A cycle class map from Chow groups with modulus to relative KK-theory

Let $\bar{X}$ be a smooth quasi-projective $d$-dimensional variety over a field $k$ and let $D$ be an effective Cartier divisor on it. In this note, we construct cycle class maps from (a variant of) the higher Chow group with modulus of the pair $(\bar{X},D)$ in the range $(d+n, n)$ to the relative $K$-groups $K_n(\bar{X}, D)$ for every $n\geq 0$.Comment: 24 pages. Final version to appear in Documenta Mat

Torsion zero cycles with modulus on affine varieties

In this note we show that given a smooth affine variety $X$ over an algebraically closed field $k$ and an effective (possibly non reduced) Cartier divisor $D$ on it, the Kerz-Saito Chow group of zero cycles with modulus ${\rm CH}_0(X|D)$ is torsion free, except possibly for $p$-torsion if the characteristic of $k$ is $p>0$. This generalizes to the relative setting classical theorems of Rojtman (for $X$ smooth) and of Levine (for $X$ singular). A stronger version of this result, that encompasses $p$-torsion as well, was proven with a different and more sophisticated method by A. Krishna and the author in another paper.Comment: Final version. 12 pages, exposition improved. Several gaps in the proofs fixe

Relative cycles with moduli and regulator maps

Let X be a separated scheme of finite type over a field k and D a non-reduced effective Cartier divisor on it. We attach to the pair (X, D) a cycle complex with modulus, whose homotopy groups - called higher Chow groups with modulus - generalize additive higher Chow groups of Bloch-Esnault, R\"ulling, Park and Krishna-Levine, and that sheafified on $X_{Zar}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight 1. When X is smooth over k and D is such that $D_{red}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El-Zein's explicit construction. This is used to define a natural regulator map from the relative motivic complex of (X,D) to the relative de Rham complex. When X is defined over $\mathbb{C}$, the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch's regulator from higher Chow groups. Finally, when X is moreover connected and proper over $\mathbb{C}$, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus $J^r_{X|D}$ of the pair (X,D). For r= dim X, we show that $J^r_{X|D}$ is the universal regular quotient of the Chow group of 0-cycles with modulus.Comment: 46 pages. Final version: Section 9 added and material rearranged. To appear in Journal of the Inst. of Math. Jussie

Zero cycles with modulus and zero cycles on singular varieties

Given a smooth variety $X$ and an effective Cartier divisor $D \subset X$, we show that the cohomological Chow group of 0-cycles on the double of $X$ along $D$ has a canonical decomposition in terms of the Chow group of 0-cycles ${\rm CH}_0(X)$ and the Chow group of 0-cycles with modulus ${\rm CH}_0(X|D)$ on $X$. When $X$ is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of ${\rm CH}_0(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that ${\rm CH}_0(X|D)$ is torsion-free and there is an injective cycle class map ${\rm CH}_0(X|D) \hookrightarrow K_0(X,D)$ if $X$ is affine. For a smooth affine surface $X$, this is strengthened to show that $K_0(X,D)$ is an extension of ${\rm CH}_1(X|D)$ by ${\rm CH}_0(X|D)$.Comment: 62 pages. Final version to appear in Compositio Mat

Torsion and divisibility for reciprocity sheaves and 0-cycles with modulus

The notion of modulus is a striking feature of Rosenlicht-Serre's theory of generalized Jacobian varieties of curves. It was carried over to algebraic cycles on general varieties by Bloch-Esnault, Park, R\"ulling, Krishna-Levine. Recently, Kerz-Saito introduced a notion of Chow group of $0$-cycles with modulus in connection with geometric class field theory with wild ramification for varieties over finite fields. We study the non-homotopy invariant part of the Chow group of $0$-cycles with modulus and show their torsion and divisibility properties. Modulus is being brought to sheaf theory by Kahn-Saito-Yamazaki in their attempt to construct a generalization of Voevodsky-Suslin-Friedlander's theory of homotopy invariant presheaves with transfers. We prove parallel results about torsion and divisibility properties for them.Comment: 15 pages, exposition improve

Experimental evaluation of shear and compression strength of masonry wall before and after reinforcement: Deep repointing

Masonry presents some inadequacies due to its almost total lack of tensile strength. Typical damage to multiple leaf walls during earthquakes is the loss of bond between the leaves with consequent collapse of the external leaf. Retrofitting or repair of this damage is a very difficult task. In many cases grout injection or wall jacketing fail due to incompatibility with the construction technique of the walls. A complementary technique to the grouting has been proposed by the authors. Experimental results and applications of the technique on site have shown positive characteristics and the results of tests carried out on site show, in some cases, increases in shear strength and stiffness of the masonry walls

Economic Citizenship in India : A Socio-Legal Comparison of Two Cases

The author discusses the evolving concept of Economic Citizenship and relates a socio-legal dimension to transnational migration to and from India. The paper explores economic citizenship by developing the definition first identified by T.H. Marshall and then uses two case studies to show contrasted applications of Marshall’s definition. Marshall states that there are three types of rights needed for an individual’s development so that s/he can exist in, participate in and contribute to society. 1.Civil rights protect personal freedom. One is entitled to exercise freedom of speech, thought and faith, own property, conclude valid contracts, and have recourse to justice. 2.Political rights allow participation and franchise rights in political environments. 3.Social rights are the right to defend and assert all of one’s rights on the same terms as other members of society and by due process of law. They relate to the “whole range from the right to a modicum of economic welfare and security to the right to share to the full in the social heritage and to live the life of a civilised being according to the standards prevailing in the society.” The paper observes that presence alone of economic opportunity in a society does not mean that the state has discharged its responsibilities to its citizens. Economic opportunity should also be legally accessible to the individual. The legal tie between economic and social aims supports the ensuing right for members of society to earn their livelihoods through the right to work. A denial of these rights should let the individual have political recourse to judicial and legislative redress. Case Study One analsyes economic citizenship in the resettlement dispute in Arunachal Pradesh of Chakma and Hajong tribes from the Chittagong Hill Tracts. These residents are indigenous people of India and are entitled to Indian citizenship by the Citizenship Act 1995, but they lack legal recognition as citizens. The State and Central Governments formally and systematically refuse rights to these individuals - in breach of the right to life guaranteed by the Constitution of India. Case Study Two analyses the economic citizenship rights for another group of individuals. It considers the role of economic citizenship as it is exercised by Non Resident Indians (NRIs), Persons of Indian Origin (PIOs) and Overseas Citizens of India (OCIs) while they live and work outside India. Some persons are granted Indian citizenship by birth, while others are not. The paper concludes on two perspectives. Today's reality is that India grants economic citizenship without full political and social rights to some categories of Indians abroad and formally and informally denies economic citizenship to political citizens living on Indian territory