Henri Gillet's research while affiliated with University of Illinois at Chicago and other places

Publications (64)

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We consider a short sequence of hermitian vector bundles on some arithmetic variety. Assuming that this sequence is exact on the generic fiber we prove that the alternated sum of the arithmetic Chern characters of these bundles is the sum of two terms, namely the secondary Bott Chern character class of the sequence and its Chern character with supp...
Article
Generalizing Atiyah extensions, we introduce and study differential abelian tensor categories over differential rings. By a differential ring, we mean a commutative ring with an action of a Lie ring by derivations. In particular, these derivations act on a differential category. A differential Tannakian theory is developed. The main application is...
Article
We associate weight complexes of (homological) motives, and hence Euler characteristics in the Grothendieck group of motives, to arithmetic varieties and Deligne–Mumford stacks; this extends the results in the paper [H. Gillet, C. Soulé, Descent, motives and K-theory, J. Reine Angew. Math. 478 (1996) 127–176], where a similar result was proved for...
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The arithmetic Chow groups and their product structure are extended from the category of regular arithmetic varieties to regular Deligne-Mumford stacks over the ring of integers in a number field. Comment: 17 pages. Exposition improved. Final version, to appear in the Fields Institute Communications
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We prove an analogue in Arakelov geometry of the Grothendieck-Riemann-Roch theorem.
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The problem of defining intersection products on the Chow groups of schemes has a long history. Perhaps the first example of a theorem in intersection theory is Bézout’s theorem, which tells us that two projective plane curves C and D, of degrees c and d and which have no components in common, meet in at most cd points. Furthermore if one counts th...
Article
In this paper we construct an explicit representative for the Grothendieck fundamental class [Z] of a complex submanifold Z of a complex manifold X, under the assumption that Z is the zero locus of a real analytic section of a holomorphic vector bundle E. To this data we associate a super-connection A on the exterior algebra of E, which gives a "tw...
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In this paper we show how to use elementary methods to prove that the volume of Sl_k R / Sl_k Z is zeta(2) * zeta(3) * ... * zeta(k) / k. Using a version of reduction theory presented in this paper, we can compute the volumes of certain unbounded regions in Euclidean space by counting lattice points and then appeal to the machinery of Dirichlet ser...
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Two results in Differential Algebra, Kolchin's Irreducibility Theorem, and a result on descent of projective varieties (due to Buium) are proved using methods of "modern" or "Grothendieck style" algebraic geometry.
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We develop a formalism of direct images for metrized vector bundles in the context of the non-archimedean Arakelov theory introduced in our previous joint work with S. Bloch, and we prove a Riemann-Roch-Grothendieck theorem for this direct image.
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map jZj F Gamma! jY j is a quasifibration, and the fiber over any point in the open simplex of jY j corresponding to ae is homeomorphic to jZ ae j. it reads Then the map jjA 7! jZ(A; Delta)j jj F Gamma! jY j is a quasifibration, and the fiber over any point in the open simplex of jjY jj corresponding to ae is homeomorphic to jZ ae j. The proof then...
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Let $E$ be a holomorphic vector bundle on a compact K\"ahler manifold $X$. If we fix a metric $h$ on $E$, we get a Laplace operator $\Delta$ acting upon smooth sections of $E$ over $X$. Using the zeta function of $\Delta$, one defines its regularized determinant $det'(\Delta)$. We conjectured elsewhere that, when $h$ varies, this determinant $det'(...
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To an arbitrary variety over a field of characteristic zero, we associate a complex of Chow motives, which is, up to homotopy, unique and bounded. We deduce that any variety has a natural Euler characteristic in the Grothendieck group of Chow motives. We show that the cohomology with integer coefficients of any singular variety over the complex num...
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Using arithmetic intersection theory, a theory of heights for projective varieties over rings of algebraic integers is developed. These heights are generalizations of those considered by Weil, Schmidt, Nesterenko, Philippon, and Faltings. Several of their properties are proved, including lower bounds and an arithmetic Bézout theorem for the height...
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Consider a compact 3-manifold M with boundary consisting of a single torus. The papers (CS1), (CS2) and (CGLS) discuss the variety of characters of SL2(C) representations of …1(M), and some of the ways in which the topological structure of M is re∞ected in the algebraic geometry of the character variety. We will describe in this paper a certain a-n...
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A celebrated question of Lyndon [L] as reinterpreted by Chiswell [C], Alperin and Moss [A-M], and Imrich [I], is whether one can classify the groups which act freely on ℝ-trees. It is a classical theorem that any group which acts freely, without inversions, on a simplicial tree is free. If T is a Λ-tree for Λ ⊂ ℝ a subgroup (possibly equal to ℝ its...
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The purpose of this note is to remark that Theorem 3.7 in [1], when combined with the work of Bismut and Freed [2], leads, in the algebraic case, to an improvement of both results concerning the holonomy of determinant line bundles.
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This paper sketches the relationship between the arithmetic Chow groups introduced by the authors, and the theory of differential characters due to Cheeger and Simons. Applications given are computing the holonomy of the Quillen connection and studying the Abel-Jacobi homomorphism.
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In this paper, we prove that in the case of holomorphic locally Khler fibrations, the analytic and algebraic geometry constructions of determinant bundles for direct images coincide. We calculate the curvature of the holomorphic Hermitian connection for the Quillen metric on the determinant bundle. We study the behavior of the Quillen metric under...
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In this paper, we derive the main properties of Kähler fibrations. We introduce the associated Levi-Civita superconnection to construct analytic torsion forms for holomorphic direct images. These forms generalize in any degree the analytic torsion of Ray and Singer. In the case of acyclic complexes of holomorphic Hermitian vector bundles, such form...
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We attach secondary invariants to any acyclic complex of holomorphic Hermitian vector bundles on a complex manifold. These were first introduced by Bott and Chern [Bot C]. Our new definition uses Quillen's superconnections. We also give an axiomatic characterization of these classes. These results will be used in [BGS2] and [BGS3] to study the dete...
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We correct an error in (1), and provide a new shorter proof of Theorem B0.
Article
The problem of defining intersection products on the Chow groups of schemes has a long history. Perhaps the first example of a theorem in intersection theory is Bézout’s theorem, which tells us that two projective plane curves C and D, of degrees c and d and which have no components in common, meet in at most cd points. Furthermore if one counts th...

