Limit stable objects on Calabi-Yau 3-folds

Abstract

In this paper, we introduce new enumerative invariants of curves on Calabi-Yau 3-folds via certain stable objects in the derived category of coherent sheaves. We introduce the notion of limit stability on the category of perverse coherent sheaves, a subcategory in the derived category, and construct the moduli spaces of limit stable objects. We then define the counting invariants of limit stable objects using Behrend's constructible functions on that moduli spaces. It will turn out that our invariants are generalizations of counting invariants of stable pairs introduced by Pandharipande and Thomas. We will also investigate the wall-crossing phenomena of our invariants under change of stability conditions. I(X) = {I C ⊂ O X | C ⊂ X is a subscheme with dim C ≤ 1}.

Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

“…Note that universal Atiyah classes were studied via the full cotangent complex in [4], but for the truncated version our approach is much more elementary. A topic of current interest [9,10,16,17,18] is defining invariants of threefolds X (and particularly Calabi-Yau threefolds) using virtual cycles on moduli spaces of objects of D b (X) satisfying some natural conditions (such as stability or simplicity). This paper fills the gap mentioned in [16,Sect.…”

Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture

“…Stability conditions at the large-volume limit had been previously constructed in [Bay09] and [Tod09a] as "polynomial" or "limit" stability condition. As an additional confirmation that the heart A ω,B seems to give the right construction, we prove: Proposition 1.3.2 (Lemma 6.2.1 and Lemma 6.2.2).…”

Stable pairs and BPS invariants

“…Wall-crossing. Arend Bayer [1] and Yukinoba Toda [33] have made the beautiful observation that (2.4) should be seen as a wall-crossing formula. In fact, the wall-crossing is much simpler than the wall-crossing conjectured in [26] to equate the invariants P n,β to the reduced DT invariants of [23].…”

“…If instead we use Bayer's polynomial stability conditions or Toda's limit stability conditions, then the analysis can be made precise. These stability conditions have been constructed, and the stable objects are as claimed above [1,33].…”

“…Perhaps (2.14) is the first nontrivial example of a wall-crossing formula in the derived category that can be rigorously proved. Toda [33] has gone further with wall-crossings for arbitrary (rather than irreducible) stable pairs. Using the work of Joyce [14], he proves analogues of Theorems 1 and 2 for the Euler characteristics of the moduli spaces of stable pairs.…”