Variation of geometric invariant theory quotients. (With an appendix: “An example of a thick wall” by Nicolas Ressayre).

*(English)*Zbl 1001.14018Let \(X\) be a projective algebraic variety acted on by some reductive algebraic group \(G\), both of them being defined over the field of complex numbers. In general, the orbit space \(X/G\) is not an algebraic variety, which is due to the fact that there might exist non-closed orbits. However, the methods of geometric invariant theory allow to construct an “algebraic quotient” \(X//G\) i.e., a suitable algebraic variety with quotient properties, from this set-up. This construction is based upon the choice of a very ample \(G\)-linearized line bundle \(L\) on \(X\) and the associated sets \(X^{ss}(L)\) and \(X^s(L)\) of semi-stable and stable points, respectively, in \(X\). Then there is a regular map \(f:X\to X^{ss}//G\) whose domain is the open subvariety \(X^{ss}:= X^{ss} (L)\), and where \(X^{ss}\to X^{ss}//G\) is a good categorical quotient. Moreover, \(X^s:=X^s(L)\) is a maximal \(G\)-invariant Zariski open subset such that the restriction of \(f\) to \(X^s\) gives an algebraic quotient \(X^s/G\subset X^{ss} //G\). As there are infinitely many isomorphism classes of ample \(G\)-linearized line bundles \(L\) on \(X\), at least two natural questions are worthwile to be investigated more closely:

(A) Is the set of non-isomorphic quotients \(X^{ss} (L)//G\) obtained in this way finite and, if so, how can it be described?

(B) How does the quotient \(X^{ss}(L)//G\) change if \(L\) varies in the group \(\text{Pic}^G(X)\) of isomorphism classes of \(G\)-linearized line bundles on \(X\)?

The fundamental comparison problen for different geometric invariant theory quotients was apparently first addressed by M. Goresky and R. MacPherson in 1986, when these authors constructed the first natural morphisms between various quotients of this type. On the other hand, a similar comparison problem exists in symplectic geometry, where the variation of symplectic reductions of a symplectic manifold \(M\) with respect to the action of a compaet Lie group \(K\) is of crucial importance.

In the present paper, the authors take up these variation problems, mainly so with regard to the above mentioned questions (A) and (B), and extend the existing partial results by M. Brion and C. Procesi [in: Operator algebras, unitary representations enveloping algebras and invariant theory, Proc. Coll. Hon. Dixmier 1989, Prog. Math. 92, 509-539 (1990; Zbl 0741.14028)], Y. Hu [Duke Math. J. 68, No. 1, 151-184 and No. 3, 609 (1992; Zbl 0812.14031 and Zbl 0815.14032)], V. Guillemin and S. Sternberg [Invent. Math. 97, No. 3, 485-522 (1989; Zbl 0683.53033)], and others in the purely algebraic case.

Roughly speaking, the authors show, by a very subtle analysis, that, among all ample \(G\)-linearized line bundles \(L\) on \(X\) which define projective geometric quotients, there are indeed only finitely many equivalence classes, and that these classes span certain convex subsets, the so-called “chambers”, in a certain convex cone in Euclidean space. Moreover, they show that, when crossing a “wall” separating one chamber from another one, the corresponding quotients change by a birational transformation which looks very much like a Mori flip in higher-dimensional birational geometry.

In order to get these very precise and far-reaching results on the variation of geometric quotients of projective varieties, the authors have invented (and applied) various new ideas, methods and techniques, in the four chapters of their work, which are centered at the study of symplectic reductions in symplectic geometry. Hilbert scheme methods and algebraic moduli problems, so-called limit quotients, and \(G\)-ample cones in the Néron-Severi group of the fixed projective variety \(X\).

All together, this is a very important and pioneering contribution towards the study of geometric quotients and moduli spaces in algebraic geometry, which also extends many notions and methods from geometric invariant theory to the geometry of Kähler manifolds. Of course, the latter fact is highly interesting and enlightening purely as such.

In the meantine, as the authors point out, there has appeared a paper by M. Thaddeus entitled “Geometric invariant theory and flips” [J. Am. Math. Soc. 9, No. 3, 691-723 (1996; Zbl 0874.14042)], where some of the results (e.g., the finiteness theorem 0.2.3) presented in the paper under review were reproved, and partially enriched, by different methods. In particular, M. Thaddeus obtained some further information about the structure of the flip-like maps arising, in the present work, from the variation process of crossing walls between chambers.

