Related papers: Cusps of the K\"ahler moduli space and stability c…

We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In…

We show that for many moduli spaces M of torsion sheaves on K3 surfaces S, the functor D(S) -> D(M) induced by the universal sheaf is a P-functor, hence can be used to construct an autoequivalence of D(M), and that this autoequivalence can…

In this paper, we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions. The resulting spaces have chamber structures such that each chamber corresponds to a…

Let $(X,H)$ be a smooth, projective, polarized surface over $\mathbb{C}$, and let $v \in K_{\mathrm{num}}(X)$ be a class of positive rank. We prove that for certain Bridgeland stability conditions $\sigma = (\mathcal{A}, Z)$ "on the…

The space of Bridgeland stability conditions is a complex manifold that can be attached to a triangulated category, of which it encodes some homological properties. These notes are an introduction to this topic, with a focus on examples…

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the…

Tom Bridgeland constructed explicit stability conditions on K3 surfaces. This paper attempts to shed more light on these particular examples, especially on the hearts of the underlying t-structures. We prove that two K3 surfaces X and X'…

This is a survey article on mirror symmetry and Fourier-Mukai partners of Calabi-Yau threefolds with Picard number one based on recent works by the authors [HoTa1,2,3,4]. For completeness, mirror symmetry and Fourier-Mukai partners of K3…

On a smooth projective threefold, we show that there are only two isomorphism types for the moduli of stable objects with respect to Bayer's standard polynomial Bridgeland stability - the moduli of Gieseker-stable sheaves and the moduli of…

Let X be a K3 surface with a polarization H of degree H^2=2rs and with a primitive Mukai vector (r,H,s). The moduli space of sheaves over X with the isotropic Mukai vector (r,H,s) is again a K3 surface Y. We prove that Y\cong X, if Picard…

We prove that a stability condition on a K3 surface is determined by the masses of spherical objects up to a natural $\mathbb{C}$-action. This is motivated by the result of Huybrechts and the recent proposal of Bapat-Deopurkar-Licata on the…

We introduce an analogue of Bridgeland's stability conditions for polarised varieties. Much as Bridgeland stability is modelled on slope stability of coherent sheaves, our notion of Z-stability is modelled on the notion of K-stability of…

We give a structure result on the set of locally constant stability conditions, $\operatorname{Stab}(\mathcal{D}/R)$, defined by Halpern-Leistner-Robotis showing that it has the structure of a complex manifold, in total analogy with…

We review a proof of the well know result stating that moduli spaces of stable sheaves with fixed Chern character on a polarized $K3$ surface are deformations of a hyperk\"ahler variety of Type $K3^{[n]}$ (if a suitable numerical hypothesis…

By using a Fourier-Mukai transform for sheaves on K3 surfaces we show that for a wide class of K3 surfaces $X$ the punctual Hilbert schemes $\Hilb^n(X)$ can be identified, for all $n\geq 1$, with moduli spaces of Gieseker stable vector…

For any smooth projective moduli space $M$ of Gieseker stable sheaves on a complex projective K3 surface (or an abelian surface) S, we prove that the Chow motive $\mathfrak{h}(M)$ becomes a direct summand of a motive $\bigoplus…

We show that the K-moduli spaces of log Fano pairs $(\mathbb{P}^3, cS)$ where $S$ is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily-Borel compactification of moduli of quartic K3 surfaces as $c$…

For infinitely many $d$, Hassett showed that special cubic fourfolds of discriminant $d$ are related to polarized K3 surfaces of degree $d$ via their Hodge structures. For half of the $d$, each associated K3 surface $(S,L)$ canonically…

We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds, using the one-loop semiclassical approximation. The data being parameterized includes a choice of complex structure on the manifold, as well as some ``extra…

We construct the moduli space of cubic surfaces which do not admit a Sylvester form as an arithmetic quotient, and determine the graded ring of modular forms of even weights.