Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFlat line bundles and the Cappell-Miller torsion in Arakelov geometry
106
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we extend Delignes 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 side, the idea goes back to Fays holomorphic extension of determinants of Dolbeault laplacians, and it is shown here to be equivalent to the holomorphic Cappell-Miller torsion. On the Deligne pairing side, the logarithm is a refinement of the intersection connections considered in previous work. The construction naturally leads to an Arakelov theory for flat line bundles on arithmetic surfaces and produces arithmetic intersection numbers valued in ${mathbb C}/pi i {mathbb Z}$. In this context we prove an arithmetic Riemann-Roch theorem. This realizes a program proposed by Cappell-Miller to show that the holomorphic torsion exhibits properties similar to those of the Quillen metric proved by Bismut, Gillet and Soule. Finally, we give examples that clarify the kind of invariants that the formalism captures; namely, periods of differential forms.
rate research
Read More
In this article, we construct a $theta$-density for the global sections of ample Hermitian line bundles on a projective arithmetic variety. We show that this density has similar behaviour to the usual density in the Arakelov geometric setting, where only global sections of norm smaller than $1$ are considered. In particular, we prove the analogue by $theta$-density of two Bertini kind theorems, on irreducibility and regularity respectively.
This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesnt require the variety under consideration to be generically smooth or projective. In order to construct such an approach we develop a theory of generalized rings and schemes, which include classical rings and schemes together with exotic objects such as F_1 (field with one element), Z_infty (real integers), T (tropical numbers) etc., thus providing a systematic way of studying such objects. This theory of generalized rings and schemes is developed up to construction of algebraic K-theory, intersection theory and Chern classes. Then existence of Arakelov models of algebraic varieties over Q is shown, and our general results are applied to such models.
The purpose of this book is to build up the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for the researches of arithmetic geometry in several directions.
Let $G$ be a simply-connected semisimple compact Lie group, $X$ a compact Kahler manifold homogeneous under $G$, and $L$ a negative $G$-equivariant holomorphic line bundle over $X$. We prove that all $G$-invariant Kahler metrics on the total space of $L$ arise from the Calabi ansatz. Using this, we then show that there exists a unique $G$-invariant scalar-flat Kahler metric in each Kahler class of $L$.
We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in seven dimensions, when this condition is equivalent to integrability in the sense of Duchemin. We prove that the choice of a reduction to Sp(n)H^* (or equivalently, a complement of the qc distribution) yields a unique K-connection satisfying natural conditions on torsion and curvature. We show that the choice of a compatible metric on the qc distribution determines a canonical reduction to Sp(n)Sp(1) and a canonical Sp(n)Sp(1)-connection whose curvature is almost entirely determined by its torsion. We show that its Ricci tensor, as well as the Ricci tensor of the Biquard connection, has an interpretation in terms of intrinsic torsion.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
