Homotopy principles for equivariant isomorphisms
Let be a reductive complex Lie group acting holomorphically on Stein manifolds and . Let and be the quotient mappings. When is there an equivariant biholomorphism of and ? A necessary condition is that the categorical quotients and are biholomorphic and that the biholomorphism sends the Luna strata of isomorphically onto the corresponding Luna strata of . Fix . We demonstrate two homotopy principles in this situation. The first result says that if there is a -diffeomorphism , inducing , which is -biholomorphic on the reduced fibres of the quotient mappings, then is homotopic, through -diffeomorphisms satisfying the same conditions, to a -equivariant biholomorphism from to . The second result roughly says that if we have a -homeomorphism which induces a continuous family of -equivariant biholomorphisms of the fibres and for and if satisfies an auxiliary property (which holds for most ), then is homotopic, through -homeomorphisms satisfying the same conditions, to a -equivariant biholomorphism from to . Our results improve upon those of [KLS15] and use new ideas and techniques.
Key words and phrases:Oka principle, geometric invariant theory, Stein manifold, complex Lie group, reductive group, categorical quotient, Luna stratification.
2010 Mathematics Subject Classification:Primary 32M05. Secondary 14L24, 14L30, 32E10, 32M17, 32Q28.
Let be a reductive complex Lie group. Let and be Stein manifolds (always taken to be connected) on which acts holomorphically. We have quotient mappings and where and are normal Stein spaces, the categorical quotients of and . Let , . We say that and are in the same Luna stratum of if the fibres and are -biholomorphic. The fibres are affine -varieties, not necessarily reduced. The Luna strata form a locally finite stratification of by locally closed smooth subvarieties. A necessary condition for and to be -equivariantly biholomorphic is that there is a biholomorphism which preserves the Luna strata, i.e., is -biholomorphic to for all . Suppose that such a exists. Our problem then is to find a -equivariant biholomorphism inducing . It is possible that one has made a poor choice of (see Example 4.1) or it could be that no choice of admits a lift (see Example 4.2).
Use to identify the quotients, and call the common quotient with quotient maps and . We say that and have common quotient . More specifically, we replace by . Then is a Stein -manifold whose quotient mapping is projection onto the first factor and is the common quotient. Our problem then is to find a -equivariant biholomorphism which induces , the identity map of . So we can always reduce to the case that and have a common quotient and our problem is to lift to a -biholomorphism of and . In the spirit of Gromov’s work [Gro89], we show that there is a -biholomorphic lift of if there are appropriate continuous or smooth lifts of .
where denotes the set of -biholomorphisms of and . Let denote the natural projection of to . Then is a principal homogeneous space for the group and the global sections of form a principal homogeneous space for the group of global sections of . In general, there is no reasonable structure of complex variety on or (see [KLS15, Section 3]). However, we can say what the sections of of various kinds are. Clearly a holomorphic section of over an open subset should be a -biholomorphism inducing . We are also able to define what a continuous section of over is, which we call a strong -homeomorphism (see Section 3).
Let be a -diffeomorphism inducing . We say that is strict if it induces a -biholomorphism of with for all where the subscript means that we are considering the reduced structures on the fibres (see Example 3.2). Let denote the product of the with the obvious projection to . Then the smooth sections of are the strict -diffeomorphisms. A strict -diffeomorphism is not necessarily a strong -homeomorphism (Example 3.2). Our definition of strict is more general than in [KLS15]; see Remark 5.9.
Here is our first main result.
Let and be Stein -manifolds with common quotient . Suppose that there is a strict -diffeomorphism . Then is homotopic, through strict -diffeomorphisms, to a -biholomorphism from to .
The theorem says that a smooth section of is homotopic to a holomorphic section.
