Smooth Complex Algebraic Variety Research Articles

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.

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.

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.

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.

On the Beilinson-Hodge conjecture for $H^2$ and rational varieties

The Beilinson-Hodge conjecture asserts the surjectivity of the cycle map $$H^n_M(X,\Q(n)) \to {\rm Hom}_{MHS}(\Q(-n),H^n(X,\Q))$$ for all positive integers $n$ and every smooth complex algebraic variety $X$. For $n=2$, we prove the conjecture if $X$ is rational.

Purity of boundaries of open complex varieties

We study the boundary of an open smooth complex algebraic variety U . We ask when the cohomology of the geometric boundary Z = X \U in a smooth compactification X is pure with respect to the mixed Hodge structure. Knowing the dimension of singularity locus of some singular compactification we give a bound for k above which the cohomology H(Z) is pure. The main ingredient of the proof is purity of the intersection cohomology sheaf.

Periodicity of hermitianK-groups

AbstractBott periodicity for the unitary and symplectic groups is fundamental to topologicalK-theory. Analogous to unitary topologicalK-theory, for algebraicK-groups with finite coefficients, similar results are consequences of the Milnor and Bloch-Kato conjectures, affirmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraicK-groups for any ring implies periodicity for the hermitianK-groups, analogous to orthogonal and symplectic topologicalK-theory.The proofs use in an essential way higherKSC-theories, extending those of Anderson and Green. They also provide an upper bound for the higher hermitianK-groups in terms of higher algebraicK-groups.We also relate periodicity to étale hermitianK-groups by proving a hermitian version of Thomason's étale descent theorem. The results are illustrated in detail for local fields, rings of integers in number fields, smooth complex algebraic varieties, rings of continuous functions on compact spaces, and group rings.

GENERALIZED ZARISKI–VAN KAMPEN THEOREM AND ITS APPLICATION TO GRASSMANNIAN DUAL VARIETIES

We formulate and prove a generalization of Zariski–van Kampen theorem on the topological fundamental groups of smooth complex algebraic varieties. As an application, we prove a hyperplane section theorem of Lefschetz–Zariski–van Kampen type for the fundamental groups of the complements to the Grassmannian dual varieties.

Zero loci of admissible normal functions with torsion singularities

We show that the zero locus of a normal function on a smooth complex algebraic variety S is algebraic provided that the normal function extends to an admissible normal function on a smooth compactification such that the divisor at infinity is also smooth. This result, which has also been obtained recently by M. Saito using a different method [22], generalizes a previous result proved by the authors for admissible normal functions on curves [4]

ON THE INTERSECTION OF RATIONAL TRANSVERSAL SUBTORI

AbstractWe show that under a suitable transversality condition, the intersection of two rational subtori in an algebraic torus (ℂ*)n is a finite group which can be determined using the torsion part of some associated lattice. We also give applications to the study of characteristic varieties of smooth complex algebraic varieties. As an example we discuss A. Suciu’s line arrangement, the so-called deleted B3-arrangement.

Beilinson-Hodge cycles on semiabelian varieties

Beilinson conjectured that all rational cycles of type (q,q) on the qth cohomology of a smooth complex algebraic variety should come from motivic cohomology. The purpose of this note is to prove this when the variety is a semiabelian variety or a product of curves. The proof is based on the study of invariants under the Mumford-Tate group.

On admissible rank one local systems

A rank one local system L on a smooth complex algebraic variety M is 1-admissible if the dimension of the first cohomology group H 1 ( M , L ) can be computed from the cohomology algebra H ∗ ( M , C ) in degrees ⩽2. Under the assumption that M is 1-formal, we show that all local systems, except finitely many, on a non-translated irreducible component W of the first characteristic variety V 1 ( M ) are 1-admissible, see Proposition 3.1. The same result holds for local systems on a translated component W, but now H ∗ ( M , C ) should be replaced by H ∗ ( M 0 , C ) , where M 0 is a Zariski open subset obtained from M by deleting some hypersurfaces determined by the translated component W, see Theorem 4.3. One consequence of this result is that the local systems L where the dimension of H 1 ( M , L ) jumps along a given positive-dimensional component of the characteristic variety V 1 ( M ) have finite order, see Theorem 4.7. Using this, we show in Corollary 4.9 that dim H 1 ( M , L ) = dim H 1 ( M , L − 1 ) for any rank one local system L on a smooth complex algebraic variety M.

