that \(L_d\) is globally generated and write \(S = \mathrm{Sym}(H^{0}(X, L_{d}))\). The section ring \[ R(L_{d}) = \bigoplus_{\ell \geq 0}H^{0}(X,\ell L_{d}) \] is naturally a finitely generated graded \(S\)-module and therefore admits a minimal graded free resolution \[ \dots \rightarrow E_{p} \rightarrow \dots \rightarrow E_{2} \rightarrow E_{1} \rightarrow E_{0} \rightarrow R(L_{d}) \rightarrow 0. \] Then \(L_{d}\) is said to satisfy (Property) \(N_{p}\) for \(p \geq 0\) if \(E_{0} = S\) and \(E_{i} = \oplus S(-i - 1)\) for \(1 \leq i \leq p\). The first main result of the paper is devoted to \(N_p\) properties for Calabi-Yau varieties. Recall that \(X\) is called Calabi-Yau if \(K_X \cong \mathcal{O}_{X}\) and \(h^1(\mathcal{O}_{X}) = 0\). Assume additionally that \(B\) is globally generated and denote by \(\phi_{B}\) the map associated to the complete linear system \(|B|\). A curve \(C\) cut out by \(n-1\) general global sections of \(B\) is called a curve section of \(B\). {Theorem A}. Let \(X\) be a Calabi-Yau variety of dimension \(n\geq 2\). Let \(B\) be a globally generated line bundle on \(X\) such that \(\dim \phi_{B}(X) = n\). Let \(C\) be a curve section of \(B\). When \(n\geq 3\), assume further that \(h^{2}(\mathcal{O}_{X}) = 0\). {\parindent=…
reviewed by: Piotr Pokora
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 14Q15
Mathematics Subject Classification ID: 14Q20
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 7003605
zbMATH Keywords: Calabi-Yau variety
zbMATH Keywords: variety of general type
zbMATH Keywords: syzygy
zbMATH Keywords: projective normality
MaRDI profile type: Publication
full work available at URL: j.aim.2018.12.002
OpenAlex ID: W2905312797
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Wikidata QID: Q128760572
DOI: 10.1016/J.AIM.2018.12.002
description: scientific article; zbMATH DE number 7003605