arXiv research

The paper studies the zeta-regularized determinant of a pseudo-Laplacian on a cuspidal end with flat unitary line bundle.

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…

Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.

This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the…

We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial gen…

We construct a 2+1 dimensional classical gauge theory on manifolds with spin structure whose action is a refinement of the Atiyah-Patodi- Singer eta-invariant for twisted Dirac operators. We investigate the properties of the Lagrangian field theory for closed, spun 3-manifolds and compact, spun 3-manifolds with boundar…

In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology si…

The paper compares two torsion invariants in complex vector bundles.

We construct a Fourier--Mukai transform for smooth complex vector bundles $E$ over a torus bundle $π:M \to B,$ the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles $E$ with a flat partial unitary connection, that is families or deformations of flat …

The paper studies the zeta-regularized determinant of a pseudo-Laplacian on a cuspidal end with specific boundary conditions.

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding…

Establishes uniform Hörmander estimates for flat line bundles on Kähler manifolds.

We show that a unipotent vector bundle on a non-Kaehler compact complex manifold does not admit a flat holomorphic connection in general. We also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection.

Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.

Random flat bundles on surfaces have least eigenvalues at least 1/4.

We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a half-translation surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian with von Neumann boundary conditions. As an interesting byproduct of our study, w…

We introduce a differential geometric framework for describing families of quantum error-correcting codes and for understanding quantum fault tolerance. This work unifies the notion of topological fault tolerance with fault tolerance in other kinds of quantum error-correcting codes. In particular, we use fibre bundles …

We present an alternate definition of the mod {\bf Z} component of the Atiyah-Patodi-Singer $η$ invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with t…

The study extends a theorem to bundles on manifolds with boundaries.

The paper explores a B-field transform of complex structures on complex tori.

Invariant Kähler metrics on line bundles are derived from the Calabi ansatz.

Eta invariant of (2,3,5) nilmanifolds vanishes but eta function is nontrivial.

Let $M$ be the moduli space of rank 3 parabolic vector bundles over a Riemann surface with several punctures. By the Mehta-Seshadri correspondence, this is the space of rank 3 unitary representations of the fundamental group of the punctured surface with specified conjugacy classes of the images of each boundary compon…

Unitons, i.e.\ harmonic spheres in a unitary group, correspond to \lq uniton bundles\rq, i.e.\ holomorphic bundles over the compactified tangent space to the complex line with certain triviality and other properties. In this paper, we use a monad representation similar to Donaldson's representation of instanton bundles…

This paper deforms complex tori and their mirrors using gerbes.

Using quaternionic Feix--Kaledin construction we provide a local classification of quaternion-Kähler metrics with a rotating $S^1$-symmetry with the fixed point set submanifold $S$ of maximal possible dimension. For any Kähler manifold $S$ equipped with a line bundle with a unitary connection of curvature proportional …

The paper shows how to uniquely determine a connection up to gauge.

Let K be a compact semi-simple Lie group. We classify K-invariant Kaehler structures on the space Kc/(P,P), where Kc is the complexification of K, P is a parabolic subgroup of Kc, and (P,P) the commutator subgroup. For each Kaehler structure, we study its moment map and associated pre-quantum line bundle for geometric …

We introduce a symplectic structure on the space of connections in a G-principal bundle over a four-manifold and the Hamiltonian action on it of the group of gauge transformations which are trivial on the boundary. The symplectic reduction becomes the moduli space of flat connections over the manifold. On the moduli sp…

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

The spectrum of the Laplace-Dolbeault operator for any line bundle with parallel curvature on a flat complex torus is computed. The Ray-Singer analytic torsion is then deduced, generalizing thus Bost's result for ample line bundles and Ray-Singer's ones for flat bundles, of which we a geometric interpretation is given.

We focus on the topology and dynamics of minimal sets and Levi-flats in surfaces of general type. Our method relies on the ergodic theory of Riemann surfaces laminations: we use harmonic measures and Lyapunov exponents. Our first result establishes that minimal sets have large Hausdorff dimension when a leaf is simply …

We consider the moduli space MN of flat unitary connections on an open Kaehler manifold U (complement of a divisor with normal crossings) with restrictions on their monodromy transformations. Using intersection and L2 cohomologies with degenerating coefficients we construct a natural symplectic form F on MN. When U is …

This article deals with two topics: the first, which has a general character, is a variation formula for the the determinant line bundle in non-Kählerian geometry. This formula, which is a consequence of the non-Kählerian version of the Grothendieck-Riemann Roch theorem proved recently by Bismut, gives the variation of…

Continuous metrics on ample bundles lie in infinite-dimensional cones.

Maps asymptotically embed conic transforms from circle bundles.

For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial "intersection" connection on the Deligne pairing. A relationship is…

Complex valued analytic torsion and dynamical zeta function studied on locally symmetric spaces.

Study Bergman kernels on Kähler manifolds, answering Lu-Tian's question.

We study the asymptotic expansion of the determinant of the graph Laplacian associated to discretizations of a half-translation surface endowed with a flat unitary vector bundle. By doing so, over the discretizations, we relate the asymptotic expansion of the number of spanning trees and the sum of cycle-rooted spannin…

The paper studies volumes of direct images for high tensor powers of ample bundles.

Let E be a Real or Quaternionic Hermitian vector bundle over a Klein surface M. We study the action of the gauge group of E on the space of Galois-invariant unitary connections and we show that the closure of a semi-stable orbit contains a unique unitary orbit of projectively flat, Galois-invariant connections. We then…

We show that the R/Z part of the analytically defined eta invariant of Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, we discuss the relation wi…

The paper studies Kähler-Einstein metrics on circle bundles and their obstruction flatness.

The paper studies scalar flat Kähler metrics on line bundles and proves their properties.

Let L->M be a Hermitian line bundle over a compact manifold. Write S for the space of all unitary connections in L whose curvatures define symplectic forms on M and G for the group of unitary bundle isometries of L, which acts on S by pull-back. The main observation of this note is that S carries a G-invariant symplect…

Study numerically flat bundles on Fujiki manifolds using algebraic groups.

The study classifies holomorphic projective connections on complex threefolds.