Complex Algebraic Varieties Research Articles

On a generalization of Stickelberger’s Theorem

We prove two versions of Stickelberger’s Theorem for positive dimensions and use them to compute the connected and irreducible components of a complex algebraic variety. If the variety is given by polynomials of degree ≤d in n variables, then our algorithms run in parallel (sequential) time (nlogd)O(1) (dO(n4)). In the case of a hypersurface, the complexity drops to O(n2log2d) (dO(n)). In the proof of the last result we use the effective Nullstellensatz for two polynomials, which we also prove by very elementary methods.

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.

A Note on the Projective Varieties of Almost General Type

A $\mathbf{Q}$-Cartier divisor $D$ on a projective variety $M$ is {\it almost nup}, if $(D , C) > 0$ for every very general curve $C$ on $M$. An algebraic variety $X$ is of {\it almost general type}, if there exists a projective variety $M$ with only terminal singularities such that the canonical divisor $K_M$ is almost nup and such that $M$ is birationally equivalent to $X$. We prove that a complex algebraic variety is of almost general type if and only if it is neither uniruled nor covered by any family of varieties being birationally equivalent to minimal varieties with numerically trivial canonical divisors, under the minimal model conjecture. Furthermore we prove that, for a projective variety $X$ with only terminal singularities, $X$ is of almost general type if and only if the canonical divisor $K_X$ is almost nup, under the minimal model conjecture.

Geometric motivic Poincaré series of quasi-ordinary singularities

AbstractThegeometric motivic Poincaré seriesof a germ (S, 0) of complex algebraic variety takes into account the classes in the Grothendieck ring of the jets of arcs through (S, 0). Denef and Loeser proved that this series has a rational form. We give an explicit description of this invariant when (S, 0) is an irreducible germ ofquasi-ordinary hypersurface singularityin terms of the Newton polyhedra of thelogarithmic jacobian ideals. These ideals are determined by thecharacteristic monomialsof a quasi-ordinary branch parametrizing (S, 0).

HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES

In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces.We introduce a motivic Chern class transformationmCy: K0( var /X) → G0(X) ⊗ ℤ[y], which generalizes the total λ-class λy(T*X) of the cotangent bundle to singular spaces. Here K0( var /X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G0(X) is the Grothendieck group of coherent sheaves of [Formula: see text]-modules. A first construction of mCyis based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mCyis based on some results from the theory of algebraic mixed Hodge modules due to M. Saito.We define a natural transformation Ty*: K0( var /X) → H*(X) ⊗ ℚ[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Ty*is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y = -1), the Todd class transformation in the singular Riemann-Roch theorem of Baum–Fulton–MacPherson (for y = 0) and the L-class transformation of Cappell-Shaneson (for y = 1).We also explain the relation among the "stringy version" of our characteristic classes, the elliptic class of Borisov–Libgober and the stringy Chern classes of Aluffi and De Fernex–Lupercio–Nevins–Uribe.All our results can be extended to varieties over a base field k of characteristic 0.

Counting Irreducible Components of Complex Algebraic Varieties

We present an algorithm for counting the irreducible components of a complex algebraic variety defined by a fixed number of polynomials encoded as straight-line programs (slps). It runs in polynomial time in the Blum–Shub–Smale (BSS) model and in randomized parallel polylogarithmic time in the Turing model, both measured in the lengths and degrees of the slps. Our algorithm is obtained from an explicit version of Bertini’s theorem. For its analysis we further develop a general complexity theoretic framework appropriate for algorithms in algebraic geometry.

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]

Wonderful compactification of an arrangement of subvarieties

We define the wonderful compactification of an arrangement of subvarieties. Given a complex nonsingular algebraic variety $Y$ and certain collection $\mathcal{G}$ of subvarieties of $Y$, the wonderful compactification $Y_\mathcal{G}$ can be constructed by a sequence of blow-ups of $Y$ along the subvarieties of the arrangement. This generalizes the Fulton-MacPherson configuration spaces and the wonderful models given by De Concini and Procesi. We give a condition on the order of blow-ups in the construction of $Y_\mathcal{G}$ such that each blow-up is along a nonsingular center.

Topology and geometry of cohomology jump loci

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, V_k and R_k, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of V_k and R_k are analytically isomorphic, if the group is 1-formal; in particular, the tangent cone to V_k at 1 equals R_k. These new obstructions to 1-formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasi-projective algebraic variety. This sheds new light on a classical problem of J.-P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given.

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.

Equivariant Genera of Complex Algebraic Varieties

For smooth manifolds, Atiyah and Meyer studied contributions of monodromy to usual signatures. In this note we obtain Atiyah-Meyer type formulae for equivariant Hodge-theoretic genera of complex algebraic varieties. Equivariant Hirzebruch genera y(X;g) of a quasi-projective variety X acted upon by a nite group of algebraic automorphisms are dened by combining the group action with the information encoded by the Hodge ltration of the mixed Hodge structure in cohomology. While for a projective algebraic manifold y(X;g) can by computed by the Atiyah- Singer holomorphic Lefschetz theorem, we derive a Atiyah-Meyer type formula for y(X;g) in the case when X is not necessarily smooth or compact, but just bers equivariantly (in the complex topology) over a compact algebraic manifold. These results apply to computing Hodge-theoretic invariants of orbit spaces. We also obtain some results comparing equivariant Hodge-theoretic genera of the range and domain of an equivariant algebraic map in terms of its singularities.