Finally, the paper under review comes with an appendix by N. Ressayre, entitled “An example of a thick wall”, which provides an instructive counter-example to the possible conjecture that a certain particular result for torus actions could be generalized to arbitrary reductive group actions.

(A) Is the set of non-isomorphic quotients \(X^{ss} (L)//G\) obtained in this way finite and, if so, how can it be described?

(B) How does the quotient \(X^{ss}(L)//G\) change if \(L\) varies in the group \(\text{Pic}^G(X)\) of isomorphism classes of \(G\)-linearized line bundles on \(X\)?

The fundamental comparison problen for different geometric invariant theory quotients was apparently first addressed by M. Goresky and R. MacPherson in 1986, when these authors constructed the first natural morphisms between various quotients of this type. On the other hand, a similar comparison problem exists in symplectic geometry, where the variation of symplectic reductions of a symplectic manifold \(M\) with respect to the action of a compaet Lie group \(K\) is of crucial importance.

In the present paper, the authors take up these variation problems, mainly so with regard to the above mentioned questions (A) and (B), and extend the existing partial results by M. Brion and C. Procesi [in: Operator algebras, unitary representations enveloping algebras and invariant theory, Proc. Coll. Hon. Dixmier 1989, Prog. Math. 92, 509-539 (1990; Zbl 0741.14028)], Y. Hu [Duke Math. J. 68, No. 1, 151-184 and No. 3, 609 (1992; Zbl 0812.14031 and Zbl 0815.14032)], V. Guillemin and S. Sternberg [Invent. Math. 97, No. 3, 485-522 (1989; Zbl 0683.53033)], and others in the purely algebraic case.

Roughly speaking, the authors show, by a very subtle analysis, that, among all ample \(G\)-linearized line bundles \(L\) on \(X\) which define projective geometric quotients, there are indeed only finitely many equivalence classes, and that these classes span certain convex subsets, the so-called “chambers”, in a certain convex cone in Euclidean space. Moreover, they show that, when crossing a “wall” separating one chamber from another one, the corresponding quotients change by a birational transformation which looks very much like a Mori flip in higher-dimensional birational geometry.

In order to get these very precise and far-reaching results on the variation of geometric quotients of projective varieties, the authors have invented (and applied) various new ideas, methods and techniques, in the four chapters of their work, which are centered at the study of symplectic reductions in symplectic geometry. Hilbert scheme methods and algebraic moduli problems, so-called limit quotients, and \(G\)-ample cones in the Néron-Severi group of the fixed projective variety \(X\).

All together, this is a very important and pioneering contribution towards the study of geometric quotients and moduli spaces in algebraic geometry, which also extends many notions and methods from geometric invariant theory to the geometry of Kähler manifolds. Of course, the latter fact is highly interesting and enlightening purely as such.

In the meantine, as the authors point out, there has appeared a paper by M. Thaddeus entitled “Geometric invariant theory and flips” [J. Am. Math. Soc. 9, No. 3, 691-723 (1996; Zbl 0874.14042)], where some of the results (e.g., the finiteness theorem 0.2.3) presented in the paper under review were reproved, and partially enriched, by different methods. In particular, M. Thaddeus obtained some further information about the structure of the flip-like maps arising, in the present work, from the variation process of crossing walls between chambers.

Finally, the paper under review comes with an appendix by N. Ressayre, entitled “An example of a thick wall”, which provides an instructive counter-example to the possible conjecture that a certain particular result for torus actions could be generalized to arbitrary reductive group actions.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14L24 | Geometric invariant theory |

53D30 | Symplectic structures of moduli spaces |

14L30 | Group actions on varieties or schemes (quotients) |

14D20 | Algebraic moduli problems, moduli of vector bundles |

53D20 | Momentum maps; symplectic reduction |

##### Keywords:

group actions; geometric quotients; wall; geometric invariant theory; symplectic geometry; chamber; Mori flip; symplectic reductions; algebraic moduli problems; limit quotients##### References:

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