Citations

... be the subgroup generated by sheaves whose supports have codimensions bigger than or equal to i. By a classical result of Grothendieck (see [Gil05,Theorem 3.10]), the smoothness of Y implies that the tensor product on the K-theory restricts to the map ...
... Chern classes with values in the Chow groups are a special case of more general constructions of Chern classes with values in an arbitrary oriented generalized cohomology theory. The Chern classes can also be extended to higher K-groups K * (X) with values in certain groups ofétale cohomology (see [58] and [30]) or motivic cohomology (see [31] and [52]). ...
... The total Chern character [Gil2] in the bilinear K -cohomology of the pure bimotive c * (X sv R×L ) is then given by the homomorphism: ...
... The most important input in the following is Serre's Vanishing Conjecture Thm. 5.4, which is due (independently) to Roberts [20] and Gillet-Soulé [3]. In fact, we present the theory of cycles (in our context) as a special case of [3], which thus becomes our main reference. ...
... 2, we obtain several results on secondary characteristic classes for direct images in any degree. In particular an analogue of [BGS 1, Theorem 0.3] is proved in Theorem 2.21 in any degree, and is related to work by Gillet and Soule [GS1,2] on direct images in Arakelov theory. Theorem 2.21 will be used in [BGS 3] to prove [BGS 1, Theorem 0.3]. ...
... Let L = (L, · φ ) and N = (N , · ψ ) be two C ∞ -hermitian line bundles on X . If L is ample, we have the following asymptotic expansion (2)χ(H 0 (X , kL + N ), · sup,kφ+ψ ) = 1 (n + 1)! deg( c 1 (L) n+1 )k n+1 + o(k n+1 ), (see [1,10,22]). As a consequence, ...
... The mainstay of our results is a bound on the degree and size of the coefficients of the polynomials NL d,g . The determination of these bounds is based on (54) and involves the theory of heights of multiprojective varieties as developped by D' Andrea et al. (2013), and, before them, Bost et al. (1991), Krick et al. (2001), Philippon (1995), and Rémond (2001a,b), among others. We recall here the results that we need, following D' Andrea et al. (2013). ...
... This coincides with the notion of relative dimension given in [Ful98,Section 20.1] plus 1. We choose to work with this absolute notion because it coincides with the one discussed in [BGS95], which will be our main reference in a subsequent article. A reason for the name "S-absolute dimension" is given in Remark 3.2. ...
... The height function H is almost the same as the height of pure motives in [22] and in Koshikawa [29] which generalizes the height of abelian varieties defined by Faltings [10]. The height function H ♦ is almost the same as the height of mixed motives defined in [23] and is closely related to the height parings of Beilinson [3], Bloch [5] and Gillet-Soulé [14,15], and to Beilinson regulators in [2]. ...