On the complexity of counting components of algebraic varieties

We give a uniform method for the two problems of counting the connected and irreducible components of complex algebraic varieties. Our algorithms are purely algebraic, i.e., they use only the field structure of C . They work in parallel polynomial time, i.e., they can be implemented by algebraic circuits of polynomial depth. The design of our algorithms relies on the concept of algebraic differential forms. A further important building block is an algorithm of Szántó computing a variant of characteristic sets. Furthermore, we use these methods to obtain a parallel polynomial time algorithm for computing the Hilbert polynomial of a projective variety which is arithmetically Cohen–Macaulay.

Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers

The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasi-projective case results of Green–Lazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers h q , 0 of congruence covers in any dimension, generalizing results of E. Hironaka and Sakuma. We extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems. In particular, we obtain a generalization of the polynomial periodicity of Betti numbers of unbranched congruence covers due to Sarnak–Adams. We derive a geometric characterization of finite abelian covers, which recovers the classic one and the one of Pardini. We use this, for example, to prove a conjecture of Libgober about Hodge numbers of abelian covers.

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.

Vanishing theorems for Dolbeault cohomology of log homogeneous varieties

We consider a complete nonsingular complex algebraic variety having a normal crossing divisor such that the associated logarithmic tangent bundle is generated by its global sections. We obtain an optimal vanishing theorem for logarithmic Dolbeault cohomology of nef line bundles in that setting. This implies a vanishing theorem for ordinary Dolbeault cohomology which generalizes results of Broer for flag varieties, and of Mavlyutov for toric varieties.

Tropical Geometry

This is the first appearance of “Tropical Geometry” as the topic of an Oberwolfach workshop, though the subject figured earlier at the Forschungsinstitut as an Oberwolfach Seminar on “Tropical Algebraic Geometry” in October 2004. The workshop was organized by Eva-Maria Feichtner (U Bremen), Andreas Gathmann (U Kaiserslautern), Ilia Itenberg (U Strasbourg) and Thorsten Theobald (U Frankfurt) and brought together about 50 mathematicians from the many fields Tropical Geometry touches upon. The workshop came timely as the subject develops into a field of its own with numerous connections to diverse branches of mathematics. Tropical Geometry can be considered as an algebraic geometry over the semifield ({\mathbb R}, \max, +) . The name was coined by French computer scientists to honor the pioneering work of their Brazilian colleague Imre Simon on the max-plus semiring. Alternatively, Tropical Geometry can be understood as the geometry resulting from complex geometry by a certain degeneration process: complex toric varieties are replaced by real linear spaces and, more generally, complex algebraic varieties are replaced by polyhedral complexes, i.e., by piecewise-linear objects. The roots of Tropical Geometry extend at least to Bergman's logarithmic limit sets (in the 70's), Viro's patchworking construction (in the late 70's), Maslov's dequantization of positive real numbers (in the 80's), and to the use of idempotent semirings in applications to optimization, control theory, and max-plus operators (in the 90's). In recent years, the various research directions have been fruitfully merged, generalized and advanced to what is now called Tropical Geometry. Along the way, deep connections to numerous branches of pure and applied mathematics have been unveiled; among them e.g. algebraic geometry, symplectic geometry, complex analysis, dynamical systems, geometric combinatorics, as well as computer algebra, algebraic statistics, and phylogenetics. Despite of the short time span of its recent development, Tropical Geometry has already been widely recognized as an important discipline and as a unifying viewpoint for the transition of algebro-geometric problems to combinatorial ones. The aim of this Oberwolfach workshop was to furnish this newly emerging field with an outstanding communication platform and to foster the interaction between the various research directions that are involved. Moreover, Tropical Geometry is still in a phase of forming its foundations: fundamental concepts, such as abstract tropical varieties, and basic definitions, such as a proper notion of tropical intersection multiplicity, are still under construction. It was therefore an additional goal of this workshop to advance the field through discussions among people with different views of the subject. In both respects, the workshop met all expectations: in 22 fifty-minutes talks the current state of development was outlayed from various viewpoints. An afternoon with half-hour talks by graduate students and an evening session featuring a software presentation completed the picture. This part of the workshop is well accounted for by the following collection of abstracts. Most importantly – and only forthcoming activities and publications will indirectly report on this – numerous informal discussions evolved between participants with most different mathematical backgrounds. A fruitful atmosphere of exchange developed during the week, notably thanks to the excellent facilities at the Mathematisches Forschungsinstitut. On behalf of the participants, we wish to express our sincere thanks to the Institute for hosting this workshop.

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.

Remainder Wheels and Group Theory

Lawrence Brenton (brenton@math.wayne.edu) was educated at the University of Pennsylvania, the University of Washington in Seattle, and the University of Bonn, Germany. He has served on the faculty of Wayne State University for 33 years, specializing in singularity theory in complex algebraic and analytic varieties. His teaching activities include directing the undergraduate research program for many years as well as mentoring graduate students in mathematics and mathematics education.

Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n−1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multiseparability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions, and with quasiexactly solvable systems. Here, we announce a complete classification of nondegenerate (i.e., four-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly ten nondegenerate potentials.

Degeneracy of holomorphic curves into algebraic varieties

Applying the Second Main Theorem of [J. Noguchi, J. Winkelmann, K. Yamanoi, The second main theorem for holomorphic curves into semi-Abelian varieties II, Forum Math., in press, e-print archive, math.CV/0405492], we deal with the algebraic degeneracy of entire holomorphic curves f : C → X from the complex plane C into a complex algebraic normal variety X of positive log Kodaira dimension that admits a finite proper morphism to a semi-Abelian variety. We will also discuss applications to the Kobayashi hyperbolicity problem.