en>BattyBot: fixed CS1 errors: dates & General fixes using AWB (9832)

2014-01-05T02:36:04Z

fixed CS1 errors: dates & General fixes using AWB (9832)

New page

In mathematics, a '''Rosati involution''', named after [[Carlo Rosati]], is an involution of the rational [[endomorphism ring]] of an [[abelian variety]] induced by a polarization.

Let <math>A</math> be an [[abelian variety]], let <math>\hat A=\mathrm{Pic}^0(A)</math> be the [[dual abelian variety]], and for <math>a\in A</math>, let <math>T_a:A\to A</math> be the translation-by-<math>a</math> map, <math>T_a(x)=x+a</math>. Then each divisor <math>D</math> on <math>A</math> defines a map <math>\phi_D:A\to\hat A</math> via <math>\phi_D(a)=[T_a^*D-D]</math>. The map <math>\phi_D</math> is a polarization, i.e., has finite kernel, if and only if <math>D</math> is [[ample divisor|ample]]. The Rosati involution of <math>\mathrm{End}(A)\otimes\mathbb{Q}</math> relative to the polarization <math>\phi_D</math> sends a map <math>\psi\in\mathrm{End}(A)\otimes\mathbb{Q}</math> to the map <math>\psi'=\phi_D^{-1}\circ\hat\psi\circ\phi_D</math>, where <math>\hat\psi:\hat A\to\hat A</math> is the dual map induced by the action of <math>\psi^*</math> on <math>\mathrm{Pic}(A)</math>.

Let <math>\mathrm{NS}(A)</math> denote the [[Néron–Severi group]] of <math>A</math>. The polarization <math>\phi_D</math> also induces an inclusion <math>\Phi:\mathrm{NS}(A)\otimes\mathbb{Q}\to\mathrm{End}(A)\otimes\mathbb{Q}</math> via <math>\Phi_E=\phi_D^{-1}\circ\phi_E</math>. The image of <math>\Phi</math> is equal to <math>\{\psi\in\mathrm{End}(A)\otimes\mathbb{Q}:\psi'=\psi\}</math>, i.e., the set of endomorphisms fixed by the Rosati involution. The operation <math>E\star F=\frac12\Phi^{-1}(\Phi_E\circ\Phi_F+\Phi_F\circ\Phi_E)</math> then gives <math>\mathrm{NS}(A)\otimes\mathbb{Q}</math> the structure of a formally real [[Jordan algebra]].

==References==

*{{Citation | last1=Mumford | first1=David | author1-link=David Mumford | title=Abelian varieties | origyear=1970 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Tata Institute of Fundamental Research Studies in Mathematics | isbn=978-81-85931-86-9 | oclc=138290 | id={{MR|0282985}} | year=2008 | volume=5}}
*{{Citation | last1=Rosati | first1=Carlo | title=Sulle corrispondenze algebriche fra i punti di due curve algebriche. | language=Italian | doi=10.1007/BF02419717 | year=1918 | journal=Annali di Matematica Pura ed Applicata | volume=3 | issue=28 | pages=35–60}}

[[Category:Algebraic geometry]]

Circuit satisfiability problem - Revision history

en>BattyBot: fixed CS1 errors: dates & General fixes using AWB (9832)