Derived Quot schemes

Realizing a part of the Derived Deformation Theory program, we construct a "derived" analog of the Grothendieck's Quot scheme parametrizing subsheaves in a given coherent sheaf F on a smooth projective variety X. This analog is a differential graded manifold RQuot_h(F) (so it is always smooth in an appropriate sense) whose tangent space at a point represented by a subsheaf K in F, is a cochain complex quasiisomorphic to RHom(K, F/K).Comment: 46 pages, AMS-TeX. Revised version, to appear in Ann. Sci. EN

GLSM's for partial flag manifolds

In this paper we outline some aspects of nonabelian gauged linear sigma models. First, we review how partial flag manifolds (generalizing Grassmannians) are described physically by nonabelian gauged linear sigma models, paying attention to realizations of tangent bundles and other aspects pertinent to (0,2) models. Second, we review constructions of Calabi-Yau complete intersections within such flag manifolds, and properties of the gauged linear sigma models. We discuss a number of examples of nonabelian GLSM's in which the Kahler phases are not birational, and in which at least one phase is realized in some fashion other than as a complete intersection, extending previous work of Hori-Tong. We also review an example of an abelian GLSM exhibiting the same phenomenon. We tentatively identify the mathematical relationship between such non-birational phases, as examples of Kuznetsov's homological projective duality. Finally, we discuss linear sigma model moduli spaces in these gauged linear sigma models. We argue that the moduli spaces being realized physically by these GLSM's are precisely Quot and hyperquot schemes, as one would expect mathematically.Comment: 57 pp, LaTeX; v3: refs added, material on weighted Grassmannians adde

Orbifold Quasimap Theory

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).Comment: 42 pages, Add an extended result (see remark 5.7), To appear in Mathematische Annale