arXiv research

Study nilpotent Higgs bundles and Calabi-Yau moduli metrics.

Study rigidity of Ricci flow limits on nilpotent bundles with zero curvature.

The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.

Study shows unique harmonic metrics for certain Higgs bundles over non-compact surfaces.

Study families of flat connections with nilpotent Higgs fields, showing similar monodromy to regular Higgs bundles.

The purpose of this note is to present a short elementary proof of a theorem due to Faltings and Laumon, saying that the global nilpotent cone is a Lagrangian substack in the cotangent bundle of the moduli space of G-bundles on a complex compact curve. This result plays a crucial role in the Geometric Langlands program…

New method constructs degenerate Sasakian manifolds from hyperkähler bundles.

We show that almost nonnegatively curved m-dimensional manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m)-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanif…

Let $E$ be a holomorphic vector bundle. Let $θ$ be a Higgs field, that is a holomorphic section of $End(E)\otimesΩ^{1,0}_X$ satisfying $θ^2=0$. Let $h$ be a pluriharmonic metric of the Higgs bundle $(E,θ)$. The tuple $(E,θ,h)$ is called a harmonic bundle. Let $X$ be a complex manifold, and $D$ be a normal crossing divi…

Study quadratic one-forms on logarithmic Higgs bundles on pointed curves.

CR embeddings in complex spaces for specific Lie groups.

The Gaiotto locus for Sp(2n) is shown to lie in the nilpotent cone.

The study characterizes wobbly rank-2 bundles on Riemann surfaces using spectral curves.

In this paper we attempt to give a systematic account on privileged coordinates and the nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies …

In the description of the instanton Floer homology of a surface times a circle due to Muñoz, we compute the nilpotency degree of the endomorphism $u^2-64$. We then compute the framed instanton homology of a surface times a circle with non-trivial bundle, which is closely related to the kernel of $u^2-64$. We discuss th…

Study heterotic G2-system on 2-step nilmanifolds with torus bundles.

For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse-Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in a twisted category of holomorphic bundles. The critical points that globally minimi…

We compute cup product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency d…

Study linear perturbations of Spin(7) metrics, finding only rank one nilpotent matrices.

The paper extends a theorem to manifolds with local Ricci bounded covering geometry.

The paper studies minimal surfaces in complex hyperbolic spaces, showing moduli space connected components.

In this note, we propose an approach to the study of the analogue for unipotent harmonic bundles of Schmid's Nilpotent Orbit Theorem. Using this approach, we construct harmonic metrics on unipotent bundles over quasi-compact Kähler manifolds with carefully controlled asymptotics near the compactifying divisor; such a m…

Let $(\mathcal{E}, φ)$ be a rank two co-Higgs vector bundles on a Kähler compact surface $X$ with $φ\in H^0(X,End(\mathcal{E})\otimes T_X)$ nilpotent. If $(\mathcal{E}, φ)$ is semi-stable, then one of the following holds up to finite \' etale cover: $i)$ $X$ is uniruled. $ii)$ $X$ is a torus and $(\mathcal{E}, φ)$ is s…

We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold $M$ with a distribution $H\subseteq TM$ analysts use explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim in this p…

Study very stable Higgs bundles on Riemann surfaces, linking to multiplicity and mirror symmetry.

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 mod…

Motivated by generalized geometry, we discuss differential geometric structures on the total space $\mathfrak{T}M$ of the bundle $TM\oplus T^*M$, where $M$ is a differentiable manifold; $\mathfrak{T}M$ is called a big-tangent manifold. The vertical leaves of the bundle are para-Hermitian vector spaces. The big-tangent …

Study expanding Ricci solitons on vector bundles, reducing to Higgs bundle equations.

This paper is a sequel of arxiv:1709.09045 and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single…

The paper disproves a generalized toral rank conjecture with various counter-examples.

We study a generalization of Hodge structures which first appeared in the work of Cecotti and Vafa. It consists of twistors, that is, holomorphic vector bundles on P^1, with additional structure, a flat connection on C^*, a real subbundle and a pairing. We call these objects TERP-structures. We generalize to TERP-struc…

Smooth bundles with rough data maintain Hodge kernel isomorphism.

Starting from an even definite lattice, we construct a principal circle bundle covered by a certain three-step nilpotent Lie group G. On the base space, which is again a nilmanifold, we then study the Dirac operator twisted by the associated complex line bundles. Noting that the whole situation fibers over the circle, …

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

An absolute parallelism for $2$-nondegenerate CR manifolds $M$ of hypersurface type was recently constructed independently by Isaev-Zaitsev, Medori-Spiro, and Pocchiola in the minimal possible dimension ($\dim M=5$), and for $\dim M=7$ in certain cases by the first author. We develop a bigraded analog of Tanaka's prolo…

A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of …

Study integrability of geodesic flow on specific Lie groups.

We introduce a new geometric structure on differentiable manifolds. A \textit{Contact} \textit{Pair}on a manifold $M$ is a pair $(α,η)$ of Pfaffian forms of constant classes $2k+1$ and $2h+1$ respectively such that $α\wedge dα^{k}\wedgeη\wedge dη^{h}$ is a volume form. Both forms have a characteristic foliation whose …

We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle $T^*G$ of a 2n-dimensional Lie group $G$, which are left invariant with respect to the Lie group structure on $T^*G$ induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized …

Nilpotent quandles have simple characterizations and are closely related to nilpotency.

We review some basic facts on vector fields, in the complex-analytic setting, thus, obtaining a rationality result and an extension of the Birkhoff-Grothendieck theorem, as follows: (1) Let $Z$ be a compact complex manifold endowed with a very ample line bundle $L$. Denote by $\mathfrak{g}_L$ the extended Lie algebra o…

Nilpotent Lie algebras obtained from ordered sets and quivers are algebraic Ricci solitons.

New method constructs nilpotent Lie algebras from quivers.

Study knot invariants using automorphism groups of free nilpotent groups.

Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.

We exhibit pseudo Riemannian manifolds which are Szabó nilpotent of arbitrary order, or which are Osserman nilpotent of arbitrary order, or which are Ivanov-Petrova nilpotent of order 3.

Groupoids provide a more appropriate framework for differential geometry than principal bundles. Synthetic differential geometry is the avant-garde branch of differential geometry, in which nilpotent infinitesimals are available in abundance. The principal objective in this paper is to show within our favorite framewor…

Classifies complex structures on specific nilpotent Lie algebras.