Related papers: Uhlenbeck-Donaldson compactification for framed sh…
Let M be the moduli scheme of canonically polarized manifolds with Hilbert polynomial h. We construct for a given finite set I of natural numbers m>1 with h(m)>0 a projective compactification M' of the reduced scheme underlying M such that…
We prove that any flat family $(\mathcal{ F}_u)_{u\in U}$ of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus $U^{\mathrm {ss}}:=\{u\in U \ |\ \mathcal{ F}_u\hbox{ is slope semi-stable}\}$…
The space of smooth curves admits a beautiful compactification by the moduli space of Deligne-Mumford stable curves. In this paper, we undertake a systematic investigation of alternate modular compactifications of the space of smooth…
We generalize the construction of a moduli space of semistable pairs parametrizing isomorphism classes of morphisms from a fixed coherent sheaf to any sheaf with fixed Hilbert polynomial under a notion of stability to the case of projective…
We construct a compactification of the moduli spaces of abelian differentials on Riemann surfaces with prescribed zeroes and poles. This compactification, called the moduli space of multi-scale differentials, is a complex orbifold with…
We present a novel notion of stable objects in a triangulated category. This Postnikov-stability is preserved by equivalences. We show that for the derived category of a projective variety this notion includes the case of semistable…
A curve, that is, a connected, reduced, projective scheme of dimension 1 over an algebraically closed field, admits two types of compactifications of its (generalized) Jacobian: the moduli schemes of P-quasistable torsion-free, rank-1…
Let X be a smooth projective variety over C. We find the natural notion of semistable orthogonal bundle and construct the moduli space, which we compactify by considering also orthogonal sheaves, i.e. pairs (E,\phi), where E is a torsion…
We construct modular compactifications of the universal Jacobian stack over the moduli stack of reduced curves with marked points depending on stability parameters obtained out of fixing a vector bundle on the universal curve. When…
We generalize the construction of M. Lieblich for the compactification of the moduli stack of $\PGL_r$-bundles on algebraic spaces to the moduli stack of Tanaka-Thomas $\PGL_r$-Higgs bundles on algebraic schemes. The method we use is the…
We study moduli spaces of sheaves over non-projective K3 surfaces. More precisely, if $v=(r,\xi,a)$ is a Mukai vector on a K3 surface $S$ with $r$ prime to $\xi$ and $\omega$ is a "generic" K\"ahler class on $S$, we show that the moduli…
We construct a relative compactification of Dolbeault moduli spaces of Higgs bundles for reductive algebraic groups on families of projective manifolds that is compatible with the Hitchin morphism.
We construct a new compactification of the moduli space of maps from pointed nonsingular projective stable curves to a nonsingular projective variety with prescribed ramification indices at the points. It is shown to be a proper…
We classify the Deligne-Mumford stacks M compactifying the moduli space of smooth $n$-pointed curves of genus one under the condition that the points of M represent Gorenstein curves with distinct markings. This classification uncovers new…
We construct proper good moduli spaces for moduli stacks of Bridgeland semistable orthosymplectic complexes on a complex smooth projective variety, which we propose as a candidate for compactifying moduli spaces of principal bundles for the…
We present a method for compactifying stacks of $\PGL_n$-torsors (Azumaya algebras) on algebraic spaces. In particular, when the ambient space is a smooth projective surface we use our methods to show that various moduli spaces are…
Let $X$ be a compact Calabi-Yau 3-fold, and write $\mathcal M,\bar{\mathcal M}$ for the moduli stacks of objects in coh$(X),D^b$coh$(X)$. There are natural line bundles $K_{\mathcal M}\to\mathcal M$, $K_{\bar{\mathcal M}}\to\bar{\mathcal…
We study the behavior of slope-stability of reflexive twisted sheaves over a normal projective variety $X$ under pullback along a cover. Slope-stability is always preserved if the cover does not factor via a quasi-\'etale cover. Fixing the…
A projective moduli space of pairs (C,E) where E is a slope- semistable torsion free sheaf of uniform rank on a Deligne- Mumford stable curve C is constructed via G.I.T. There is a natural SL x SL action on the relative Quot scheme over the…
We study the following question: Let $(X,g)$ be a compact Gauduchon surface, $(E,h)$ be a differentiable rank $r$ vector bundle on $X$, ${\mathcal{D}}$ be a fixed holomorphic structure on $D:=\det(E)$ and $a$ be the Chern connection of the…