Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeneralized Deligne-Hitchin Twistor Spaces: Construction and Properties
76
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we generalize the construction of Deligne-Hitchin twistor space by gluing two certain Hodge moduli spaces. We investigate such generalized Deligne-Hitchin twistor space as a complex analytic manifold, more precisely, we show it admits a global smooth trivialization such that the induced product metric is balanced, and it carries a semistable holomorphic tangent bundle. Moreover, we study the automorphism groups of the Hodge moduli spaces and the generalized Deligne-Hitchin twistor space.
rate research
Read More
Fix integers $ggeq 3$ and $rgeq 2$, with $rgeq 3$ if $g=3$. Given a compact connected Riemann surface $X$ of genus $g$, let $MDH(X)$ denote the corresponding $text{SL}(r, {mathbb C})$ Deligne--Hitchin moduli space. We prove that the complex analytic space $MDH(X)$ determines (up to an isomorphism) the unordered pair ${X, overline{X}}$, where $overline{X}$ is the Riemann surface defined by the opposite almost complex structure on $X$.
We develop a theory of twistor spaces for supersingular K3 surfaces, extending the analogy between supersingular K3 surfaces and complex analytic K3 surfaces. Our twistor spaces are obtained as relative moduli spaces of twisted sheaves on universal gerbes associated to the Brauer groups of supersingular K3 surfaces. In rank 0, this is a geometric incarnation of the Artin-Tate isomorphism. Twistor spaces give rise to curves in moduli spaces of twisted supersingular K3 surfaces, analogous to the analytic moduli space of marked K3 surfaces. We describe a theory of crystals for twisted supersingular K3 surfaces and a twisted period morphism from the moduli space of twisted supersingular K3 surfaces to this space of crystals. As applications of this theory, we give a new proof of the Ogus-Torelli theorem modeled on Verbitskys proof in the complex analytic setting and a new proof of the result of Rudakov-Shafarevich that supersingular K3 surfaces have potentially good reduction. These proofs work in characteristic 3, filling in the last remaining gaps in the theory. As a further application, we show that each component of the supersingular locus in each moduli space of polarized K3 surfaces is unirational.
Let $X$ be a compact Riemann surface $X$ of genus at--least two. Fix a holomorphic line bundle $L$ over $X$. Let $mathcal M$ be the moduli space of Hitchin pairs $(E ,phiin H^0(End(E)otimes L))$ over $X$ of rank $r$ and fixed determinant of degree $d$. We prove that, for some numerical conditions, $mathcal M$ is irreducible, and that the isomorphism class of the variety $mathcal M$ uniquely determines the isomorphism class of the Riemann surface $X$.
We describe the relation between supersymmetric sigma-models on hyperkahler manifolds, projective superspace, and twistor space. We review the essential aspects and present a coherent picture with a number of new results.
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 Deligne-Mumford stacks. We study the deformation and obstruction theories of stable pairs, and then prove the existence of virtual fundamental classes for some cases of dimension two and three. This leads to a definition of Pandharipande-Thomas invariants on three-dimensional smooth projective Deligne-Mumford stacks.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
