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The Ktheory of normal surfaces
"... Abstract. We relate the negative Ktheory of a normal surface to a resolution of singularities. The only nonzero Kgroups are K−2, which counts loops in the exceptional fiber, and K−1, which is related to the divisor class groups of the complete local rings at the singularities. We also verify two c ..."
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Abstract. We relate the negative Ktheory of a normal surface to a resolution of singularities. The only nonzero Kgroups are K−2, which counts loops in the exceptional fiber, and K−1, which is related to the divisor class groups of the complete local rings at the singularities. We also verify two conjectures of Srinivas about K0regularity and K−1 of a surface. This paper gives a geometric interpretation for the negative Ktheory of a normal surface X. If R is the semilocal ring of X at its finitely many singularities, then the negative Ktheory of X and R are the same by [W1, 1.2]. Thus we also describe the negative Ktheory of any excellent 2dimensional normal semilocal ring R. As we shall explain in a few columncentimeters, our results also give an almost complete classification of projective R[T]modules. Our interpretation relates the groups K−j(X) to a resolution of singularities ˜X → X of the surface. For starters (see 4.4), we show that K−j(X) = 0 for all j ≥ 3, even if X is singular. (This confirms a guess in [W0, p. 180].) If X is normal and j = 2, we prove that K−2(X) ∼ = Z λ, where λ denotes the number of “loops”
The universal regular quotient of the Chow group of points on projective varieties
 Invent. Math
, 1999
"... Let X be a projective variety of dimension n defined over an algebraically closed field k. For X irreducible and nonsingular, Matsusaka [Ma] constructed an abelian variety Alb (X) and a morphism α: X → Alb (X) (called the Albanese variety and mapping respectively), depending on the choice of a base ..."
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Let X be a projective variety of dimension n defined over an algebraically closed field k. For X irreducible and nonsingular, Matsusaka [Ma] constructed an abelian variety Alb (X) and a morphism α: X → Alb (X) (called the Albanese variety and mapping respectively), depending on the choice of a basepoint on X, which is universal among the morphisms to abelian varieties (see Lang [La], Serre [Se] for other constructions). Over the field of complex numbers the existence of Alb (X) and α was known before, and has a purely Hodgetheoretic description (see Igusa [I] for the Hodge theoretic construction). Incidentally, the terminology “Albanese variety ” was introduced by A. Weil, for reasons explained in his commentary on the article [1950a] of Volume I of his collected works (see [W]), one of which is that the paper [Alb] of Albanese defines it (for a surface) as a quotient of the group of 0cycles of degree 0 modulo an equivalence relation. Let CH n (X)deg 0 denote the Chow group of 0cycles of degree 0 on X modulo rational equivalence. When X is irreducible and nonsingular, a remarkable feature of the Albanese morphism α is that it factors through a regular homomorphism
From Jacobians to onemotives: exposition of a conjecture of Deligne, The arithmetic and geometry of algebraic cycles
, 1998
"... Abstract. Deligne has conjectured that certain mixed Hodge theoretic invariants of complex algebraic invariants are motivic. This conjecture specializes to an algebraic construction of the Jacobian for smooth projective curves, which was done by A. Weil. The conjecture (and onemotives) are motivate ..."
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Abstract. Deligne has conjectured that certain mixed Hodge theoretic invariants of complex algebraic invariants are motivic. This conjecture specializes to an algebraic construction of the Jacobian for smooth projective curves, which was done by A. Weil. The conjecture (and onemotives) are motivated by means of Jacobians, generalized Jacobians of Rosenlicht, and Serre’s generalized Albanese varieties. We discuss the connections with the Hodge and the generalized Hodge conjecture. We end with some applications to number theory by providing partial answers to questions of Serre, Katz and Jannsen. Parmi toutes les choses mathématiques que j’avais eu le privilège de découvrir et d’amener au jour, cette realité des motifs m’apparaît encore comme la plus fascinante, la plus chargée de mystère au cœur même de l’identité profonde entre la “geométrie ” et l ’ “arithmétique”. Et le “yoga des motifs ” auquel m’a conduit cette réalité longtemps ignorée est peutêtre le plus puissant instrument de découverte que j’ai dégagé dans cette première période de ma vie de mathématicien. — Alexandre Grothendieck
ALBANESE AND PICARD MOTIVES OF SCHEMES
, 1998
"... Abstract. The aim of this paper is to define Albanese 1motives and Picard 1motives of schemes over a perfect field. For smooth proper schemes, these are the classical Albanese and Picard varieties. For a curve over an algebraically closed field, these are the homological 1motive of Lichtenbaum an ..."