There is also a version of the theorem for continuous sections of , but we need an additional assumption. Let be a vector field on . We say that is strata preserving if for all Luna strata of and , . We say that has the infinitesimal lifting property if every holomorphic strata preserving vector field defined on a neighbourhood of has a lift to a -invariant holomorphic vector field on where is a neighbourhood of contained in . This means that for all . The infinitesimal lifting property really only depends upon the isomorphism classes of the fibres of ; equivalently, on the slice representations of (see Section 2). For most representations of reductive groups, the infinitesimal lifting property holds (Remark 2.1) and for most representations all holomorphic vector fields on the quotient automatically preserve the strata [Sch13].
Here is our second main result.
Let and be Stein -manifolds with common quotient . Suppose that there is a strong -homeomorphism . If has the infinitesimal lifting property, then is homotopic, through strong -homeomorphisms, to a -biholomorphism from to .
See Section 3 for the definition of a homotopy of strong -homeomorphisms. The theorem says that a continuous section of is homotopic to a holomorphic section, provided that (equivalently, ) has the infinitesimal lifting property.
Our proofs of Theorems 1.1 and 1.2 have two steps, where we first reduce our homotopy principles to Oka principles of the form considered by Grauert. Let , and be as before. We say that and are locally -biholomorphic over if there is an open cover of and -biholomorphisms inducing the identity on . This condition says that there are no local obstructions to the existence of a global -biholomorphism inducing .
Let and be Stein -manifolds with common quotient . Suppose that one of the following holds.
There is a strict -diffeomorphism from to .
There is a strong -homeomorphism from to and has the infinitesimal lifting property.
Then and are locally -biholomorphic over .
Once we have no local obstructions we are able to establish the following versions of Grauert’s Oka principle.
Let and be Stein -manifolds locally -biholomorphic over a common quotient .
Any strict -diffeomorphism is homotopic, through strict -diffeomorphisms, to a -biholomorphism from to .
Any strong -homeomorphism is homotopic, through strong -homeomorphisms, to a -biholomorphism from to .
Note that Theorems 1.3 and 1.4 establish Theorems 1.1 and 1.2. The proof of Theorem 1.4 is along the lines of Grauert’s Oka principle for principal bundles of complex Lie groups (Section 10). A main result of our previous paper [KLS15] is a weaker version of Theorem 1.4. In (1) and (2) we were only able to state the existence of a -biholomorphism, but not that it was homotopic to . Also, we had to assume that (equivalently ) is generic, which means that the set of closed orbits with trivial isotropy group is open in and that the complement (which is a closed -stable subvariety of ) has codimension at least two.
We briefly mention here the Linearisation Problem. Suppose that and that is a -module such that we have a -biholomorphism of and . Then the -action on is linearisable, i.e., there is a biholomorphic automorphism of such that is linear for every . The problem of linearising actions of reductive groups on has attracted much attention both in the algebraic and holomorphic settings ([Huc90], [Kra96]). The first counterexamples for the algebraic linearisation problem were constructed by Schwarz [Sch89] for . His examples are holomorphically linearisable. Derksen and Kutzschebauch [DK98] showed that for nontrivial, there is an such that there are nonlinearisable actions of on for all . Their method was to construct actions whose stratified quotients cannot be isomorphic to the stratified quotient of a linear action. In [KLS], we show that, most of the time, a holomorphic -action on is linearisable if and only if the stratified quotient is isomorphic to the stratified quotient of a -module.
