Equivariant stable sheaves and toric GIT

For $(X,\,L)$ a polarized toric variety and $G\subset \mathrm {Aut}(X,\,L)$ a torus, denote by $Y$ the GIT quotient $X/\!\!/G$ . We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on $Y$ to the category of torus equivariant reflexive sheaves on $X$ . We show, under a genericity assumption on $G$ , that slope stability is preserved by these functors if and only if the pair $((X,\,L),\,G)$ satisfies a combinatorial criterion. As an application, when $(X,\,L)$ is a polarized toric orbifold of dimension $n$ , we relate stable equivariant reflexive sheaves on certain $(n-1)$ -dimensional weighted projective spaces to stable equivariant reflexive sheaves on $(X,\,L)$ .

Comparison of Poisson structures on moduli spaces

Let X be a complex irreducible smooth projective curve, and let $${{\mathbb {L}}}$$ L be an algebraic line bundle on X with a nonzero section $$\sigma _0$$ σ 0 . Let $${\mathcal {M}}$$ M denote the moduli space of stable Hitchin pairs $$(E,\, \theta )$$ ( E , θ ) , where E is an algebraic vector bundle on X of fixed rank r and degree $$\delta $$ δ , and $$\theta \, \in \, H^0(X,\, {\mathcal {E}nd}(E)\otimes K_X\otimes {{\mathbb {L}}})$$ θ ∈ H 0 ( X , E n d ( E ) ⊗ K X ⊗ L ) . Associating to every stable Hitchin pair its spectral data, an isomorphism of $${\mathcal {M}}$$ M with a moduli space $${\mathcal {P}}$$ P of stable sheaves of pure dimension one on the total space of $$K_X\otimes {{\mathbb {L}}}$$ K X ⊗ L is obtained. Both the moduli spaces $${\mathcal {P}}$$ P and $${\mathcal {M}}$$ M are equipped with algebraic Poisson structures, which are constructed using $$\sigma _0$$ σ 0 . Here we prove that the above isomorphism between $${\mathcal {P}}$$ P and $${\mathcal {M}}$$ M preserves the Poisson structures.

Moduli spaces of stable sheaves over quasi-polarized surfaces, and the relative Strange Duality morphism

The main result of the present paper is a construction of relative moduli spaces of stable sheaves over the stack of quasipolarized projective surfaces. For this, we use the theory of good moduli spaces, whose study was initiated by Alper. As a corollary, we extend the relative Strange Duality morphism to the locus of quasipolarized K3 surfaces.

Towards refining the topological strings on compact Calabi-Yau 3-folds

We make a proposal for calculating refined Gopakumar-Vafa numbers (GVN) on elliptically fibered Calabi-Yau 3-folds based on refined holomorphic anomaly equations. The key examples are smooth elliptic fibrations over (almost) Fano surfaces. We include a detailed review of existing mathematical methods towards defining and calculating the (unrefined) Gopakumar-Vafa invariants (GVI) and the GVNs on compact Calabi-Yau 3-folds using moduli of stable sheaves, in a language that should be accessible to physicists. In particular, we discuss the dependence of the GVNs on the complex structure moduli and on the choice of an orientation. We calculate the GVNs in many instances and compare the B-model predictions with the geometric calculations. We also derive the modular anomaly equations from the holomorphic anomaly equations by analyzing the quasi-modular properties of the propagators. We speculate about the physical relevance of the mathematical choices that can be made for the orientation.

The Kodaira dimension and singularities of moduli of stable sheaves on some elliptic surfaces

Let [Formula: see text] be an elliptic surface over [Formula: see text] with [Formula: see text], and [Formula: see text] be the moduli scheme of rank-two stable sheaves [Formula: see text] on [Formula: see text] with [Formula: see text] in [Formula: see text]. We look into defining equations of [Formula: see text] at its singularity [Formula: see text], partly because if [Formula: see text] admits only canonical singularities, then the Kodaira dimension [Formula: see text] can be calculated. We show the following: (A) [Formula: see text] is at worst canonical singularity of [Formula: see text] if the restriction of [Formula: see text] to the generic fiber of [Formula: see text] has no rank-one subsheaf, and if the number of multiple fibers of [Formula: see text] is a few. (B) We obtain that [Formula: see text] and the Iitaka program of [Formula: see text] can be described in purely moduli-theoretic way for [Formula: see text], when [Formula: see text], [Formula: see text] has just two multiple fibers, and one of its multiplicities equals [Formula: see text]. (C) On the other hand, when [Formula: see text] has a rank-one subsheaf, it may be insufficient to look at only the degree-two part of defining equations to judge whether [Formula: see text] is at worst canonical singularity or not.

Moduli of stable sheaves supported on curves of genus three contained in a quadric surface

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the variation of the moduli spaces of α-semi-stable pairs. We classify the stable sheaves using locally free resolutions or extensions. We give a global description: the moduli space is obtained from a certain flag Hilbert scheme by performing two flips followed by a blow-down.

T-dual solutions of the Hull–Strominger system on non-Kähler threefolds

We construct new examples of solutions of the Hull–Strominger system on non-Kähler torus bundles over K3 surfaces, with the property that the connection {\nabla} on the tangent bundle is Hermite–Yang–Mills. With this ansatz for the connection {\nabla}, we show that the existence of solutions reduces to known results about moduli spaces of slope-stable sheaves on a K3 surface, combined with elementary analytical methods. We apply our construction to find the first examples of T-dual solutions of the Hull–Strominger system on compact non-Kähler manifolds with different topology.

Non-reduced moduli spaces of sheaves on multiple curves

Some coherent sheaves on projective varieties have a non-reduced versal deformation space; for example, this is the case for most unstable rank 2 vector bundles on ℙ2, see [18]. In particular, some moduli spaces of stable sheaves are non-reduced. We consider some sheaves on ribbons (double structures on smooth projective curves): let E be a quasi locally free sheaf of rigid type and let 𝓔 be a flat family of sheaves containing E. We find that 𝓔 is a reduced deformation of E when some canonical family associated to 𝓔 is also flat. We consider also a deformation of the ribbon to reduced projective curves with two components, and find that E can be deformed in two distinct ways to sheaves on the reduced curves. In particular some components M of the moduli spaces of stable sheaves deform to two components of the moduli spaces of sheaves on the reduced curves, and M appears as the “limit” of varieties with two components, whence the non-reduced structure of M.