##
**Moduli space of principal sheaves over projective varieties.**
*(English)*
Zbl 1079.14018

Let \(X\) be a smooth projective variety of dimension \(n\) over \(\mathbb{C}\), and let \(G\) be a connected algebraic reductive group. The aim of this article is to generalize Ramanathan’s notion of (semi)stability of principal \(G\)-bundles to \(X\), that is, to prove that there exists a projective moduli space for these bundles when allowing also principal sheaves on \(X\). A principal \(\text{GL}(R,\mathbb{C})\)-bundle over \(X\) is equivalent to a vector bundle of rank \(R\). When \(X\) is a curve, M. S. Narasimhan and C. S. Seshadri [Ann. Math. (2) 82, 540–567 (1965; Zbl 0171.04803)] constructed the moduli space of vector bundles, and later D. Gieseker [Ann. Math. (2) 106, 45–60 (1977; Zbl 0381.14003)], M. Maruyama [J. Math. Kyoto Univ. 17, 91–126 (1977; Zbl 0374.14002)] and C. T. Simpson [Publ. Math., Inst. Hautes Étud. Sci. 79, 47–129 (1994; Zbl 0891.14005)] proved that the moduli space of torsion free sheaves on a curve is projective. A. Ramanathan [Math. Ann. 213, 129–152 (1975; Zbl 0284.32019)] defined a notion of stability for principal \(G\)-bundles, and constructed the projective moduli space of such bundles on a curve. The authors of the article define the notion of {(semi)stable \(G\)-sheaves} in terms of filtrations of the associated adjoint bundle of (Killing) orthogonal algebras. They then construct a projective coarse moduli space for semistable principal \(G\)-sheaves. This truly generalizes Ramanathans notion, and gives a simpler proof of his results in the case where \(X\) is a curve.

Let \(G^\prime=[G,G]\) be the commutator subgroup of \(G\), and write \(\mathfrak g=\mathfrak z\oplus\mathfrak g^\prime\) for the Lie algebra of \(G\) where \(\mathfrak g^\prime\) is the semisimple part and \(\mathfrak z\) is the center. Let \(U\) be an open set with codim\(X\backslash U\geq 2\). Let \(P(\mathfrak g)=P(\mathfrak z\oplus\mathfrak g^\prime)=\mathfrak z U\oplus P(\mathfrak g^\prime)\) be the vector bundle associated to \(P\) by the adjoint representation of \(G\) in \(\mathfrak g\). A principal \(G\)-sheaf \(\mathcal{P}\) over \(X\) is a triple \(\mathcal{P}=(P,E,\psi)\) consisting of a torsion free sheaf \(E\) on \(X\), a principal \(G\)-bundle \(P\) on the maximal open set \(U_E\) where \(E\) is locally free, and an isomorphism of vector bundles \[ \psi:P(\mathfrak g^\prime)\overset\cong\rightarrow E|_{U_E}. \] Using the induced orthogonal Killing structure, it is possible to define an orthogonal algebra filtration \(0\subset E_{-l}\subseteq E_{-l+1}\subset\cdots\subset E_l=E\) of \(E\). Then a Hilbert polynomial for this filtration is defined, and is used to define (semi)stability in a way similar to the usual. Using the geometric invariant theory (of Mumford), there is a set of numerical data \(\tau=(d_1,\dots,d_q,c_i)\), and as in GIT, values of \(\tau\) determine (semi)stability. To construct the announced coarse moduli space, the authors define families of semistable principal \(G\)-sheaves, and an equivalence of such. Again, when \(X\) is a curve, this is a generalization of Ramanathan’s notion. Also a sheafification of the family-functor is given: \(\widetilde{F}^\tau_G\) is the sheafification of the family functor with numerical data \(\tau\). This is to tell about the relations with stacks. The main result of the article is: For a polarized complex smooth projective variety \(X\) there is a coarse projective moduli space of \(S\)-equivalence classes of semistable \(G\)-sheaves on \(X\) with fixed numerical invariants.

