Stability conditions in families
The main aim of the paper is to study behaviour of Bridgeland's stability conditions in flat families of projective varieties. Let us fix a (numerical) stability condition \(\sigma\) in \(D^b (X)\) over \(S\). The authors prove that the stack parametrizing \(\sigma\)-semistable objects with fixed numerical class is algebraic of finite type over \(S\). Then one can deduce existence of moduli spaces in the relative case. Here current technology allows to get this result only in the characteristic \(0\) case. The problem with even defining the above stack is that it is not clear what a stability condition is in the relative case, i.e., for (some) derived categories of coherent sheaves on the fibers of a flat family \(X\to S\) of projective schemes over some base scheme. One of the main aims of the paper is to define and study this notion, taking into account also fibers over non-closed points of the base. The definition of stability condition in the relative case includes good openness of geometric stability in families of objects, boundedness of semistable objects etc. that are usually non-trivial theorems in the case of moduli spaces of semistable sheaves. One of the advantages of the authors' approach is that they can prove a generalization of Bridgeland's deformation theorem in the relative case. Another important result says that for a polarized flat family of smooth projective varieties one gets existence of relative stability conditions when the corresponding construction is known fiberwise. Currently this is restricted to low dimensions: \(1\), \(2\) and sometimes \(3\). They use this result to show deformation-invariance of Donaldson-Thomas invariants for some Calabi-Yau threefolds (e.g., for quintic threefolds). The authors show also some other applications of their result. One of the most important is the generalization of Mukai's theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to a new proof of the integral Hodge conjecture and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type. Reviewer's remark: Even though the paper is very long, some standard arguments are only sketched. To shorten the paper the authors define stacks as ``functors'' from schemes to groupoids meaning pseudo-functors and they consider ``subfunctors'', etc. So an unexperienced reader who read about stacks in other sources should be rather careful.
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