Master spaces for stable pairs.

*(English)*Zbl 0981.14007The master spaces, appearing in the title, are coarse moduli spaces for stable oriented pairs on a smooth projective variety \(X\) over \(\mathbb C\). An oriented pair \(({\mathcal E}, \varepsilon, \varphi)\) consists of a torsion free coherent sheaf \({\mathcal E}\) on \(X\), a morphism (“framing”) \(\varphi: {\mathcal E} \rightarrow {\mathcal E}_0\) to a fixed reference sheaf \({\mathcal E}_0\) and an orientation \(\varepsilon\) of \(\det({\mathcal E})\) (i.e. a morphism to a fixed representative of the isomorphism class of \(\det({\mathcal E})\), given by the choice of a Poincaré sheaf on Pic\((X)\times X\)).

These master spaces are projective and they come with a natural \(\mathbb C^\ast\)-action. It is interesting to study the fixed points and quotients of this \(\mathbb C^\ast\)-action. In particular, one obtains insight into the birational geometry of the moduli spaces of \(\delta\)-semistable pairs in the sense of D. Huybrechts and M. Lehn [Int. J. Math. 6, 297-324 (1995; Zbl 0865.14004)], where \(\delta\) is a polynomial with positive leading coefficient. Master spaces in a more abstract setting appear already in the basic work of M. Thaddeus [J. Am. Math. Soc. 9, 691-725 (1996; Zbl 0874.14042)] where he studies systematically how GIT quotients depend on the polarization.

In the first part of the paper under review, the authors study GIT quotients with respect to different linearizations of a given \(\mathbb C^\ast\)-action on a projective space. In particular, they study the behavior under such quotients of (semi-)stable points with respect to an auxiliary reductive algebraic group \(G\). These results are applied in the second part of the paper to the construction of master spaces. The construction follows the usual lines of GIT constructions but the difficulty lies in identifying the (semi-)stable points in \(\mathbb P(U\oplus V)\), where \(U\) and \(V\) are linear representations of the reductive algebraic group \(G\). At this point, the authors introduce the \(\mathbb C^\ast\)-action given by the trivial action on \(U\) and the multiplication with \(z\in \mathbb C^\ast\) on \(V\). Then they can apply the results of the first part.

This paper provides the technical details of the construction of master spaces, solving a specific moduli problem. The construction of these spaces is motivated by a variety of applications, which are not carried out in the present paper. Possible applications are related to the Verlinde formula, Gromov-Witten invariants for Grassmannians and the study of the relation between Seiberg-Witten invariants and Donaldson invariants [see e.g. Ch. Okonek and A. Teleman in: Several complex variables, Math. Sci. Res. Inst. Publ. 37, 391-428 (1999; Zbl 0978.57028)].

These master spaces are projective and they come with a natural \(\mathbb C^\ast\)-action. It is interesting to study the fixed points and quotients of this \(\mathbb C^\ast\)-action. In particular, one obtains insight into the birational geometry of the moduli spaces of \(\delta\)-semistable pairs in the sense of D. Huybrechts and M. Lehn [Int. J. Math. 6, 297-324 (1995; Zbl 0865.14004)], where \(\delta\) is a polynomial with positive leading coefficient. Master spaces in a more abstract setting appear already in the basic work of M. Thaddeus [J. Am. Math. Soc. 9, 691-725 (1996; Zbl 0874.14042)] where he studies systematically how GIT quotients depend on the polarization.

In the first part of the paper under review, the authors study GIT quotients with respect to different linearizations of a given \(\mathbb C^\ast\)-action on a projective space. In particular, they study the behavior under such quotients of (semi-)stable points with respect to an auxiliary reductive algebraic group \(G\). These results are applied in the second part of the paper to the construction of master spaces. The construction follows the usual lines of GIT constructions but the difficulty lies in identifying the (semi-)stable points in \(\mathbb P(U\oplus V)\), where \(U\) and \(V\) are linear representations of the reductive algebraic group \(G\). At this point, the authors introduce the \(\mathbb C^\ast\)-action given by the trivial action on \(U\) and the multiplication with \(z\in \mathbb C^\ast\) on \(V\). Then they can apply the results of the first part.

This paper provides the technical details of the construction of master spaces, solving a specific moduli problem. The construction of these spaces is motivated by a variety of applications, which are not carried out in the present paper. Possible applications are related to the Verlinde formula, Gromov-Witten invariants for Grassmannians and the study of the relation between Seiberg-Witten invariants and Donaldson invariants [see e.g. Ch. Okonek and A. Teleman in: Several complex variables, Math. Sci. Res. Inst. Publ. 37, 391-428 (1999; Zbl 0978.57028)].

Reviewer: Bernd Kreußler (Kaiserslautern)

##### MSC:

14D22 | Fine and coarse moduli spaces |

14L24 | Geometric invariant theory |

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |

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

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