The article contains eleven pages with preliminaries where the concepts above are being defined. Chapter 1 contains the definition of Lie-sheaves and the construction of the moduli spaces of various tensors. Chapter 2 and 3 makes use of the category of complex analytic spaces to construct a subspace of a product of Jacobians parametrizing reductions to \(G\). Chapter 4 contains the construction of the moduli space announced in the main result. This construction involves a lemma given by Ramanathan, proving the existence of good quotients. Chapter 5 contains results proving that the constructions given here are true generalizations of Ramanathans results. Also, several other details are skipped to this chapter.

It remains to say that this article is rather deep. The proofs require a deep understanding of geometric invariant theory, analytic theory, and the theory of sheaves. Also noncommutative theory is involved in the definition of Lie-sheaves. Diving into some of the proofs even involves a deep knowledge of deformation theory. It will also help the reader to be familiar to the concept of stacks.

Let \(G^\prime=[G,G]\) be the commutator subgroup of \(G\), and write \(\mathfrak g=\mathfrak z\oplus\mathfrak g^\prime\) for the Lie algebra of \(G\) where \(\mathfrak g^\prime\) is the semisimple part and \(\mathfrak z\) is the center. Let \(U\) be an open set with codim\(X\backslash U\geq 2\). Let \(P(\mathfrak g)=P(\mathfrak z\oplus\mathfrak g^\prime)=\mathfrak z U\oplus P(\mathfrak g^\prime)\) be the vector bundle associated to \(P\) by the adjoint representation of \(G\) in \(\mathfrak g\). A principal \(G\)-sheaf \(\mathcal{P}\) over \(X\) is a triple \(\mathcal{P}=(P,E,\psi)\) consisting of a torsion free sheaf \(E\) on \(X\), a principal \(G\)-bundle \(P\) on the maximal open set \(U_E\) where \(E\) is locally free, and an isomorphism of vector bundles \[ \psi:P(\mathfrak g^\prime)\overset\cong\rightarrow E|_{U_E}. \] Using the induced orthogonal Killing structure, it is possible to define an orthogonal algebra filtration \(0\subset E_{-l}\subseteq E_{-l+1}\subset\cdots\subset E_l=E\) of \(E\). Then a Hilbert polynomial for this filtration is defined, and is used to define (semi)stability in a way similar to the usual. Using the geometric invariant theory (of Mumford), there is a set of numerical data \(\tau=(d_1,\dots,d_q,c_i)\), and as in GIT, values of \(\tau\) determine (semi)stability. To construct the announced coarse moduli space, the authors define families of semistable principal \(G\)-sheaves, and an equivalence of such. Again, when \(X\) is a curve, this is a generalization of Ramanathan’s notion. Also a sheafification of the family-functor is given: \(\widetilde{F}^\tau_G\) is the sheafification of the family functor with numerical data \(\tau\). This is to tell about the relations with stacks. The main result of the article is: For a polarized complex smooth projective variety \(X\) there is a coarse projective moduli space of \(S\)-equivalence classes of semistable \(G\)-sheaves on \(X\) with fixed numerical invariants.

The article contains eleven pages with preliminaries where the concepts above are being defined. Chapter 1 contains the definition of Lie-sheaves and the construction of the moduli spaces of various tensors. Chapter 2 and 3 makes use of the category of complex analytic spaces to construct a subspace of a product of Jacobians parametrizing reductions to \(G\). Chapter 4 contains the construction of the moduli space announced in the main result. This construction involves a lemma given by Ramanathan, proving the existence of good quotients. Chapter 5 contains results proving that the constructions given here are true generalizations of Ramanathans results. Also, several other details are skipped to this chapter.

It remains to say that this article is rather deep. The proofs require a deep understanding of geometric invariant theory, analytic theory, and the theory of sheaves. Also noncommutative theory is involved in the definition of Lie-sheaves. Diving into some of the proofs even involves a deep knowledge of deformation theory. It will also help the reader to be familiar to the concept of stacks.

Reviewer: Arvid Siqveland (Kongsberg)

### MSC:

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

14D22 | Fine and coarse moduli spaces |

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

14L24 | Geometric invariant theory |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14A22 | Noncommutative algebraic geometry |