arXiv research

Same cohomology for curves with levels, proving stable range.

We realize Stasheff's multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves.

We construct certain operations on stable moduli spaces and use them to compare cohomology of moduli spaces of closed manifolds with tangential structure. We obtain isomorphisms in a stable range provided the $p$-adic valuation of the Euler characteristics agree, for all primes $p$ not invertible in the coefficients fo…

The paper studies the moduli space of Higgs pairs and their geometric properties.

The study shows boundedness and constructs a moduli space for Calabi-Yau fibrations.

Constructs a space for stable holomorphic submersions over a fixed base.

Smooth torus actions on moduli spaces of super stable curves and maps of genus zero.

We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We show that evaluation maps, forgetful maps are real morphisms. We analyze the real part of the moduli space.

In this paper, certain natural and elementary polygonal objects in Euclidean space, {\it the stable polygons}, are introduced, and the novel moduli spaces ${\bfmit M}_{{\bf r}, ε}$ of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and i…

Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and equivalently, the universal moduli space of semistable holomorphic G^c-bundles. Th…

Given a one parameter flat family of polarized algebraic varieties, we show that any K-stable limit is unique. In particular, moduli spaces of K-stable polarized varieties are automatically Hausdorff when they exist. We also give a characterization of K-stable limits in terms of the CM line bundle, and some application…

We describe the moduli space of stable rank 2 parabolic bundles over an elliptic curve with 3 marked points.

We give a method to construct stable vector bundles whose rank divides the degree over curves of genus bigger than one. The method complements the one given by Newstead. Finally, we make some systematic remarks and observations in connection with rationality of moduli spaces of stable vector bundles.

We give definitions of moduli spaces of framed, r-Spin and Pin surfaces. We apply earlier work of the author to show that each of these moduli spaces exhibits homological stability, and we identify the stable integral homology with that of certain infinite loop spaces in each case. We further show that these moduli spa…

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 an…

Homotopy theory for $(2n+1)$-dimensional manifold triads with fixed boundary.

New Poisson structures found on Higgs bundle moduli spaces.

In this paper we consider the topological side of a problem which is the analogue of Sen's S-duality testing conjecture for Hitchin's moduli space of rank 2 stable Higgs bundles of fixed determinant of odd degree over a Riemann surface. We prove that all intersection numbers in the compactly supported cohomology vanish…

We show that the prequantum line bundle on the moduli space of flat $SU(2)$ connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of fl…

Constructs moduli spaces for Calabi-Yau cones and Sasaki-Einstein manifolds.

Paper connects volumes of moduli spaces of super Riemann surfaces to integrals over stable Riemann surfaces.

Study shows non-polyhedral structure in moduli spaces for n≥8.

We prove that the CM line bundle is ample on the proper moduli space which parametrizes KSBA stable varieties.

New approach to proving Chen-Donaldson-Sun theorem with examples.

We define and count lattice points in the moduli space of stable genus g curves with n labeled points. This extends a construction of the second author for the uncompactified moduli space. The enumeration produces polynomials with top degree coefficients tautological intersection numbers on the compactified moduli spac…

The paper studies Lagrangian structures in Higgs bundle moduli spaces and their conformal limits.

For $L \hookrightarrow X$ a Lagrangian embedding associated with a real homogeneous space, we construct the moduli space of stable holomorphic discs mapping to $(X,L)$ as an orbifold with corners equipped with a group action. Some essential constructions involving orbifolds with corners are also discussed, including th…

Moduli space of genus zero stable maps to the projective three-space naturally carries a real structure such that the fixed locus is a moduli space for real rational spatial curves with real marked points. The latter is a normal projective real variety. The singular locus being in codimension at least two, a first Stie…

We study the topology of the tropical moduli space parametrizing stable tropical curves of genus g with n marked points in which the bounded edges have total length 1, and prove that it is highly connected. Using the identification of this space with the dual complex of the boundary in the moduli space of stable algebr…

In this paper, we prove that the tangent bundle of the moduli space $\cSU_C(r,d)$ of stable bundles of rank $r>2$ and of fixed determinant of degree $d$ (such that $(r,d)=1$), on a smooth projective curve $C$ is always stable, in the sense of Mumford-Takemoto. This verifies a well-known conjecture, and is related to a …

This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of the moduli space leads to interesting questions of a combinatorial nature.

We study the natural Kähler metrics on moduli spaces of stable oriented pairs in a very general framework, and we prove a universal formula expressing the Kähler class of such a moduli space in terms of characteristic classes of the universal bundle. We use these results to compute explicitly the volumina of certain Qu…

The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have 0, 1, ..., 4 complex-conjugate pairs of roots respectively. We show that the GIT-stable completion of each of these five components admits the structure of an arithmetic rea…

We study actions of finite groups on moduli spaces of stable holomorphic vector bundles and relate the fixed-point sets of those actions to representation varieties of certain orbifold fundamental groups.

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 sta…

Geometric Invariant Theory applied to Kähler manifolds yields analytic models for vector bundles.

In this paper we modify the coordinate construction in our previous paper on the universal moduli space of pair consisting of a Riemann Surfaces and a stable holomorphic bundles on the Riemann Surface, so as to produce a new set of coordinates, which are in fact K\" ahler coordinates on this universal moduli space. Fur…

We investigate the geometry and topology of a standard moduli space of stable bundles on a Riemann surface, and use a generalization of the Verlinde formula to derive results on intersection pairings.

We determine the quantum cohomology of the moduli space of odd degree rank two stable vector bundles over a Riemann surface $Σ$ of any genus. This work together with dg-ga/9710029 prove that this quantum cohomology is isomorphic to the instanton Floer cohomology of the three manifold $Σ\times S^1$. (Note: There is some…

We present an explicit construction of the moduli spaces of rank 2 stable parabolic bundles of parabolic degree 0 over the Riemann sphere, corresponding to "optimum" open weight chambers of parabolic weights in the weight polytope. The complexity of the different moduli space' weight chambers is understood in terms of …

In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that …

Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's "Yang-Mills over a Riemann Surface" to compute Z/2-Betti numbers of these spaces, proving formulas recently obt…

We describe partial semi-simplicial resolutions of moduli spaces of surfaces with tangential structure. This allows us to prove a homological stability theorem for these moduli spaces, which often improves the known stability ranges and give explicit stability ranges in many new cases. In each of these cases the stable…

Two types of differentials are shown equivalent for compactifying moduli spaces.

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective…

The moduli space of stable bundles of rank 2 and degree 1 on a Riemann surface has rational cohomology generated by the so-called universal classes. The work of Baranovsky, King-Newstead, Siebert-Tian and Zagier provided a complete set of relations between these classes, expressed in terms of a recursion in the genus. …

Constructs geometric models for moduli spaces of Higgs bundles over Riemann sphere.

Constructs a new mathematical structure for Riemann surfaces with projective structures.