Thin monodromy in Sp(4). (March 2014)

• Record Type: Journal Article
• Authors: Brav, Christopher; Thomas, Hugh
• Is Part Of: Compositio mathematica. Volume 150:Number 3(2014:May)
• Page Start: 333
• Page End: 343
• Publication Date: 2014-03
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X13007550
• Languages: English
• ISSNs: 0010-437X

Abstract

<abstract> <title>Abstract</title> <p>We show that some hypergeometric monodromy groups in <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg5f5bm1wr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\rm Sp}(4, \mathbf{Z})$]]></tex-math></alternatives></inline-formula> split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg5f5bm27s" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$2$]]></tex-math></alternatives></inline-formula>. In particular, we show that the monodromy group of the natural quotient of the Dwork family of quintic threefolds in <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg5f5bm19c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathbf{P}^{4}$]]></tex-math></alternatives></inline-formula> splits as <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg5f5bm0xs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathbf{Z}\ast \mathbf{Z}/5\mathbf{Z}$]]></tex-math></alternatives></inline-formula>. As a consequence, for a smooth quintic threefold <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg5f5bm132" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$X$]]></tex-math></alternatives></inline-formula> we show that the group of autoequivalences <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg5f5bm1tn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$D^{b}(X)$]]></tex-math></alternatives></inline-formula> generated by the spherical twist along <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg5f5bm1s3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\mathcal{O}}_{X}$]]></tex-math></alternatives></inline-formula> and by tensoring with <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgg5f5bm14m" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\mathcal{O}}_{X}(1)$]]></tex-math></alternatives></inline-formula> is an Artin group of dihedral type.</p> </abstract>