Stability conditions in families

We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application 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 the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, 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.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

Finiteness and infiniteness results for Torelli groups of (hyper-)Kähler manifolds

The Torelli group $$\mathcal T(X)$$ T ( X ) of a closed smooth manifold X is the subgroup of the mapping class group $$\pi _0(\mathrm {Diff}^+(X))$$ π 0 ( Diff + ( X ) ) consisting of elements which act trivially on the integral cohomology of X. In this note we give counterexamples to Theorem 3.4 by Verbitsky (Duke Math J 162(15):2929–2986, 2013) which states that the Torelli group of simply connected Kähler manifolds of complex dimension $$\ge 3$$ ≥ 3 is finite. This is done by constructing under some mild conditions homomorphisms $$J: \mathcal T(X) \rightarrow H^3(X;\mathbb Q)$$ J : T ( X ) → H 3 ( X ; Q ) and showing that for certain Kähler manifolds this map is non-trivial. We also give a counterexample to Theorem 3.5 (iv) in (Verbitsky in Duke Math J 162(15):2929–2986, 2013) where Verbitsky claims that the Torelli group of hyperkähler manifolds are finite. These examples are detected by the action of diffeomorphsims on $$\pi _4(X)$$ π 4 ( X ) . Finally we confirm the finiteness result for the special case of the hyperkähler manifold $$K^{[2]}$$ K [ 2 ] .

Transverse hilbert schemes, bi-hamiltonian systems and hyperkähler geometry

Dedicated to the memory of Sir Michael Francis Atiyah (1929-2019) We give a characterization of Atiyah’s and Hitchin’s transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperkähler manifolds arising from the transverse Hilbert scheme construction, with particular attention paid to the monopole moduli spaces.