Here is a brief summary of the contents of the paper. In Section 2 we review general results about quotients and the Luna stratification. In Section 3 we recall facts about -finite functions and use them to define the notion of strong -homeomorphism. Section 4 gives examples showing problems that can arise in finding local or global lifts of strata preserving biholomorphisms of quotients. In Section 5 we establish Theorem 1.3. Here we use two techniques: deforming an automorphism of to a liftable automorphism and lifting homotopies on the quotient by lifting associated vector fields. After establishing Theorem 1.3 we are able to assume that and are locally -biholomorphic over . In Section 6 we define a type of -diffeomorphism from to , those of type , which roughly are those -diffeomorphisms inducing whose restriction to each fibre has a biholomorphic -equivariant extension to a neighbourhood of . We also define the notion of a -invariant vector field on of type . These are roughly the smooth -invariant vector fields, annihilating the -invariant holomorphic functions, whose restrictions to each fibre extend in a neighbourhood of to a -invariant holomorphic vector field annihilating the -invariant holomorphic functions. We establish important properties of the -diffeomorphisms of type (assuming the results of Section 7). In Section 7 we prove several technical results, among them the fact that the -invariant vector fields of type are closed in the Fréchet space of all smooth -invariant vector fields on . In Section 8 we show that any strong -homeomorphism from to is homotopic, through strong -homeomorphisms from to , to one of type . The analogous result for strict -diffeomorphisms follows similarly. In Sections 9 and 10 we modify the techniques of Cartan [Car58] to show that any -diffeomorphism from to of type is homotopic, through -diffeomorphisms of type , to a -biholomorphism from to , completing the proof of Theorem 1.4.
Acknowledgement. We thank E. Bierstone for useful discussions.
For details of what follows see [Lun73] and [Sno82, Section 6]. Let be a normal Stein space with a holomorphic action of a reductive complex Lie group . The categorical quotient of by the action of is the set of closed orbits in with a reduced Stein structure that makes the quotient map the universal -invariant holomorphic map from to a Stein space. When is understood, we drop the subscript in and . Since is normal, is normal. If is an open subset of , then induces isomorphisms of -algebras and . We say that a subset of is -saturated if it is a union of fibres of . If is an affine -variety, then is just the complex space corresponding to the affine algebraic variety with coordinate ring . If is a -module and is the quotient mapping, then the fibre is the null cone of .
If is a closed orbit in , then the stabiliser (or isotropy group) is reductive. We say that closed orbits and have the same isotropy type if is -conjugate to . Thus we get the isotropy type stratification of with strata whose labels are conjugacy classes of reductive subgroups of .
Assume that is smooth and connected, and let be a closed orbit. Then we can consider the slice representation which is the action of on . We say that closed orbits and have the same slice type if they have the same isotropy type and, after arranging that , the slice representations are isomorphic representations of . The stratification by slice type is finer than that by isotropy type, but the slice type strata are unions of irreducible components of the isotropy type strata [Sch80, Proposition 1.2]. Hence if the isotropy type strata are irreducible, the slice type strata and isotropy type strata are the same. This occurs for the case of a -module [Sch80, Lemma 5.5]. Let and be the slice representation as above. Write where is an -module. The Zariski tangent space to the fibre at is isomorphic to as -module, and
so that the fibre determines the slice representation (and vice versa). Hence the Luna stratification of the introduction is the same as the slice type stratification.
There is a unique open stratum , corresponding to the closed orbits with minimal stabiliser. We call this the principal stratum and the closed orbits above are called principal orbits. The isotropy groups of principal orbits are called principal isotropy groups. By definition, is generic when the principal isotropy groups are trivial and has codimension at least two in .
If is simple, then, up to isomorphism, all but finitely many -modules with are generic and have the infinitesimal lifting property [Sch95, Corollary 11.6 (1)]. The same result holds for semisimple groups but one needs to assume that every irreducible component of is a faithful module for the Lie algebra of [Sch95, Corollary 11.6 (2)]. A “random” -module is generic and has the infinitesimal lifting property, although infinite families of counterexamples exist. More precisely, a faithful -dimensional -module without zero weights is generic (and has the infinitesimal lifting property) if and only if it has at least two positive weights and at least two negative weights and no weights have a common prime divisor. Finally, is generic (or has the infinitesimal lifting property) if and only if every slice representation does. Hence these properties only depend upon the Luna stratification of . If one is in the situation where all slice representations are orthogonal, then [Sch80, Theorems 3.7 and 6.7] shows that one has the infinitesimal lifting property.
