Birational Invariants Research Articles

Motivic infinite cyclic covers

We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (under certain finiteness conditions) an element in the Grothendieck ring K0(VarCμˆ), which we call motivic infinite cyclic cover, and show its birational invariance. Our construction provides a unifying approach for the Denef–Loeser motivic Milnor fiber of a complex hypersurface singularity germ, and the motivic Milnor fiber of a rational function, respectively.

Stable birational invariants and the Lüroth problem

We describe recent progress on rationality and stable rationality questions. We discuss the cohomological or Chow decomposition of the diagonal, a very strong stably birationally invariant property which controls many of the previously defined stable birational invariants. On the other hand, it behaves very well under specialization and desingularization of mild singularities.

Stable Birational Invariants with Galois Descent and Differential Forms

I show that the cohomology of the generic points of algebraic complex varieties becomes {\sl stable} birational invariant, when considered `modulo the cohomology of the generic points of the affine spaces'.

The unstable slice filtration

The main goal of this paper is to construct an analogue of Vo- evodsky's slice filtration in the motivic unstable homotopy category. The construction is done via birational invariants, this is motivated by the exis- tence of an equivalence of categories between the orthogonal components for Voevodsky's slice filtration and the birational motivic stable homotopy cat- egories constructed in (9). Another advantage of this approach is that the slices appear naturally as homotopy fibres (and not as in the stable setting, where they are defined as homotopy cofibres) which behave much better in the unstable setting.

Birational invariants and 𝔸1-connectedness

We study some aspects of the relationship between 𝔸1-homotopy theory and birational geometry. We study the so-called 𝔸1-singular chain complex and zeroth 𝔸1-homology sheaf of smooth algebraic varieties over a field k. We exhibit some ways in which these objects are similar to their counterparts in classical topology and similar to their motivic counterparts (the (Voevodsky) motive and zeroth Suslin homology sheaf). We show that if k is infinite, the zeroth 𝔸1-homology sheaf is a birational invariant of smooth proper varieties, and we explain how these sheaves control various cohomological invariants, e.g., unramified étale cohomology. In particular, we deduce a number of vanishing results for cohomology of 𝔸1-connected varieties. Finally, we give a partial converse to these vanishing statements by giving a characterization of 𝔸1-connectedness by means of vanishing of unramified invariants.

Invariants birationnels dans la suite spectrale de Bloch-Ogus

AbstractFor a field k of cohomological dimension d we prove that the groups , (l, car.k) = 1, are birational invariants of smooth projective geometrically integral varieties over k of dimension n. Using the Kato conjecture, which has been recently established by Kerz and Saito [18], we obtain a similar result over a finite field for the groups . We relate one of these invariants with the cokernel of the l-adic cycle class map , which gives an analogue of a result of Colliot-Thélène and Voisin [5] 3.11 over ℂ for varieties over a finite field.

On the Birational Invariance of the BCOV Torsion of Calabi-Yau Threefolds

Fang et al. (J. Diff. Geom. 80(2):175–259, 2008, Sect. 4, Conj. 4.17) conjecture that a certain spectral string-theoretic invariant of Calabi-Yau threefolds (the BCOV torsion) is a birational invariant. We prove a weak form of this conjecture. The proof combines the arithmetic Riemann-Roch theorem in Arakelov geometry with some inputs from motivic integration theory.

Statistical Learning Theory of Quasi-Regular Cases

Many learning machines such as normal mixtures and layered neural networks are not regular but singular statistical models, because the map from a parameter to a probability distribution is not one-to-one. The conventional statistical asymptotic theory can not be applied to such learning machines because the likelihood function can not be approximated by any normal distribution. Recently, new statistical theory has been established based on algebraic geometry and it was clarified that the generalization and training errors are determined by two birational invariants, the real log canonical threshold and the singular fluctuation. However, their concrete values are left unknown. In the present paper, we propose a new concept, a quasi-regular case in statistical learning theory. A quasi-regular case is not a regular case but a singular case, however, it has the same property as a regular case. In fact, we prove that, in a quasi-regular case, two birational invariants are equal to each other, resulting that the symmetry of the generalization and training errors holds. Moreover, the concrete values of two birational invariants are explicitly obtained, the quasi-regular case is useful to study statistical learning theory.

Birational invariants defined by Lawson homology

New birational invariants for a projective manifold are defined by using Lawson homology. These invariants can be highly nontrivial even for projective threefolds. Our techniques involve the weak factorization theorem of Wlodarczyk and tools developed by Friedlander, Lawson, Lima-Filho and others. A blowup formula for Lawson homology is given in a separate section. As an application, we show that for each n\geq 5, there is a smooth rational variety X of dimension n such that the Griffiths groups Griff}_p(X) are infinitely generated even modulo torsion for all p with 2\leq p\leq n-3.

THE COHOMOLOGICAL BIRATIONAL INVARIANTS FOR 4-DIMENSIONAL TORI

This paper is the first systematic step to obtain the birational classificationof 4-dimensional tori. All the cohomological birational invariants of this tori arecalculated. All the 4-dimensional tori with nontrivial invariants are described.

