Vortices and coupled vortices arise from Yang-Mills-Higgs theories and can be viewed as generalizations or analogues to Yang-Mills connections and, in particular, Hermitian-Yang-Mills connections. We proved an analytic compactification of the moduli spaces of vortices and coupled vortices on hermitian vector bundles over compact Kähler manifolds. In doing so we introduced the concept of ideal coupled vortices and characterized the singularities of ideal coupled vortices as well as Hermitian-Yang-Mills connections.
We exhibit the first non-trivial concrete examples of Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds in all complex dimensions bigger than two (Fano K-moduli spaces). We also discuss potential applications to explicit study of moduli spaces of K-stable Fano manifolds with large anti-canonical volume. Our arguments are based on recent progress about the geometry of metric tangent cones and on related ideas about the algebro-geometric study of singularities of K-stable Fano varieties.
We give a characterization of toroidal (resp., semi-toric) compactifications due to Ash-Mumford-Rapoport-Tai (resp., Looijenga) as log minimal models and apply it to study weak K-moduli compactifications, giving a different proof to a theorem of Alexeev-Engel. We also discuss towards further generalization, in particular revisit Shah-Sterk compactification of moduli of polarized Enriques surfaces to show compatibility with log K-stability.
We consider the set of solutions to the rho-vortex equations over a Kahler surface and prove a Uhlenbeck compactness result, namely that a sequence of solutions with the same energy converge to the sum of a solution of smaller energy and deltas of Dirac.
This note is a summary of our work [OO] which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to discuss their Gromov-Hausdorff limits along any sequences, which are even not necessarily "maximally degenerating". Our results also give a proof of Kontsevich-Soibelman [KS04, Conjecture 1] (cf., [GW00, Conjecture 6.2]) in the case of K3 surfaces as a byproduct.
We provide a moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics via compactification of moduli varieties of Morgan-Shalen and Satake type. In patricular, we use it to study the Gromov-Hausdorff limits of hyperKahler metrics with fixed diameters, especially for K3 surfaces.
For moduli of polarized smooth K-trivial a.k.a., Calabi-Yau varieties in a general sense, we revisit a classical problem of constructing its "weak K-moduli" compactifications which parametrizes K-semistable (i.e., semi-log-canonical K-trivial) degenerations. Although weak K-moduli is not unique in general, they always contain a unique partial compactification (K-moduli). Our main theorem is the log minimality of their normalizations, under some conditions. Partially to confirm that known examples satisfy the conditions, we also include an appendix on the algebro-geometric reconstruction of Kul…
In this paper we consider a canonical compactification of Hitchin's moduli space of stable Higgs bundles with fixed determinant of odd degree over a Riemann surface, producing a projective variety by gluing in a divisor at infinity. We give a detailed study of the compactified space, the divisor at infinity and the moduli space itself. In doing so we reprove some assertions of Laumon and Thaddeus on the nilpotent cone.
In this paper we study the relationship between three compactifications of the moduli space of Hermitian-Yang-Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson-Uhlenbeck-Yau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by Gieseker-Maruyama semistable torsion-free sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves…
The aim of this paper is to construct the parabolic version of the Donaldson--Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algenraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non--emptiness of the moduli space of parabolic stable bundles of rank 2 and also prove the existence of components with smooth points.
This survey covers earlier work of the author as well as recent work on Riemann's moduli space, its canonical cell decomposition and compactification, and the related operadic structure of arc complexes.
For any degenerating Calabi-Yau family, we introduce new limit space which we call galaxy, whose dense subspace is the disjoint union of countably infinite open Calabi-Yau varieties, parametrized by the rational points of the Kontsevich-Soibelman's essential skeleton, while dominated by the Huber adification over the Puiseux series field. Other topics include: projective limits of toroidal compactifications, locally modelled on what we call the limit toric varieties, the way to attach tropicalized family to given Calabi-Yau family, which are weakly related to each other.
We compactify the classical moduli variety of compact Riemann surfaces by attaching moduli of (metrized) graphs as boundary. The compactifications do not admit the structure of varieties and patch together to form a big connected moduli space in which $\sqcup_{g} M_{g}$ is open dense. The metrized graphs, which are often studied as "tropical curves", are obtained as Gromov-Hausdorff collapse by fixing diameters of the hyperbolic metrics of the Riemann surfaces. This phenomenon can be also seen as an archemidean analogue of the tropicalization of Berkovich analytification of $M_{g}$ (cf., [ACP]…
We completely determine the moduli space M_{N,k} of k-vortices in U(N) gauge theory with N Higgs fields in the fundamental representation. Its open subset for separated vortices is found as the symmetric product (C x CP^{N-1})^k / S_k. Orbifold singularities of this space correspond to coincident vortices and are resolved resulting in a smooth moduli manifold. Relation to Kahler quotient construction is discussed.