3. -finite functions and strong homeomorphisms
Let and be Stein -manifolds. Despite the fact that we can state our main theorems in the case that is the identity, our proofs (especially in Section 5) require us to consider the case that is an arbitrary strata preserving biholomorphism.
If is a subset of , we denote by , and for is defined analogously. The group acts on , , where , , , . Let denote the holomorphic functions such that the span of is finite dimensional. They are called the -finite holomorphic functions on and obviously form an -algebra. If is a smooth affine -variety, then the techniques of [Sch80, Proposition 6.8, Corollary 6.9] show that for open and Stein we have
Let be a reductive subgroup of and let be an -saturated neighbourhood of the origin of an -module . We always assume that is Stein, in which case is also Stein. Let (or ) denote the quotient of by the (free) -action sending to for , and . We denote the image of in by . By the slice theorem, is locally -biholomorphic to such tubes . If is an irreducible nontrivial -module, let denote the elements of contained in a copy of , and similarly define . Then generates over . By Nakayama’s Lemma, restrict to minimal generators of the -module for some neighbourhood of if and only if the restrictions of the to form a basis of .
Let be a -biholomorphism inducing . Let , let be elements of whose restrictions to span and let be elements of whose restrictions to span . Then the generate over a neighbourhood of and the generate over a neighbourhood of . Over a neighbourhood of we have where the are in . The are generally not unique. However, if the and are linearly independent when restricted to and , respectively, then and the matrix is unique and invertible. It follows that over a neighbourhood of where the matrix valued function equals .
Let be a -equivariant homeomorphism inducing a strata preserving biholomorphism . Let and the , and be as above. We say that is strong for over at if where the are continuous in a neighbourhood of , inducing an isomorphism . Note that this condition is independent of our choice of the and . We say that is strong over at if is strong over for at for all irreducible nontrivial . One does not actually need to worry about all . Let be irreducible nontrivial -modules such that is generated by . If is strong over for the at , then it is strong over at . Note that if is strong over at , then it is strong over at for sufficiently close to and that is strong over at . Finally, we say that is a strong -homeomorphism over if it is strong over at all . When we omit the phrase “over ” we mean that and as in the introduction. The strong -homeomorphisms of form a group under composition. As we saw above, -biholomorphisms inducing are strong.
If is a strong -homeomorphism, then the various determine -isomorphisms of the (not necessarily reduced) algebraic -varieties and , so one may consider as a continuous family of isomorphisms of the fibres and , . If is a family of strong -homeomorphisms for , we say that the family is a homotopy of strong -homeomorphisms if the corresponding can be chosen to be continuous in and .
We consider strong -homeomorphisms to be the continuous analogues of -biholomorphisms of and . This is especially evident in the case that the automorphism group scheme associated to exists. For open, let denote the group of -biholomorphisms of inducing . Suppose that there is a complex space over whose fibres are complex Lie groups such that we have a canonical identification of the holomorphic sections with for all open in . Then we say that is the group scheme associated to the Stein -manifold . Most of the time does not exist (see [KLS15, Section 3]). However, it does exist in case is flat (compare [KS92, Chapter III, Section 2]).
This example is from Kraft-Schwarz [KS92, Chapter III, Section 2]. Let with coordinate functions and , and let be the normaliser of the diagonal matrices in . Let and . Then (so ), and is generated by which has minimal generators , , and . An element sends to for some , and it sends to . Then where . Hence . Conversely, given , such that , there is a corresponding with and . We can see our automorphisms as sections of a group scheme , as follows. As variety,
with projection sending to . The group structure on induces a group structure on the fibres of :
The inverse to is . The group is isomorphic to the group of holomorphic sections of . The strong -homeomorphisms of are the same thing as the continuous sections of .
Let be as in the previous example. Choose the branch of the square root in a neighbourhood of with and let , . Then is smooth in a neighbourhood of . Let be the -diffeomorphism which sends to and to for near . Then the corresponding is which has no limit at . The fibre of over is not reduced, and the reduced fibre is on which the restriction of is the identity. Thus is a strict -diffeomorphism, but it is not a strong -homeomorphism.
