Morphisms between Cremona groups, and characterization of rational varieties

作者：Cantat Serge^*

来源：Compositio Mathematica, 2014, 150(7): 1107-1124.

DOI：10.1112/S0010437X13007835

• Cremona groups
• rational varieties
• algebraic transformation groups

<jats:title>Abstract</jats:title><jats:p>We classify all (abstract) homomorphisms from the group<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007835_inline1" /><jats:tex-math>$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\sf PGL}_{r+1}(\mathbf{C})$</jats:tex-math></jats:alternatives></jats:inline-formula>to the group<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007835_inline2" /><jats:tex-math>${\sf Bir}(M)$</jats:tex-math></jats:alternatives></jats:inline-formula>of birational transformations of a complex projective variety<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007835_inline3" /><jats:tex-math>$M$</jats:tex-math></jats:alternatives></jats:inline-formula>, provided that<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007835_inline4" /><jats:tex-math>$r\geq \dim _\mathbf{C}(M)$</jats:tex-math></jats:alternatives></jats:inline-formula>. As a byproduct, we show that: (i)<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007835_inline5" /><jats:tex-math>${\sf Bir}(\mathbb{P}^n_\mathbf{C})$</jats:tex-math></jats:alternatives></jats:inline-formula>is isomorphic, as an abstract group, to<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007835_inline6" /><jats:tex-math>${\sf Bir}(\mathbb{P}^m_\mathbf{C})$</jats:tex-math></jats:alternatives></jats:inline-formula>if and only if<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007835_inline7" /><jats:tex-math>$n=m$</jats:tex-math></jats:alternatives></jats:inline-formula>; and (ii)<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007835_inline8" /><jats:tex-math>$M$</jats:tex-math></jats:alternatives></jats:inline-formula>is rational if and only if<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007835_inline9" /><jats:tex-math>${\sf PGL}_{\dim (M)+1}(\mathbf{C})$</jats:tex-math></jats:alternatives></jats:inline-formula>embeds as a subgroup of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X13007835_inline10" /><jats:tex-math>${\sf Bir}(M)$</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>

• 出版日期2014-7