로자티 비자발성

수학에서 카를로 로사티의 이름을 딴 로사티 비자발성은 양극화에 의해 유도된 아벨계 다양성의 이성적 내형성 고리를 비자발적으로 표현한 것이다.

Let ${\displaystyle A}$ be an abelian variety, let ${\displaystyle {\hat {A}}=\mathrm {Pic} ^{0}(A)}$ be the dual abelian variety, and for ${\displaystyle a\in A}$, let ${\displaystyle T_{a}:$$A\to A}$은(는) ${\displaystyle a}$ 지도$$, T ${\displaystyle a_{}}$( ${\displaystyle x}$)${\displaystyle }$= ${\displaystyle x}$+ ${\displaystyle T_{a}(x)=x+a}$.그런 다음 $A{\displaystyle }$ ${\displaystyle A}$의 각 divisor $D{\displaystyle }$ ${\displaystyle D}$은(는) 지도 $\varphi {\displaystyle {}_{}}$ ${\displaystyle D_{}}$: ${\displaystyle A}$→ ${\displaystyle a\stackrel{}{}}$ ${\displaystyle \stackrel{}{^}}$ ${\$를 정의한다$$.${\displaystyle A_{}}$$\to {\$${\displaystyle \mathrm{hat}}$${A$${\displaystyle \}}$=${\displaystyle }$[ ${\displaystyle {}_{}^{}}$ ${\displaystyle a_{}^{}}$ ${\displaystyle {}_{}^{}}$ ${\displaystyle {}_{}^{}}$- ${\displaystyle D}$ ${\displaystyle ]}$ ${\displaystyle \phi _{D}(a)=$$[T_{a}^{*}D-D$ 지도 $map{\displaystyle {}_{}}$ ${\displaystyle D_{}}$ ${\$는 D ${\displaystyle D}$가 넉넉하면 양극화다$$.The Rosati involution of ${\displaystyle \mathrm {End} (A)\otimes \mathbb {Q} }$ relative to the polarization ${\displaystyle \phi _{D}}$ sends a map ${\displaystyle \psi \in \mathrm {End} (A)\otimes \mathbb {Q} }$ to the map ${$$\displaystyle \psi '=\phi _{D}^{-1}\circ {\hat {\psi }}\circ \phi _{D}}$, where ${\displaystyle {\hat {\psi }}:{\hat {A}}\to {\hat {A}}}$ is the dual map induced by the action of ${\displaystyle \psi ^{*}}$ on ${\displaystyle \mathrm {Pic} (A)}$.

$N{\displaystyle \mathrm{}}$ ${\displaystyle \mathrm{S}}$( ${\displaystyle A}$) ${\displaystyle \mathrm {NS} (A)}$이(가) $A{\displaystyle }$ ${\displaystyle A}$의 Néron-Seberi 그룹을 나타내도록$$ 하십시오$.$The polarization ${\displaystyle \phi _{D}}$ also induces an inclusion ${\displaystyle \Phi :\mathrm {NS} (A)\otimes \mathbb {Q} \to \mathrm {End} (A)\otimes \mathbb {Q} }$ via ${\displaystyle \Phi _{E}=\phi _{D}^{-1}\circ \phi _{E}}$.The image of ${\displaystyle \Phi }$ is equal to ${\displaystyle \{\psi \in \mathrm {End} (A)\otimes \mathbb {Q} :\psi '=\psi \}}$, i.e., the set of endomorphisms fixed by the Rosati involution.작업 $E{\displaystyle }$ ${\displaystyle \star F}$ = ${\displaystyle 1\frac{\mathrm{}}{}}$ ${\displaystyle 2{\mathrm{}}^{}}$ ${\displaystyle {-}^{}}$ - ${\displaystyle 1^{}}$ ($$ ${\displaystyle E_{}}$ ${\displaystyle {}_{}}$ ∘${\displaystyle {}_{}}$+ F + ${\displaystyle {\mathrm{}}_{}}$ F ${\displaystyle {}_{}}$ ∘ ${\displaystyle E_{}}$) {\$displaystyle E\$star F={\$frac {1}{2}}\\\\$$Phi ^{-1}(\Phi _{E}\circ \Phi _{F}+\Phi _{F}\circ \Phi _{E})}$ then gives ${\displaystyle \mathrm {NS} (A)\otimes \mathbb {Q} }$ the structure of a formally real Jordan algebra.

참조

• Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290
• Rosati, Carlo (1918), "Sulle corrispondenze algebriche fra i punti di due curve algebriche.", Annali di Matematica Pura ed Applicata (in Italian), 3 (28): 35–60, doi:10.1007/BF02419717, S2CID 121620469