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Abstract. The aim of this paper is to define Albanese 1motives and Picard 1motives of schemes over a perfect field. For smooth proper schemes, these are the classical Albanese and Picard varieties. For a curve over an algebraically closed field, these are the homological 1motive of Lichtenbaum and the motivic H 1 of Deligne. This paper settles a conjecture of Deligne about providing an algebraic description, via 1motives, of the first homology and cohomology groups of a complex algebraic variety. It also contains a purely algebraic proof that the Albanese and the Picard 1motives of a scheme are dual. For a curve C over an algebraically closed field, the 1motive H 1 m (C) of Deligne is dual to the 1motive h1(C) of Lichtenbaum. Special NOTE: This paper is a revised version of my Ph.D thesis of 1996 at Brown University. My partial results in this direction were announced in a preprint “Albanese and Picard 1motives ” circulated in March 1995. After this work was completed, I became aware of the announcement without proofs (math.AG/9803112) of similar results by BarbieriViale and Srinivas. They focus on the groups H 1 and H 2n−1. They define their Albanese 1motives to be the dual of the Picard 1motives. Whereas, here, we focus on H 1 and H1 and provide separate definitions of the Albanese and Picard 1motives and later prove that they are dual. Our definition of the Albanese makes evident the connections with the algebraic singular homology of SuslinVoevodsky. The definition of the Albanese 1motive and the Albanese scheme is inspired by an email message of Serre (May 16, 1995) in response to my preprint of March 1995. A shorter version of this preprint has been submitted for publication. 1 2 NIRANJAN RAMACHANDRAN
A CONJECTURE OF DELIGNE ON ONEMOTIVES
, 2001
"... We introduce new motivic invariants of arbitrary varieties over a perfect field. These cohomological invariants take values in the category of 1motives (considered up to isogeny). The algebraic definition of these invariants presented here proves, up to isogeny, a conjecture of Deligne. Applicatio ..."
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We introduce new motivic invariants of arbitrary varieties over a perfect field. These cohomological invariants take values in the category of 1motives (considered up to isogeny). The algebraic definition of these invariants presented here proves, up to isogeny, a conjecture of Deligne. Applications include some cases of conjectures of Serre, Katz and Jannsen on the independence of ℓ of parts of the étale cohomology of arbitrary varieties over number fields and finite fields.
Albanese and Picard 1motives
"... Let X be an ndimensional algebraic variety over a field of characteristic zero. We describe algebraically defined Deligne 1motives Alb + (X), Alb − (X), Pic + (X) and Pic − (X) which generalize the classical Albanese and Picard varieties of a smooth projective variety. We compute Hodge, ℓadic and ..."
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Let X be an ndimensional algebraic variety over a field of characteristic zero. We describe algebraically defined Deligne 1motives Alb + (X), Alb − (X), Pic + (X) and Pic − (X) which generalize the classical Albanese and Picard varieties of a smooth projective variety. We compute Hodge, ℓadic and De Rham realizations proving Deligne’s conjecture for H 2n−1, H2n−1, H 1 and H1. We investigate functoriality, universality, homotopical invariance and invariance under formation of projective bundles. We compare our cohomological and homological 1motives for normal schemes. For proper schemes, we obtain an AbelJacobi map from the (LevineWeibel) Chow group of zero cycles to our cohomological Albanese 1motive which is the universal regular homomorphism to semiabelian varieties. By using this universal property we get
On algebraic mixed Hodge substructures of H 2
, 1999
"... For X a complete algebraic variety we introduce an algebraically defined 1motive whose QHodge realization is the largest Hodge substructure of H 2 /W0 of level ≤ 1. We relate this 1motive to the NéronSeveri group of X. If the nonnormal locus of X is finite we show that the NéronSeveri group is ..."
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For X a complete algebraic variety we introduce an algebraically defined 1motive whose QHodge realization is the largest Hodge substructure of H 2 /W0 of level ≤ 1. We relate this 1motive to the NéronSeveri group of X. If the nonnormal locus of X is finite we show that the NéronSeveri group is the subgroup of H 2 (X) of classes which are Zariski locally trivial and are in F 1 of the Hodge filtration. 0