arXiv research

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 $\hat c_1$ …

Formula for analytic torsion forms in fibrations by projective curves.

This is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula "`a la Bott" for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable tor…

We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivarian…

In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the ``closed fibers at infinity''. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with …

Mathematical analysis of Riemann surfaces and their moduli spaces using hybrid Laplacians.

Paper proves various types of varieties minimize a specific energy.

The study shows that symplectic Lefschetz fibrations can have infinitely many sections.

Paper introduces hybrid curves moduli space and canonical measures.

In this note we give geometric formulations and proofs of three results of S. Morita. These results relate certain two dimensional cohomology classes of various moduli spaces of curves. We also give a geometric interpretation of a fourth result of Morita. One motivation of this work is to facilitate the application of …

Given a choice of metric on the Riemann surface, the regularized determinant of Laplacian (analytic torsion) is defined via the complex power of elliptic operators: $\det(Δ)=\exp(-ζ'(0))$ In this paper we gave an asymptotic effective estimate of analytic torsion under Arakelov metric. In particular, after taking th…

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…

Let $π: {\mathbb C}_g \to {\mathbb M}_g$ be the universal family of compact Riemann surfaces of genus $g \geq 1$. We introduce a real-valued function on the moduli space ${\mathbb M}_g$ and compute the first and the second variations of the function. As a consequence we relate the Chern form of the relative tangent bun…

We give a new proof of a slightly weaker form of a theorem of P. Colmez. This theorem gives a formula for the Faltings height of abelian varieties with complex multiplication by a C.M. field whose Galois group over $\bf Q$ is abelian; it reduces to the formula of Chowla and Selberg in the case of elliptic curves. We sh…

This paper introduces complex Chern-Simons bundles in families setting and proves their crystalline nature.

Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.

We prove sharp inequalities for determinants of Toeplitz operators and twisted Laplace operators on the two-sphere, generalizing the Moser-Trudinger-Onofri inequality. In particular a sharp version of conjectures of Gillet-Soule and Fang motivated by Arakelov geometry is obtained; applications to SU(2)-invariant determ…

We extend the notion of canonical measures to all (possibly non-compact) metric graphs. This will allow us to introduce a notion of "hyperbolic measures" on universal covers of metric graphs. Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in rela…

We discuss analogies between number theory and the theory of dynamical systems on spaces with a one-codimensional foliation. The emphasis is on comparing the "explicit formulas" of analytic number theory with certain dynamical Lefschetz trace formulas. We also point out a possible relation between an Arakelov-Euler cha…

We prove a result of Chern-Weil type for canonically metrized line bundles on one-parameter families of smooth complex curves. Our result generalizes a result due to J.I. Burgos Gil, J. Kramer and U. Kühn that deals with a line bundle of Jacobi forms on the universal elliptic curve over the modular curve with full leve…

For a one parameter family of Calabi-Yau threefolds, Green, Griffiths and Kerr have expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper, we show that the total singularities can be expressed by the sum of asymptotic values of BCOV invariants, s…

The work is motivated by a result of Manin, which relates the Arakelov Green function on a compact Riemann surface to configurations of geodesics in a 3-dimensional hyperbolic handlebody with Schottky uniformization, having the Riemann surface as conformal boundary at infinity. A natural question is to what extent the …

Holomorphic curves in moduli spaces are quasi-isometrically immersed.

We revisit three results due to Morita expressing certain natural integral cohomology classes on the universal family of Riemann surfaces C_g, coming from the parallel symplectic form on the universal jacobian, in terms of the Miller-Morita-Mumford classes e and e_1. Our discussion will be on the level of the natural 2…

This article presents the constructions of new infinite families of smooth 4-manifolds with the property that any two manifolds in the same family are homeomorphic and, from their construction, seem to be quite different, but cannot be distinguished by Seiberg-Witten invariants. Whether these manifolds are, or are not,…

Constructs a connection for Hodge theoretic projective structures on Riemann surfaces.

Computes canonical heights for arithmetic log surfaces using Hurwitz zeta function.

Study surfaces with constant ratio of principal curvatures in Euclidean and isotropic geometries.

It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwe…

We define $(p,q)$ hermitian geometry as the target space geometry of the two dimensional $(p,q)$ supersymmetric sigma model. This includes generalised Kähler geometry for $(2,2)$, generalised hyperkähler geometry for $(4,2)$, strong Kähler with torsion geometry for $(2,1)$ and strong hyperkähler with torsion geometry f…

Non-lorentzian geometry reviewed, including Lie algebras and Klein geometries.

Simpler method derived for path geometries on surfaces, characterizing projective path geometries.

New definition of Born geometry connects to known geometries.

We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.

The paper extends group constructions to coset geometries, creating new ways to combine geometries.

New symmetries found in Riemann-Cartan geometries.

Survey explores interactions between convex and complex geometry.

Lecture notes on geodesics in differential geometry.

Lecture notes on Finslerian geometry.

Spin(7) geometry linked to multisymplectic geometry.

The extension functors between categories of Cartan geometries can be used to define different categories of Cartan geometries with additional morphisms. The Cartan geometries modeled on skeletons can be used for the description of such categories of Cartan geometries and therefore we develop the theory of Cartan geome…

A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spac…

Paper introduces a generalized Bures-Wasserstein geometry for SPD matrices.

We show that a car, viewed as a nonholonomic system, provides an example of a flat parabolic geometry of type $({\bf SO}(2,3),P_{12})$, where $P_{12}$ is a Borel parabolic subgroup in ${\bf SO}(2,3)$. We discuss the relations of this geometry of a car with the geometry of circles in the plane (a low dimensional Lie sph…

The study sets limits on the complexity of Klein geometries.

Surveying probabilistic real algebraic geometry.

Develops Weyl structures for path geometries, simplifying their study.

Ray-marching method visualizes 8 Thurston geometries in real-time.