Wall-crossing in genus-zero hybrid theory

The hybrid model is the Landau–Ginzburg-type theory that is expected, via the Landau–Ginzburg/ Calabi–Yau correspondence, to match the Gromov–Witten theory of a complete intersection in weighted projective space. We prove a wall-crossing formula exhibiting the dependence of the genus-zero hybrid model on its stability parameter, generalizing the work of [21] for quantum singularity theory and paralleling the work of Ciocan-Fontanine–Kim [7] for quasimaps. This completes the proof of the genus-zero Landau– Ginzburg/Calabi–Yau correspondence for complete intersections of hypersurfaces of the same degree, as well as the proof of the all-genus hybrid wall-crossing [11].

Birational geometry of moduli spaces of perverse coherent sheaves on blow-ups

In order to study the wall-crossing formula of Donaldson type invariants on the blown-up plane, Nakajima–Yoshioka constructed a sequence of blow-up/blow-down diagrams connecting the moduli space of torsion free framed sheaves on projective plane, and that on its blow-up. In this paper, we prove that Nakajima–Yoshioka’s diagram realizes the minimal model program. Furthermore, we obtain a fully-faithful embedding between the derived categories of these moduli spaces.

Yang-Yang functions, monodromy and knot polynomials

We derive a structure of ℤ[t, t−1]-module bundle from a family of Yang-Yang functions. For the fundamental representation of the complex simple Lie algebra of classical type, we give explicit wall-crossing formula and prove that the monodromy representation of the ℤ[t, t−1]-module bundle is equivalent to the braid group representation induced by the universal R-matrices of Uh(g). We show that two transformations induced on the fiber by the symmetry breaking deformation and respectively the rotation of two complex parameters commute with each other.

Virtual cycles of gauged Witten equation

In this paper, we construct virtual cycles on moduli spaces of solutions to the perturbed gauged Witten equation over a fixed smooth r-spin curve, under the framework of [G. Tian and G. Xu, Analysis of gauged Witten equation, J. reine angew. Math. 740 (2018), 187–274]. Together with the wall-crossing formula proved in the companion paper [G. Tian and G. Xu, A wall-crossing formula for the correlation function of gauged linear σ-model, preprint], this paper completes the construction of the correlation function for the gauged linear σ-model announced in [G. Tian and G. Xu, Correlation functions in gauged linear σ-model, Sci. China Math. 59 (2016), 823–838] as well as the proof of its invariance.

Localizing virtual structure sheaves for almost perfect obstruction theories

Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the K-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a K-theoretic wall-crossing formula for simple $\mathbb{C} ^\ast $ -wall crossings and define K-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.

The wall-crossing behavior for Bridgeland’s stability conditions on abelian and K3 surfaces

The wall-crossing behavior for Bridgeland’s stability conditions on the derived category of coherent sheaves on K3 or abelian surface is studied. We introduce two types of walls. One is called the wall for categories, where thet-structure encoded by stability condition is changed. The other is the wall for stabilities, where stable objects with prescribed Mukai vector may get destabilized. Some fundamental properties of walls and chambers are studied, including the behavior under Fourier–Mukai transforms. A wall-crossing formula of the counting of stable objects will also be derived. As an application, we will explain previous results on the birational maps induced by Fourier–Mukai transforms on abelian surfaces. These transformations turns out to coincide with crossing walls of certain property.

MOTIVIC DONALDSON–THOMAS THEORY AND THE ROLE OF ORIENTATION DATA

In this paper we introduce and motivate the concept of orientation data, as it appears in the framework for motivic Donaldson–Thomas theory built by Kontsevich and Soibelman. By concentrating on a single simple example we explain the role of orientation data in defining the integration map, a central component of the wall crossing formula.

A wall-crossing formula for degrees of Real central projections

The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.