Coring stabilizers for a Hopf algebra coaction

The coring stabilizer Stab ( P ) is introduced for any prime ideal P of a right H-comodule algebra A such that the factor ring A / P is either right or left Goldie. This notion is used to obtain Hopf algebraic analogs of two category equivalences associated with a homogeneous space. The category of linearized quasicoherent sheaves on a noncommutative homogeneous space is interpreted as a suitable quotient category of the category of Hopf modules. Birational invariance of such quotient categories is proved. It is shown that for a birational H-coequivariant extension B of A properly defined subsets of prime ideals of A and B correspond to each other bijectively.

Tempered Fundamental Group and Metric Graph of a Mumford Curve

The aim of this paper is to give some general results on the tempered fundamental group of p -adic smooth algebraic varieties (which is a sort of analog of the topological fundamental group of complex algebraic varieties in the p -adic world). The main result asserts that one can recover the metric structure of the graph of the stable model of a Mumford curve from the tempered fundamental group of the curve. We also prove the birational invariance, invariance under algebraically closed extensions and a Kunneth formula for the tempered fundamental group. We describe the tempered fundamental group of an abelian variety and link the tempered fundamental group of a curve to the tempered fundamental group of its Jacobian variety.

Dimension Polynomials of Intermediate Fields and Krull-type Dimension of Finitely Generated Differential Field Extensions

Let K be a differential field of zero characteristic and let L be a finitely generated differential field extension of K. We introduce the concept of a transcendence type and dimension of the extension L/K considering chains of intermediate differentials fields of this extension. Using the technique of differential dimension polynomials we obtain relationships between the transcendence type and dimension of L/K and differential birational invariants of this extension carried by its dimension polynomials.

On maximal curves with Frobenius dimension 3

Frobenius dimension is one of the most important birational invariants of maximal curves. In this paper, a characterization of maximal curves with Frobenius dimension equal to 3 is provided. Our main tool is the Natural Embedding Theorem for maximal curves. As an application, maximal curves with Frobenius dimension 3 defined over the fields with 16 and 25 elements are completely classified.

Blow-up formulas and smooth birational invariants

We prove that the blow-up formula for the singular homology of a complex smooth projective variety with a smooth center respects two natural filtrations, namely the topological and the geometric filtrations. This then enables us to establish some smooth birational invariants.

A geometric approach to problems in birational geometry

A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties. A fundamental question is the following: Given two mildly singular projective varieties with some of the first variety's pseudonormed spaces being isometric to the corresponding ones of the second variety's, can one construct a birational map between them that induces these isometries? In this work, a positive answer to this question is given for varieties of general type. This can be thought of as a theorem of Torelli type for birational equivalence.

Higher elliptic genera

We show that elliptic classes introduced in \cite{annals} for spaces with infinite fundamental groups yield Novikov's type higher elliptic genera which are invariants of K-equivalence. This include, as a special case, the birational invariance of higher Todd classes studied recently by J.Rosenberg and J.Block-S.Weinberger. We also prove the modular properties of these genera, show that they satisfy a McKay correspondence, and consider their twist by discrete torsion.

An analogue of the Novikov Conjecture in complex algebraic geometry

We introduce an analogue of the Novikov on signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain higher Todd genera are birational invariants. This implies birational invariance of certain extra combinations of Chern classes (beyond just the classical Todd genus) in the case of varieties with large fundamental group (in the topological sense). We prove the conjecture under the assumption of the strong Novikov Conjecture for the fundamental group, which is known to be correct for many groups of geometric interest. We also show that, in a certain sense, our conjecture is best possible.

On certain birational invariants of the Fermat curves

For an integer m ≥ 3 let K m = Q ( x , y ) ( x m + y m = 1 ) be the m -th Fermat field over Q . We study, in the case of Fermat fields, the groups of integral differentials introduced by Kähler [E. Kähler, Geometria aritmetica, Annali di Mat. 45 (1958)] and Bost [R. Berndt, Arithmetisch ganze Differentiale, Abh. Math. Sem. Univ. Hamburg 47 (1978) 249–270] for arithmetic function fields and compute them for small m . An essential step of our considerations is the explicit description of the discrete valuation rings with quotient field K m which are essentially of finite type and smooth over Z .

Stringy Chern classes of singular varieties

Motivic integration [M. Kontsevich, Motivic integration, Lecture at Orsay, 1995] and MacPherson's transformation [R. MacPherson, Chern classes for singular varieties, Ann. of Math. 100 (1974) 423–432] are combined in this paper to construct a theory of “stringy” Chern classes for singular varieties. These classes enjoy strong birational invariance properties, and their definition encodes data coming from resolution of singularities. The singularities allowed in the theory are those typical of the minimal model program; examples are given by quotients of manifolds by finite groups. For the latter an explicit formula is proven, assuming that the canonical line bundle of the manifold descends to the quotient. This gives an expression of the stringy Chern class of the quotient in terms of Chern–Schwartz–MacPherson classes of the fixed-point set data.