arXiv research

Tautological classes, or generalised Miller-Morita-Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend…

Study Euler and Chern classes of tautological line bundles on polygon moduli spaces.

The tautological ring of strata of differentials is characterized based on the presence of poles.

Study smooth bundles over BSU(2) not induced by actions.

Constructs differential characters on nonlinear Graßmannians.

The study explores tautological classes and their vanishing/nontriviality for manifolds with odd dimensions.

Starting from the description of Segre forms as direct images of (powers of) the first Chern form of the (anti)tautological line bundle on the projectivized bundle of a holomorphic hermitian vector bundle, we derive a version of the pointwise Kobayashi-Lübke inequality.

The paper shows how uniform RC-positivity on manifolds implies projectivity and rational connectedness.

For $G$ a finite subgroup of ${\rm SL}(3,{\mathbb C})$ acting freely on ${\mathbb C}^3{\setminus} \{0\}$ a crepant resolution of the Calabi-Yau orbifold ${\mathbb C}^3\!/G$ always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid H…

Surface bundles' signatures tied to their Euler characteristics.

Given an $n$-gon, the poset of all collections of pairwise non-crossing diagonals is isomorphic to the face poset of some convex polytope called \textit{associahedron}. We replace in this setting the $n$-gon (viewed as a disc with $n$ marked points on the boundary) with an arbitrary oriented surface with a number of la…

We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern …

The paper proves properties of strata of differentials, showing they are affine and extremal.

Study of tautological forms on curve moduli spaces.

We present a formula for the full Cheeger-Chern-Simons class of the tautological flat complex vector bundle of rank two over BSL(2,\C^δ). Our formula improves the formula by Dupont and Zickert, where the class is only computed modulo 2-torsion.

This paper investigates the projectivization of real vector bundles over small covers. We first give a necessary and sufficient condition for such a projectivization to be a small cover. Then associated with moment-angle manifolds, we further study the structure of such a projectivization as a small cover. As an applic…

Extends metric construction for k-differentials, showing flat area's canonical hermitian metric.

Diagonalizes tautological classes of definite 4-manifolds.

We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back operation involving the tautological universal bundle and the classifying morphism o…

We show that a conjectural extension of a fixed point formula in Arakelov geometry implies results about a tautological subring in the arithmetic Chow ring of bases of abelian schemes. Among the results are an Arakelov version of the Hirzebruch proportionality principle and a formula for a critical power of $\hat c_1$ …

Establishes metrics with positive curvature on projective line bundles.

For a connected Lie group G, we show that a complex structure on the total space TG of the tangent bundle of G that is left invariant and has the property that each left translation G-orbit is a totally real submanifold is induced from a smooth immersion of TG into the complexification of G. For G compact and connected…

The paper studies tautological rings of strata of differentials, proving cohomological stability and no relations in specific degrees.

There is a natural filtration on the space of degree-$k$ homogeneous polynomials in $n$ independent variables with coefficients in the algebra of smooth functions on the Grassmannian $\mathrm{Gr}(n,s)$, determined by the tautological bundle. In this paper we show that the space of $s$-dimensional integral elements of a…

New operators in Khovanov-Rozansky homology exhibit symmetry.

No non-constant holomorphic maps between certain complex manifolds with specific properties.

Fix a simple complex Lie group G and a principal sl(2,C) subalgebra of Lie(G). Then the moduli space of semi-stable, topologically trivial G-Higgs bundles on a hyperbolic, spin Riemann surface acquires a marked point. This is the unique C*-fixed point on the Hitchin section. We describe a universal analytic family of d…

In this paper we prove a conjecture by Feldman-Ilmanen-Knopf in \cite{FIK} that the gradient shrinking soliton metric they constructed on the tautological line bundle over $\CP^1$ is the uniform limit of blow-ups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups…

New method connects complex geometry with twistor spaces.

Calculates volumes of linear subvarieties in moduli spaces of Abelian differentials.

The curvature of web curves is studied in 3D manifolds.

The paper proves a conjecture linking Higgs bundles and Lie algebra actions.

Study characteristic classes for manifold bundles, focusing on fiber families.

Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to a cohomology class of a homotopy colimit BSO of classifying spaces BSO_n. Simil…

The paper constructs cohomology classes on curve strata.

Researchers establish a connection between knot homology and Lie algebra actions.

The paper studies curvature properties of vector bundles and their applications to quasi-Fuchsian space.

The expression (-1/u) times the Hessian of u transforms as a symmetric (0,2) tensor under projective coordinate transformations, so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold M, the section u can be regarded as a metric potential analogous to the local potential …

The paper proves a pointwise Gysin formula for vector bundles and applies it to show positivity of polynomials.

The paper proves a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.

We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…

The materials accompany a lecture short course presented at the 2011 Park City Mathematics Institute, Graduate Summer School on Moduli Spaces of Riemann Surfaces. The lectures were part of/coordinated with an overall program, including lectures by Ursula Hamenstadt on Teichmueller Theory, Andy Putman on Mapping Class a…

Completed volumes match with combinatorial classes of the double ramification cycle.

Following a suggestion made by J.-P. Demailly, for each $k\ge 1$, we endow, by an induction process, the $k$-th (anti)tautological line bundle $\mathcal O_{X_k}(1)$ of an arbitrary complex directed manifold $(X,V)$ with a natural smooth hermitian metric. Then, we compute recursively the Chern curvature form for this me…

We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we …

Alexander self-dual complexes yield smooth compactifications of $M_{0,n}$ with explicit Chow rings.

On a natural circle bundle T(M) over a 4-dimensional manifold M equipped with a split signature metric g, whose fibers are real totally null selfdual 2-planes, we consider a tautological rank 2 distribution D obtained by lifting each totally null plane horizontally to its point in the fiber. Over the open set where g i…

We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute t…