We determine all restrictions on the dimension of the fixed locus of a diagonalizable group acting on a smooth projective variety that arise from the Chern numbers of the ambient variety. We reduce the problem to finding lower bounds for actions of p-groups, which we achieve by analyzing the equivariant cobordism ring with the help of the concentration theorem. To do so, we construct enough explicit examples of actions that realize the expected lower bound. We then prove that this family is maximal in the equivariant cobordism ring, in an appropriate sense.
1. Introduction 1.1. Existence of fixed points 1.2. Fixed locus dimension 1.3. Remarks 1.4. Strategy of the proof 1.5. Organization of the paper 2. The Lazard ring 2.1. The algebraic Hurewicz morphism 2.2. Cobordism classes of varieties 2.3. Landweber ideals 3. Actions on cobordism generators 3.1. Group actions on varieties 3.2. Generators of the Landweber ideals 3.3. Milnor hypersurfaces 3.4. The q-filtration 4. Algebraic cobordism 4.1. Cobordism as a cohomology theory 4.2. Cobordism of graded vector bundles 5. Cobordism of diagonalizable groups 5.1. Algebraic preliminaries 5.2. Equivariant cobordism 1. Introduction 1.1. Existence of fixed points. When does a group action on an algebraic variety admit fixed points? This fundamental question is typically answered by means of fixed-point theorems, which provide sufficient conditions for the existence of fixed points, based on nonequivariant information on the ambient variety.
In this paper, the information considered will be the collection of the Chern numbers of the variety, which are certain numerical invariants indexed by the partitions of its dimension. These are constructed using intersection theory, and roughly speaking quantify the geometric complexity of the variety. Let us fix a base field, assumed to be algebraically closed of characteristic zero for this introduction. Identifying smooth projective varieties having the same Chern numbers yields a ring L called the Lazard ring. Thus a smooth projective variety X has a cobordism class X ∈ L, which exactly encodes its Chern numbers.
A complete determination of the fixed-point theorems involving Chern numbers is equivalent to the computation of the subgroup J(G) ⊂ L generated by the cobordism classes of varieties admitting a fixed-point-free action of a given algebraic group G.
In this paper we consider finite type diagonalizable groups G, which means, given our assumptions on the field k, that G is the product of a split torus and a (constant) finite abelian group. Borel’s fixed-point theorem implies that J(G × G m ) = J(G), so we are reduced to considering finite abelian groups G. For such groups, the only case of real interest is when G is a finite abelian p-group for some prime p: indeed it turns out that otherwise J(G) = L (see (8.1.7)).
Our first result, proved as (8.1.8), identifies the ideal J(G) in this case, providing an algebraic analogue of a theorem of tom Dieck in topology [tD70]. By the rank of a finite abelian group, we mean its number of cyclic factors.
Theorem 1. Let G be a finite abelian p-group of rank r. Then J(G) is the subgroup I p (r) of L generated by the classes of smooth projective varieties of dimension < p r-1 having all Chern numbers divisible by p.
The subgroup I p (r) is the r-th Landweber ideal, which plays a fundamental role in many aspects of cobordism theory. As an ideal, it is generated by the classes of r explicit varieties, which turn out to admit fixed-point-free G-actions. The first of these is the disjoint union of p points, and so I p (1) = p L. When r > 1, the ideal I p (r) can be thought of as a higher version of the ideal p L. For instance, the ideal I 2 (2) is generated by 2 and P 1 , and a fixed-point-free action of Z/2 × Z/2 on P 1 is given by the commuting involutions x → -x and x → x -1 .
Concretely, this result allows for the detection of fixed points: any G-action on a variety whose cobordism class does not lie in I p (r) must have a fixed point. The theorem additionally expresses the fact that no other (nonequivariant) cobordism invariant can be used to detect fixed points.
Observe that Theorem 1 implies that the action of an abelian p-group on a variety having a Chern number prime to p must fix a point, a result previously obtained in [Hau19,(1.1.2)]. Note that the structure of the abelian p-group G (including its rank r) played no role in that particular result. Theorem 1 also recovers, for r = 1, the fixed-point theorem for actions of cyclic p-groups involving the arithmetic genus proved in [Hau19, (1.2.2)]. Theorem 1 can thus be thought of as interpolating between those two extreme special cases.
Finally, note that Theorem 1 provides a new definition of the Landweber ideal I p (r), in terms of fixed-point-free actions, instead of Chern numbers.
1.2. Fixed locus dimension. Beyond detecting fixed points, one may attempt to control the dimension of the fixed locus. To this end, for each integer d, we denote by ∆ d (G) ⊂ L the subgroup of the Lazard ring generated by the classes of varieties admitting a G-action with a fixed locus of dimension d. We observe that ∆ d (G) contains the classes of all varieties admitting a G-action with a fixed locus of dimension ≤ d (see (8.2.2)). Consequently, the cobordism class of the ambient variety imposes only lower bounds on the fixed locus dimension.
We prove in (8.2.7) that ∆ d (G) = L when G contains the multiplicative group G m . As J(G) ⊂ ∆ d (G), the considerations of §1.1 imply that ∆ d (G) = L
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