We compactify the classical moduli variety $A_g$ of principally polarized abelian varieties of complex dimension $g$ by attaching the moduli of flat tori of real dimensions at most $g$ in an explicit manner. Equivalently, we explicitly determine the Gromov-Hausdorff limits of principally polarized abelian varieties. This work is analogous to the first of our series (available at arXiv:1406.7772v2), which compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov-Hausdorff limits along holomorphic family of abelian varieties and sho…
The moduli space metric and its Kahler potential for well-separated non-Abelian vortices are obtained in U(N) gauge theories with N Higgs fields in the fundamental representation.
There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface. The Deligne-Mumford compactification of the moduli space of curves then suggests a partial compactification of the moduli space of convex real projective structures: Allow the Riemann surface to degenerate to a stable nodal curve on which there is a regular cubic differential. We construct convex real projective structures on open surfaces corresponding to t…
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 pair $(\mathcal{D},\det(h))$. Does the complex space structure on ${\mathcal{M}}_a^{\mathrm{ASD}}(E)^*$ induced by the Kobayashi-Hitchin correspondence extend to a complex space structure on the Donaldson-Uhlenbeck compactification $\overline{\mathcal{M}}_a^\mathrm{ASD}(E)$? Our results answer this question in detail for the moduli spaces of $\mathrm{SU}(2)$…
Let $S$ be a closed, oriented surface with a finite (possibly empty) set of points removed. In this paper we relate two important but disparate topics in the study of the moduli space $\M(S)$ of Riemann surfaces: Teichmüller geometry and the Deligne-Mumford compactification. We reconstruct the Deligne-Mumford compactification (as a metric stratified space) purely from the intrinsic metric geometry of $\M(S)$ endowed with the Teichmüller metric. We do this by first classifying (globally) geodesic rays in $\M(S)$ and determining precisely how pairs of rays asymptote. We construct an "iterated ED…
We investigate the local geometry on the moduli space of G_2 structures that arises in compactifications of M-theory on holonomy G_2 manifolds. In particular, we determine the homogeneity properties of couplings of the associated N=1, D=4 supergravity under the scaling of moduli space coordinates. We then find some brane solitons of N=1, D=4 supergravity that are associated with wrapping M-branes on cycles of the compact space. These include cosmic strings and domain walls that preserve half of supersymmetry of the four-dimensional theory, and non-supersymmetric electrically and magnetically c…
In this paper, we give an equivariant compactification of the space PFlat(S) of homothety classes of half-translation structures on a compact, connected, orientable surface S. We introduce the space PMix(S) of homothety classes of mixed structures on S, that are CAT(0) tree-graded spaces in the sense of Drutu and Sapir, with pieces which are R-trees and completions of surfaces endowed with half-translation structures. Endowing Mix(S) with the equivariant Gromov topology, and using asymptotic cone techniques, we prove that PMix(S) is an equivariant compactification of PFlat(S), thus allowing us…
In this thesis, we study moduli in compactifications of ten-dimensional heterotic supergravity. We consider supersymmetric compactifications to four-dimensional maximally symmetric space, commonly referred to as the Strominger system. The compact part of space-time $X$ is a six-dimensional manifold of what we refer to as a heterotic $SU(3)$-structure. We show that this system can be put in terms of a holomorphic operator $\bar D$ on a bundle $\mathcal{Q}=T^*X\oplus\mathrm{End}(TX)\oplus\mathrm{End}(V)\oplus TX$, defined by a series of extensions. We proceed to compute the infinitesimal deforma…
In this note we show that for the group G = U(N) the space of Hecke modifications of a rank N vector bundle over a Riemann surface C coincides with the moduli space of solutions of certain non-abelian vortex equations over C . Through the recent work of Kapustin and Witten this then leads to an isomorphism between the moduli space of vortices and the moduli space of singular monopoles on the product of C with a closed interval I .
We make a detailed study of the moduli space of winding number two (k=2) axially symmetric vortices (or equivalently, of co-axial composite of two fundamental vortices), occurring in U(2) gauge theory with two flavors in the Higgs phase, recently discussed by Hashimoto-Tong (hep-th/0506022) and Auzzi-Shifman-Yung (hep-th/0511150). We find that it is a weighted projective space WCP^2_(2,1,1)=CP^2/Z_2. This manifold contains an A_1-type (Z_2) orbifold singularity even though the full moduli space including the relative position moduli is smooth. The SU(2) transformation properties of such vortic…
