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Register a new userBimeromorphic automorphisms groups of certain conic bundles
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We study the groups of biholomorphic and bimeromorphic automorphisms of conic bundles over certain compact complex manifolds of algebraic dimension zero.
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A group $G$ is called Jordan if there is a positive integer $J=J_G$ such that every finite subgroup $mathcal{B}$ of $G$ contains a commutative subgroup $mathcal{A}subset mathcal{B}$ such that $mathcal{A}$ is normal in $mathcal{B}$ and the index $[mathcal{B}:mathcal{A}] le J$ (V.L. Popov). In this paper we deal with Jordaness properties of the groups $Bir(X)$ of birational automorphisms of irreducible smooth projective varieties $X$ over an algebraically closed field of characteristic zero. It is known (Yu. Prokhorov - C. Shramov) that $Bir(X)$ is Jordan if $X$ is non-uniruled. On the other hand, the second named author proved that $Bir(X)$ is not Jordan if $X$ is birational to a product of the projective line and a positive-dimensional abelian variety. We prove that $Bir(X)$ is Jordan if (uniruled) $X$ is a conic bundle over a non-uniruled variety $Y$ but is not birational to a product of $Y$ and the projective line. (Such a conic bundle exists only if $dim(Y)ge 2$.) When $Y$ is an abelian surface, this Jordaness property result gives an answer to a question of Prokhorov and Shramov.
Let G be a Lie group, $T^*G$ its cotangent bundle with its natural Lie group structure obtained by performing a left trivialization of T^*G and endowing the resulting trivial bundle with the semi-direct product, using the coadjoint action of G on the dual space of its Lie algebra. We investigate the group of automorphisms of the Lie algebra of $T^*G$. More precisely, amongst other results, we fully characterize the space of all derivations of the Lie algebra of $T^*G$. As a byproduct, we also characterize some spaces of operators on G amongst which, the space J of bi-invariant tensors on G and prove that if G has a bi-invariant Riemannian or pseudo-Riemannian metric, then J is isomorphic to the space of linear maps from the Lie algebra of G to its dual space which are equivariant with respect to the adjoint and coadjoint actions, as well as that of bi-invariant bilinear forms on G. We discuss some open problems and possible applications.
We show that if a group automorphism of a Cremona group of arbitrary rank is also a homeomorphism with respect to either the Zariski or the Euclidean topology, then it is inner up to a field automorphism of the base-field. Moreover, we show that a similar result holds if we consider groups of polynomial automorphisms of affine spaces instead of Cremona groups.
In this paper we study principally polarized abelian varieties that admit an automorphism of prime order $p>2$. It turns out that certain natural conditions on the multiplicities of its action on the differentials of the first kind do guarantee that those polarized varieties are not jacobians of curves.
A conic bundle is a contraction $Xto Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov which predicts that, if $Xto Z$ is a conic bundle such that $X$ has canonical singularities and $Z$ is $mathbb{Q}$-Gorenstein, then $Z$ is always $frac{1}{2}$-lc, and the multiplicities of the fibers over codimension $1$ points are bounded from above by $2$. Both values $frac{1}{2}$ and $2$ are sharp. This is achieved by solving a more general conjecture of Shokurov on singularities of bases of lc-trivial fibrations of relative dimension $1$ with canonical singularities.
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