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We extend to orbifolds the quasimap theory of arXiv:0908.4446 and arXiv:1106.3724, as well as the genus zero wall-crossing results from arXiv:1304.7056 and arXiv:1401.7417. As a consequence, we obtain generalizations of orbifold mirror theorems, in particular, of the mirror theorem for toric orbifolds recently proved independently by Coates, Corti, Iritani, and Tseng (arXiv:1310.4163).
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We construct relative moduli spaces of semistable pairs on a family of projective Deligne-Mumford stacks. We define moduli stacks of stable orbifold Pandharipande-Thomas pairs on stacks of expanded degenerations and pairs, and then show they are separated and proper Deligne-Mumford stacks of finite type. As an application, we present the degeneration formula for the absolute and relative orbifold Pandharipande-Thomas invariants.
Let $k$ be a field and $X/k$ be a smooth quasiprojective orbifold. Let $Xto underline{X}$ be its coarse moduli space. In this paper we study the Brauer group of $X$ and compare it with the Brauer group of the smooth locus of $underline{X}$.
We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T^{poly}_X on X, and is generated as a T^{poly}_X-algebra by the sum of the determinants det(N_{X^g}) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevichs formality theorem, for the cup product, does not hold for Deligne--Mumford stacks in general. We discuss relationships with orbifold cohomology, extending Ruans cohomological conjectures. This employs a trivialization of the determinants in the case of a symplectic group action on a symplectic variety X, which requires (for the cup product) a nontrivial normalization missing in previous literature.
We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle T X. In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by D and multiplicities are bounded by the ramification indices of the components of D. We estimate precisely the curvature tensor of the corresponding directed structure V D in the general orbifold case-with a special attention to the compact case D = 0 and to the logarithmic situation where the ramification indices are infinite. Using holomorphic Morse inequalities on the tautological line bundle of the projectivized orbifold Green-Griffiths bundle, we finally obtain effective sufficient conditions for the existence of global orbifold jet differentials.
We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces.
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