Complex Algebraic Varieties Research Articles

Purity, Formality, and Arrangement Complements

We prove a purity implies statement in the context of the rational homotopy theory of smooth complex algebraic varieties, and apply it to complements of hypersurface arrangements. In particular, we prove that the complement of a toric arrangement is formal. This is analogous to the classical formality theorem for complements of hyperplane arrangements, due to Brieskorn, and generalizes a theorem of De Concini and Procesi.

Topological K-theory of complex noncommutative spaces

The purpose of this work is to give a definition of a topological K-theory for dg-categories over$\mathbb{C}$and to prove that the Chern character map from algebraic K-theory to periodic cyclic homology descends naturally to this new invariant. This topological Chern map provides a natural candidate for the existence of a rational structure on the periodic cyclic homology of a smooth proper dg-algebra, within the theory of noncommutative Hodge structures. The definition of topological K-theory consists in two steps: taking the topological realization of algebraic K-theory and inverting the Bott element. The topological realization is the left Kan extension of the functor ‘space of complex points’ to all simplicial presheaves over complex algebraic varieties. Our first main result states that the topological K-theory of the unit dg-category is the spectrum$\mathbf{BU}$. For this we are led to prove a homotopical generalization of Deligne’s cohomological proper descent, using Lurie’s proper descent. The fact that the Chern character descends to topological K-theory is established by using Kassel’s Künneth formula for periodic cyclic homology and the proper descent. In the case of a dg-category of perfect complexes on a separated scheme of finite type, we show that we recover the usual topological K-theory of complex points. We show as well that the Chern map tensorized with$\mathbb{C}$is an equivalence in the case of a finite-dimensional associative algebra – providing a formula for the periodic homology groups in terms of the stack of finite-dimensional modules.

Rank One Local Systems and Forms of Degree One

Cohomology support loci of rank one local systems of a smooth quasiprojective complex algebraic variety are finite unions of torsion-translated complex subtori of the character variety of the fundamental group. Tangent spaces of the character variety are (partially) represented by logarithmic 1-forms. In this paper, we give a relation between cohomology support loci and the natural strata of 1-forms given by the dimension of the vanishing locus. This relation generalizes the one for the projective case due to Green and Lazarsfeld and also generalizes the partial relation due to Dimca in the quasi-projective case.

Degenerations of amoebae and Berkovich spaces

We prove a continuity result for the fibers of the Berkovich analytification of a complex algebraic variety with respect to the maximum of the Archimedean norm and the trivial norm. As a consequence, we obtain generalizations of a result of Mikhalkin and Rullgård about degenerations of amoebae onto tropical varieties.

The Homotopy Fixed Point Theorem and the Quillen–Lichtenbaum conjecture in Hermitian K-theory

We settle two conjectures for computing higher Grothendieck–Witt groups (also known as Hermitian K-groups) of noetherian schemes X, under some mild conditions. It is shown that the comparison map from the Hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence. We also prove that the mod 2ν comparison map between the Hermitian K-theory of X and its étale version is an isomorphism on homotopy groups in the same range as for the Quillen–Lichtenbaum conjecture in K-theory. Applications compute higher Grothendieck–Witt groups of complex algebraic varieties and rings of 2-integers in number fields, and hence values of Dedekind zeta-functions.

On Euler–Poincaré characteristics

It is well known that the Euler characteristic of the cohomology of a complex algebraic variety coincides with the Euler characteristic of its cohomology with compact support. An old result of G. Laumon asserts that a relative version of this statement is true in ℓ-adic cohomology. The purpose of this note is to extend Laumon's result to the topological setting. Some applications are also discussed.

ON REDUCTIVE AUTOMORPHISM GROUPS OF REGULAR EMBEDDINGS

Let G be a connected reductive complex algebraic group acting on a smooth complete complex algebraic variety X. We assume that X is a regular embedding, a condition satisfied in particular by smooth toric varieties and flag varieties. For any set D of G-stable prime divisors, we study the action on X of the group Aut°(X, D), the connected automorphism group of X stabilizing all elements of D. We determine a Levi subgroup A(X, D) of Aut°(X, D), and also relevant invariants of X as a spherical A(X, D)-variety. As a byproduct, we obtain a complete description of the inclusion relation between closures of A(X, D)-orbits on X.

Monodromy and the Lefschetz fixed point formula

We give a new proof - not using resolution of singularities - of a formula of Denef and the second author expressing the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. Our proof uses l-adic cohomology of non-archimedean spaces, motivic integration and the Lefschetz fixed point formula for finite order automorphisms. We also consider a generalization due to Nicaise and Sebag and at the end of the paper we discuss connections with the motivic Serre invariant and the motivic Milnor fiber.

E1–formality of complex algebraic varieties

Let X be a smooth complex algebraic variety. Morgan [Mor78] showed that the rational homotopy type of X is a formal consequence of the differential graded algebra defined by the first term of its weight spectral sequence. In the present work we generalize this result to arbitrary nilpotent complex algebraic varieties (possibly singular and/or non-compact) and to algebraic morphisms between them. The result for algebraic morphisms generalizes the Formality Theorem of [DGMS75] for compact K\"ahler varieties, filling a gap in Morgan's theory concerning functoriality over the rational numbers. As an application, we study the Hopf invariant of certain algebraic morphisms using intersection theory.