Motivic Serre invariants, ramification, and the analytic Milnor fiber

We show how formal and rigid geometry can be used in the theory of complex singularities, and in particular in the study of the Milnor fibration and the motivic zeta function. We introduce the so-called analytic Milnor fiber associated to the germ of a morphism f from a smooth complex algebraic variety X to the affine line. This analytic Milnor fiber is a smooth rigid variety over the field of Laurent series C((t)). Its etale cohomology coincides with the singular cohomology of the classical topological Milnor fiber of f; the monodromy transformation is given by the Galois action. Moreover, the points on the analytic Milnor fiber are closely related to the motivic zeta function of f, and the arc space of X. We show how the motivic zeta function can be recovered as some kind of Weil zeta function of the formal completion of X along the special fiber of f, and we establish a corresponding Grothendieck trace formula, which relates, in particular, the rational points on the analytic Milnor fiber over finite extensions of C((t)), to the Galois action on its etale cohomology. The general observation is that the arithmetic properties of the analytic Milnor fiber reflect the structure of the singularity of the germ f.

Schubert Varieties and the Fusion Products

For each A\in ℕ^n we define a Schubert variety \sh_A as a closure of the \mathrm{SL}_2(ℂ[t]) -orbit in the projectivization of the fusion product M^A . We clarify the connection of the geometry of the Schubert varieties with an algebraic structure of M^A as \mathfrak{sl}_2 \otimes ℂ[t] modules. In the case, when all the entries of A are different, \sh_A is a smooth projective complex algebraic variety. We study its geometric properties: the Lie algebra of the vector fields, the coordinate ring, the cohomologies of the line bundles. We also prove that the fusion products can be realized as the dual spaces of the sections of these bundles.

Lefschetz numbers of iterates of the monodromy and truncated arcs

We express 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. We also construct a canonical representative of the Milnor fibre in a suitable monodromic Grothendieck group.

Equivariant Poincaré Polynomials and Counting Points over Finite Fields

Suppose a finite group acts as a group of automorphisms of a smooth complex algebraic variety which is defined over a number field. We show how, in certain circumstances, an equivariant comparison theorem in l-adic cohomology may be used to convert the computation of the graded character of the induced action on cohomology into questions about numbers of rational points of varieties over finite fields. This is carried through in three applications: first, for the symmetric group acting on the moduli space of n points on a genus zero curve; second, for a unitary reflection group acting on the complement of its reflecting hyperplanes; and third, for the symmetric group action on the space of configurations of points in any smooth variety which satisfies certain strong purity conditions.

Uniform bounds and symbolic powers on smooth varieties

We show how multiplier ideals can be used to obtain uniform multiplicative bounds for certain families of ideals on a smooth complex algebraic variety. In particular we prove a quick but rather surprising result about symbolic powers of radical ideals on such a variety. Specifically, let I be a radical ideal sheaf on a smooth variety X defining a reduced subscheme Z of X and suppose that every irreducible component of Z has codimension at most e in X. Given an integer m > 0 suppose that f is a function germ that vanishes to order at least e.m at a general point of each irreducible component of Z . Then in fact f lies in the m-th power I^m of I.

On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties

We prove that for any affine variety S defined overQ there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety X(G, PO(3)) = Hom(G, PO(3))//PO(3). The subset U contains all real points of S. As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties.

Higher Bott-Chern forms and Beilinson's regulator

In this paper, we prove a Gauss-Bonnet theorem for the higher algebraic K-theory of smooth complex algebraic varieties. To each exact n-cube of hermitian vector bundles, we associate a higher Bott-Chen form, generalizing the Bott-Chern forms associated to exact sequences. These forms allow us to define characteristic classes from K-theory to absolute Hodge cohomology. Then we prove that these characteristic classes agree with Beilinson's regulator map.