Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Multiplier ideal
open-in-new
<h2 id="algebraic-geometry">Algebraic geometry</h2> <p>In algebraic geometry, the multiplier ideal of an effective Q {\displaystyle \mathbb {Q} } -<a href="/facts/Divisor_(algebraic_geometry)/V0QDypeo">divisor</a> measures singularities coming from the fractional parts of <i>D</i>. Multiplier ideals are often applied in tandem with vanishing theorems such as the <a href="/facts/Kodaira_vanishing_theorem/LzQ2n67K">Kodaira vanishing theorem</a> and the <a href="/facts/Kawamata%25E2%2580%2593Viehweg_vanishing_theorem/h4vv1a3x">Kawamata–Viehweg vanishing theorem</a>. </p><p>Let <i>X</i> be a smooth complex variety and <i>D</i> an effective Q {\displaystyle \mathbb {Q} } -divisor on it. Let μ : X ′ → X {\displaystyle \mu :X'\to X} be a log resolution of <i>D</i> (e.g., Hironaka's resolution). The multiplier ideal of <i>D</i> is </p> J ( D ) = μ ∗ O ( K X ′ / X − [ μ ∗ D ] ) {\displaystyle J(D)=\mu _{*}{\mathcal {O}}(K_{X'/X}-[\mu ^{*}D])} <p>where K X ′ / X {\displaystyle K_{X'/X}} is the relative canonical divisor: K X ′ / X = K X ′ − μ ∗ K X {\displaystyle K_{X'/X}=K_{X'}-\mu ^{*}K_{X}} . It is an ideal sheaf of O X {\displaystyle {\mathcal {O}}_{X}} . If <i>D</i> is integral, then J ( D ) = O X ( − D ) {\displaystyle J(D)={\mathcal {O}}_{X}(-D)} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Canonical_singularity/cCbkz2NR">Canonical singularity</a></li> <li><a href="/facts/Test_ideal/PJAYC8d0">Test ideal</a></li> <li><a href="/facts/Nadel_vanishing_theorem/HCuMb2rb">Nadel vanishing theorem</a></li></ul> <ul><li>Blickle, Manuel; Lazarsfeld, Robert (2004), <a href="http://www.msri.org/communications/books/Book51/contents.html">"An informal introduction to multiplier ideals"</a>, <i>Trends in commutative algebra</i>, Math. Sci. Res. Inst. Publ., vol. 51, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, pp. 87–114, <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.241.4916">10.1.1.241.4916</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FCBO9780511756382.004">10.1017/CBO9780511756382.004</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780521831956, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2132649">2132649</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:10215098">10215098</a></li> <li>Lazarsfeld, Robert (2009), "A short course on multiplier ideals", <i>2008 PCMI Lectures</i>, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0901.0651">0901.0651</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2009arXiv0901.0651L">2009arXiv0901.0651L</a></li> <li>Lazarsfeld, Robert (2004). <i>Positivity in algebraic geometry II</i>. Berlin: Springer-Verlag.</li> <li>Lipman, Joseph (1993), <a href="http://www.math.purdue.edu/~lipman/papers/polars.pdf">"Adjoints and polars of simple complete ideals in two-dimensional regular local rings"</a> (PDF), <i>Bulletin de la Société Mathématique de Belgique. Série A</i>, 45 (1): 223–244, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1316244">1316244</a></li> <li>Nadel, Alan Michael (1989), "Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature", <i><a href="/facts/Proceedings_of_the_National_Academy_of_Sciences/w75CThX0">Proceedings of the National Academy of Sciences of the United States of America</a></i>, 86 (19): 7299–7300, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1989PNAS...86.7299N">1989PNAS...86.7299N</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1073%2Fpnas.86.19.7299">10.1073/pnas.86.19.7299</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/34630">34630</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1015491">1015491</a>, <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC298048">298048</a>, <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/16594070">16594070</a></li> <li>Siu, Yum-Tong (2005), "Multiplier ideal sheaves in complex and algebraic geometry", <i>Science China Mathematics</i>, 48 (S1): 1–31, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0504259">math/0504259</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2005ScChA..48....1S">2005ScChA..48....1S</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02884693">10.1007/BF02884693</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2156488">2156488</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119163294">119163294</a></li></ul>