Du théorème de décomposition à la pureté locale

A new relation is established between the decomposition theorem and a version of the local purity theorem of Deligne and Gabber adapted to complex algebraic varieties. A reduction to the case of a fibred morphism by relative normal crossing divisors on the strata of a stratification is established.

Hodge filtered complex bordism

We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and $\A^1$-homotopy invariance. Moreover, we obtain transfer maps along projective morphisms.

Cofibrant models of diagrams: Mixed Hodge structures in rational homotopy

We study the homotopy theory of a certain type of diagram categories whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is applied to the category of mixed Hodge diagrams of differential graded algebras. Using Sullivan's minimal models, we prove a multiplicative version of Beilinson's Theorem on mixed Hodge complexes. As a consequence, we obtain functoriality for the mixed Hodge structures on the rational homotopy type of complex algebraic varieties. In this context, the mixed Hodge structures on homotopy groups obtained by Morgan's theory follow from the derived functor of the indecomposables of mixed Hodge diagrams.

Number of Jordan blocks of the maximal size for local monodromies

AbstractWe prove formulas for the number of Jordan blocks of the maximal size for local monodromies of one-parameter degenerations of complex algebraic varieties where the bound of the size comes from the monodromy theorem. In the case when the general fibers are smooth and compact, the proof calculates some part of the weight spectral sequence of the limit mixed Hodge structure of Steenbrink. In the singular case, we can prove a similar formula for the monodromy on the cohomology with compact supports, but not on the usual cohomology. We also show that the number can really depend on the position of singular points in the embedded resolution, even in the isolated singularity case, and hence there are no simple combinatorial formulas using the embedded resolution in general.

Spectral sets and distinguished varieties in the symmetrized bidisc

We show that for every pair of matrices (S,P), having the closed symmetrized bidisc Γ as a spectral set, there is a one dimensional complex algebraic variety Λ in Γ such that for every matrix valued polynomial f(z1,z2),‖f(S,P)‖⩽max(z1,z2)∈Λ‖f(z1,z2)‖. The variety Λ is shown to have the determinantal representationΛ={(s,p)∈Γ:det(F+pF⁎−sI)=0}, where F is the unique matrix of numerical radius not greater than 1 that satisfiesS−S⁎P=(I−P⁎P)12F(I−P⁎P)12. When (S,P) is a strict Γ-contraction, then Λ is a distinguished variety in the symmetrized bidisc, i.e. a one dimensional algebraic variety that exits the symmetrized bidisc through its distinguished boundary. We characterize all distinguished varieties of the symmetrized bidisc by a determinantal representation as above.

Kontsevich's conjecture on an algebraic formula for vanishing cycles of local systems

For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the corresponding locally free sheaf with integrable connection having regular singularity at innity. We also prove its local version, which may be viewed as a natural generalization of a result of E. Brieskorn in the isolated singularity case. We then generalize these to the case of the de Rham complexes of regular holonomic D-modules where we have to use the tensor product with a certain sheaf of formal microlocal dierential operators instead of the formal completion.

A-hypergeometric functions in transcendental questions of algebraic geometry

We generalize known constructions of A-hypergeometric functions. In particular, we show that the periods of middle dimension on affine or projective complex algebraic varieties are A-hypergeometric functions of the coefficients of polynomial equations of these varieties.

On the algebraicity of the zero locus of an admissible normal function

AbstractWe show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic. In Part II of the paper, which is an appendix, we compute the Tannakian Galois group of the category of one-variable admissible real nilpotent orbits with split limit. We then use the answer to recover an unpublished theorem of Deligne, which characterizes the ${\mathrm{sl} }_{2} $-splitting of a real mixed Hodge structure.

Choking Horns in Lipschitz Geometry of Complex Algebraic Varieties. Appendix by Walter D. Neumann

The paper studies the Lipschitz geometry of germs of complex algebraic varieties and introduces the notion of a choking horn. A choking horn is a family of cycles on an algebraic variety with the property that the cycles cannot be boundaries of nearby chains. The presence of choking horns is an obstruction to metric conicalness and the authors use this to prove that some classical isolated hypersurface singularities are not metrically conical. They also show that there exist countably infinitely many singular varieties, which are locally homeomorphic, but not locally subanalytically bi-Lipschitz equivalent with respect to the inner metric. The Appendix by W. Neumann uses separating sets to provide another example of the same phenomenon.

SINGULAR Q-HOMOLOGY PLANES OF NEGATIVE KODAIRA DIMENSION HAVE SMOOTH LOCUS OF NON-GENERAL TYPE

We show that if a singular Q-homology plane has negative Kodaira dimension then its smooth locus is not of general type. This generalizes the earlier result of Koras-Russell for contractible surfaces. 1. Main result We work in the category of complex algebraic varieties. Let S 0 be a singular normal surface having rational cohomology of a plane C 2 , i.e. H � (S 0 ;Q) � Q. We call S 0 a singular Q-homology plane. One of the basic invariants of S 0 is its logarithmic Kodaira dimension �(S 0 ) 2 f−1 ;0;1;2g, having the property �(S 0 −SingS 0 ) � �(S 0 ). In this paper we continue the program of classification

Weights on cohomology, invariants of singularities, and dual complexes

We study the weight filtration on the cohomology of a proper complex algebraic variety and obtain natural upper bounds on its size, when it is the exceptional divisor of a singularity. We also give bounds for the cohomology of links. The invariants of singularities introduced here gives rather strong information about the topology of rational and related singularities.