Jordan property for automorphism groups of compact varieties

Following a recent work of Oguiso, we calculate explicitly the groups of automorphisms and birational automorphisms on a Calabi-Yau manifold with Picard number two. When the group of birational automorphisms is infinite, we prove that the Cone conjecture of Morrison and Kawamata holds.

A famous problem in birational geometry is to determine when the birational automorphism group of a Fano variety is finite. The Noether–Fano method has been the main approach to this problem. The purpose of this paper is to give a new approach to the problem by showing that in every positive characteristic, there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphism group. To do this, we prove that these varieties admit ample and birationally equivariant line bundles. Our result applies the differential forms that Kollár produces on $p$-cyclic covers in characteristic $p > 0$.

We prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of these varieties are equal, and they are non-rational. The proof is based on the techniques of the method of maximal singularities, the generalized 4n2-inequality for complete intersection singularities and the technique of hypertangent divisors.

For a Zariski general (regular) hypersurface V of degree M in the (M+1)-dimensional projective space, where Mgeqslant 16, with at most quadratic singularities of rank geqslant 13, we give a complete description of the structures of rationally connected (or Fano-Mori) fibre space: every such structure over a positive-dimensional base is a pencil of hyperplane sections. This implies, in particular, that V is non-rational and its groups of birational and biregular automorphisms coincide: mathrm{Bir} V = mathrm{Aut} V. The set of non-regular hypersurfaces has codimension at least frac{1}{2}(M-11)(M-10)-10 in the natural parameter space.

A general Fano double hypersurface of index 1 is a double cover branched over a smooth divisor , here is proved to be birationally superrigid; in particular, such a hypersurface admits no non-trivial structures of a fibration into uniruled varieties, and it is non-rational. Its groups of birational and biregular automorphisms coincide.

A century ago, Camille Jordan proved that the complex general linear group ${\rm GL}_n(\Bbb{C})$ has the Jordan property: there is a Jordan constant ${\rm C}_n$ such that every finite subgroup $H\le{\rm GL}_n(\Bbb{C})$ has an abelian subgroup $H_1$ of index $[H:H_1]\le{\rm C}_n$. We show that every connected algebraic group $G$ (which is not necessarily linear) has the Jordan property with the Jordan constant depending only on $\dim G$, and that the full automorphism group ${\rm Aut}(X)$ of every projective variety $X$ has the Jordan property.

It has been recently shown by Meng and Zhang that the full automorphism group [Formula: see text] is a Jordan group for all projective varieties in arbitrary dimensions. The aim of this paper is to show that the full automorphism group [Formula: see text] is, in fact, a Jordan group even for all normal compact Kähler varieties in arbitrary dimensions. The meromorphic structure of the identity component of the automorphism group and its Rosenlicht-type decomposition play crucial roles in the proof.

Let $X$ be a complex projective variety. Suppose that the group of birational automorphisms of $X$ contains finite subgroups isomorphic to $(\mathbb{Z}/N\mathbb{Z})^r$ for $r$ fixed and $N$ arbitrarily large. We show that $r$ does not exceed $2\dim(X)$. Moreover, the equality holds if and only if $X$ is birational to an abelian variety. We also show that an analogous result holds for groups of bimeromorphic automorphisms of compact Kähler spaces under some additional assumptions.

We extend Lang's conjectures to the setting of intermediate hyperbolicity and prove two new results motivated by these conjectures. More precisely, we first extend the notion of algebraic hyperbolicity (originally introduced by Demailly) to the setting of intermediate hyperbolicity and show that this property holds if the appropriate exterior power of the cotangent bundle is ample. Then, we prove that this intermediate algebraic hyperbolicity implies the finiteness of the group of birational automorphisms and of the set of surjective maps from a given projective variety. Our work answers the algebraic analogue of a question of Kobayashi on analytic hyperbolicity.

We study automorphism groups and birational automorphism groups of compact complex surfaces. We show that the automorphism group of such a surface $X$ is always Jordan, and the birational automorphism group is Jordan unless $X$ is birational to a product of an elliptic and a rational curve.

We prove that the automorphism group of an odd dimensional Calabi-Yau manifold of Picard number 2 2 is always a finite group. This makes a sharp contrast to the automorphism groups of K3 surfaces and hyperkähler manifolds and birational automorphism groups, as we shall see. We also clarify the relation between finiteness of the automorphism group (resp. birational automorphism group) and the rationality of the nef cone (resp. movable cone) for a hyperkähler manifold of Picard number 2 2 . We will also discuss a similar conjectual relation together with the exsistence of a rational curve, expected by the cone conjecture, for a Calabi-Yau threefold of Picard number 2 2 .

On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety

The purpose of this note is to prove the following result: Theorem 1 Let X and Y be smooth projective complex varieties, and suppose that X = X1 ∪ ··· ∪ Xn and Y = Y1 ∪ ··· ∪ Yn are a disjoint union of quasiprojective subvarieties. Suppose that Xi is algebraically isomorphic to Yi, for all i. Then the Betti numbers of X and Y are equal, and in fact their Hodge numbers are equal. The result for the Betti numbers is part of the mathematical folklore; it has been used, for instance, in [1] and implicitly in [4]. Apparently it was originally noticed by Serre, who proved it by reducing the variety modulo p, counting points and using the solution to the Weil conjectures. The proof given in this paper uses mixed Hodge theory rather than the Weil conjectures; the general yoga of Hodge theory and the Weil conjectures suggest that such a proof should exist. The theorem is in fact a simple consequence of a sum formula for the Hodge numbers of the mixed Hodge structure of an arbitrary variety in terms of those of the pieces of a decomposition. (Throughout this paper, the term "variety" will mean "quasiprojective complex variety.") First, some definitions: According to [2], the rational cohomology groups of a complex algebraic variety have a weight filtration W and a Hodge filtration F, and so do its rational cohomology groups with compact support. The associated graded objects of these filtrations are denoted GrW and GrF, respectively. We refine the concept of Euler characteristic by using these filtrations: Let χ ( X ) = ∑ ( − 1 ) k dim H k ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq635.tif"/> χ m ( X ) = ∑ ( − 1 ) k dim G r m W H k ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq636.tif"/> χ p q ( X ) = ∑ ( − 1 ) k dim G r F p G r p + q W H k ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq637.tif"/> 100Similarly, define χc, χ m c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq638.tif"/> and χ c pq https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq639.tif"/> using H c k ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq640.tif"/> (cohomology with compact supports) in place of Hk(X). These graded Euler characteristics satisfy χ ( X ) = ∑ m X m ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq641.tif"/> χ m ( X ) = ∑ p + q = m X p q ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq642.tif"/> and so forth. If X is smooth of dimension n, the Poincaré duality implies that χ c p q ( X ) = χ n − p , n − q ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq643.tif"/> If X is smooth and projective, then χ m ( X ) = ( − 1 ) m dim H m ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq644.tif"/>

We slightly extend a result of Oguiso on birational or automorphism groups (resp. of Lazi\'c - Peternell on Morrison-Kawamata cone conjecture) from Calabi-Yau manifolds of Picard number two to arbitrary singular varieties X (resp. to klt Calabi-Yau pairs in broad sense) of Picard number two. When X has only klt singularities and is not a complex torus, we show that either Aut(X) is almost cyclic, or it has only finitely many connected components.

We construct a projective variety with discrete, non-finitely generated automorphism group. As an application, we show that there exists a complex projective variety with infinitely many non-isomorphic real forms.