We now establish some differentiability properties of strong -homeomorphisms inducing a strata preserving biholomorphism . Our results are a generalisation of [KLS15, Lemma 24] (see Remark 3.6 below).
Let , , , etc. be as in the beginning of this section. We always assume that is stable under multiplication by . Let . We say that has degree if , , , . We denote the elements of degree by . The elements of degree zero are the pullbacks to of the elements of where is the zero section of . For the moment, we also assume that .
By [KLS15, Lemma 23] we have
is generated by as an -algebra.
Let be nonisomorphic irreducible nontrivial -modules which appear in and such that is generated by
We may assume that are minimal with the above property. Let minimally generate where each is homogeneous and in for some .
Let and be as above. We call a standard set of generators of or a standard set of generators of .
The span of the is -stable and the are linearly independent on . Let denote the degree of , . We arrange that for , for and for .
Let denote so that . Let be the image of the point . Let be a neighbourhood of and let be a strata preserving biholomorphism. Then . Let be a strong -homeomorphism over .
Shrinking we may assume that there is an -matrix of continuous invariant functions such that . Since preserves , it preserves the closed orbit . Let denote the ideal in generated by . Note that since , .
The matrix is invertible, , .
Perhaps shrinking , we have for and .
If we restrict and the to , then is an isomorphism, with matrix , and we have (1).
Let denote the Reynolds operator, i.e., the -equivariant projection from to . It extends to a projection of to . Let us assume that some for and , say . Then and correspond to the same irreducible -module . We may assume that are the polynomials corresponding to among . Then for . Now occurs in and we have a non-degenerate pairing of with itself, which sends a pair of functions , to . Thus has a basis of cardinality . We may assume that for , . There is a homogeneous minimal generating set , of the -module . Then for . There are such that . Let . Since has degree , . Thus
Applying (1) to the matrix , , , we see that is invertible. In a -saturated neighbourhood of we can invert and then the equation above shows that , . This gives (2). ∎
We may assume that is the image of a tube . Then we have a scalar action of on as follows. Let . Then . Let , , . Let denote the fibre of at . Let denote , , , , . Finally, let denote the complex algebraic group of -vector bundle automorphisms of (see Corollary 6.31).
The following hold.
, , , .
The limit as of acting on the , , is given by the matrix with entries , , .
has a normal derivative along , and . If is the identity, then fixes .
The limit as of the is , uniformly on compact subsets of .
For (1) we compute that
Now let , so that . By Lemma 3.5, for , where is continuous. Hence, uniformly on compact subsets of ,
giving (2). For now, just consider indices between and . We have for some , and for all . The are sections of and they span each fibre of . If , , vanishes at , then for all . By (3.1), . Thus the matrix induces a linear mapping from to . Let be the dual mapping. It follows easily from (3.1) that is the normal derivative of at , and the analogous fact at other points of follows by equivariance of . Clearly, . If is the identity, then the and fix the invariants. Hence we have established (3).
We have shown that converges uniformly as on compact subsets of for , and this is trivially true for . Since the for generate , a subset of them gives local coordinates at any point of . Hence converges to uniformly on compact subsets of as and we have (4). ∎
It is not clear that exists for all and . To remedy this we replace by a matrix for which does exist.
Let . For there are polynomials
for any strong -homeomorphism with associated matrix .
By Lemma 3.3, for , , where is a polynomial with constant coefficients. We may assume that each monomial in is homogeneous of degree where we give degree 0 for and degree 1 for . Now . Replace by , . Expanding and re-expressing in terms of invariants times the we obtain
where is independent of the (which we may treat as indeterminants). ∎
By equivariance of , some of the are necessarily zero.
We break up into a sum of terms as follows. Let be a term in . Let be the sum of the for which and let be the sum of the for which . Then is homogeneous of degree . For , let be the sum of the terms in for which and . Then