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Framed sheaves on projective stacks. (English) Zbl 1315.14001

The paper under review constructs moduli spaces of framed sheaves over projective stacks. These objects are related to Donaldson’s framed \(\mathrm{SU}(r)\)-instantons on the four-dimensional real space and play a key role in physics. Different extensions of the work of Donaldson to other noncompact \(4\)-dimensional Riemannian manifolds have been widely studied, where a suitable compactification is needed as well as a convenient way to relate the data in the compatifying locus. To overcome some of the difficulties it turns out to be useful to change the base space to a projective Deligne-Mumford stack.
Let \(\mathcal{X}\) be a normal projective irreducible stack of dimension \(d\) over an algebraically closed field of characteristic zero, endowed with a coarse moduli scheme \(\pi: \mathcal{X}\rightarrow X\) and a polarization \((\mathcal{G},\mathcal{O}_{X}(1))\) on \(\mathcal{X}\). The notion of polarization for stacks is a generalization of the one of very ample line bundles for projective varieties, and allows us to relate the sheaves on the stack to those on the moduli scheme. A framed sheaf is defined as a pair \((\mathcal{E},\phi_{\mathcal{E}})\) where \(\mathcal{E}\) is a coherent sheaf on \(\mathcal{X}\) and \(\phi_{\mathcal{E}}\) is a morphism from \(\mathcal{E}\) to a fixed coherent sheaf \(\mathcal{F}\), which is called the framing.
Fixing a polynomial \(P\) of degree \(d\) and a polynomial \(\delta\) of degree \(d-1\), Theorem 1.1 shows the existence of an algebraic stack \(\mathfrak{M}^{\mathrm{s(s)}}\) of finite type over \(k\) which admits a good moduli space \(\pi: \mathfrak{M}^{\mathrm{s(s)}}\rightarrow M^{\mathrm{s(s)}}\) which is a projective scheme, and such that the associated contravariant functor \([\mathfrak{M}^{\mathrm{s(s)}}]\) (which takes any \(k\)-scheme \(S\) to the set of isomorphism classes of objects in \(\mathfrak{M}^{\mathrm{s(s)}}(S)\)) is isomorphic to the moduli functor of \(\delta\)-(semi)stable framed sheaves on \(\mathcal{X}\) with Hilbert polynomial \(P\). This construction is made by using Geometric Invariant Theory but adapting the arguments to the stack setting, essentially proving the boundedness of the class of \(\delta\)-semistable objects and constructing a Quot-scheme parametrizing them. Theorem 1.2 calculates the tangent space and the obstruction to the smoothness of the moduli space in terms of hyper-Ext groups.
When \(\mathcal{X}\) is a projective irreducible orbifold of dimension \(2\), the authors consider framings along \(1\)-dimensional smooth integral closed substacks \(\mathcal{D}\). They are proven to be \(\delta\)-stable for a suitable choice of polynomial \(\delta\), hence there exists a quasi-projective scheme which is a fine moduli space for them (Theorem 1.3). Last part of the paper applies the previous results to the theory of \(2\)-dimensional orbifolds which are toric root stacks (Theorem 1.4).
The paper finishes with some appendixes. The first three of them show technical results about coherent sheaves on projective stacks: a semicontinuity theorem, a Serre duality theorem and a characterization of the dual of a coherent sheaf. Appendix \(D\) applies the techniques and results of the paper to study in detail the example of framed sheaves on stacky Hirzebruch surfaces.

MSC:

14A20 Generalizations (algebraic spaces, stacks)
14D20 Algebraic moduli problems, moduli of vector bundles
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
14D22 Fine and coarse moduli spaces
14D23 Stacks and moduli problems
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli

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