Nash constructible functions

Nash constructible functions on a real algebraic set V are defined as linear combinations, with integer coefficients, of Euler characteristic of fibres of proper regular morphisms restricted to connected components of algebraic sets. We prove that if V is compact, these functions are sums of signs of semialgebraic arc-analytic functions (i.e. functions which become analytic when composed with any analytic arc). We also give a sharp upper bound to the number of semialgebraic arc-analytic functions which are necessary to define any given Nash constructible functions.     

Rigidity and moduli space in Real Algebraic Geometry

Let R be a real closed field and let X be an affine algebraic variety over R. We say that X is universally map rigid (UMR for short) if, for each irreducible affine algebraic variety Z over R, the set of nonconstant rational maps from Z to X is finite. A bijective map from an affine algebraic variety over R to X is called weak change of the algebraic structure of X if it is regular and φ⁻¹ is a Nash map, which preserves nonsingular points. We prove the following rigidity theorem: every affine algebraic variety over R is UMR up to a weak change of its algebraic structure. Let us introduce another notion. Let Y be an affine algebraic variety over R. We say that X and Y are algebraically unfriendly if all the rational maps from X to Y and from Y to X are trivial, i.e., Zariski locally constant. From the preceding theorem, we infer that, if dim (X)≥1, then there exists a set of weak changes of the algebraic structure of X such that, for each t,s ∈ R with t≠s, and are algebraically unfriendly. This result implies the following expected fact: For each (nonsingular) affine algebraic variety X over R of positive dimension, the natural Nash structure of X does not determine the algebraic structure of X. In fact, the moduli space of birationally nonisomorphic (nonsingular) affine algebraic varieties over R, which are Nash isomorphic to X, has the same cardinality of R. This result was already known under the special assumption that R is the field of real numbers and X is compact and nonsingular. The author is a member of GNSAGA of CNR, partially supported by MURST and European Research Training Network RAAG 2002–2006 (HPRN–CT–00271).     

Classification of real Nash fibre bundles

Let G be a compact subgroup of an orthogonal group and X an affine, real, semialgebraic Nash variety. A principal Nash G-bundle over X is said to be strongly Nash if it is induced, up to Nash equivalences, of some universal bundle under a Nash map. Not all Nash bundles are strongly Nash and we denote by S(X, G) the class of strongly Nash G-bundles over X. The principal aim of this paper is to prove the following classification theorem: two bundles of S(X, G) are Nash equivalent if and only if they are topologically equivalent; more,there exists a bijection between the family of the classes of Nash equivalent bundles of S(X, G) and , where is the sheaf of germs of the continous maps from X to G. This result leads to find the largest class of principal Nash G-bundles over X in which the topological equivalence always implies the Nash one. Well, we prove that this class is exactly S(X, G). Research partially supported by M.I.U.R.     

Logarithmic foliations on compact algebraic surfaces

LetF be a holomorphic singular foliation with an invariant compact curveS on a compact algebraic surfaceX. In this work we prove thatF is logarithmic under some generic conditions in the singularities ofF inS.Supported by CNPq (Brazil).     

Characterization of isolated homogeneous hypersurface singularities in ?4

Let V be a hypersurface with an isolated singularity at the origin in ℂ ⁿ⁺¹. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface signularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau’s theorem remains true for singularities with geometric genus equal to zero. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Algebraic Levi-Flat Hypervarieties in Complex Projective Space

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in ℂ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety.     

Extension of Fréchet valued real analytic functions from subvarieties of ℝd

It is shown that for an algebraic subvariety X of ℝ^d every Fréchet valued real analytic function on X can be extended to a real analytic function on ℝ^d if and only if X is of type (PL), i.e. all of its singularities are of a certain type. Necessity of this condition is shown for any real analytic variety. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonfattening of Mean Curvature Flow at Singularities of Mean Convex Type

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝ^{n + 1} remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. We further show that assumptions of the theorem hold provided all blowup flows are of the kind appearing in a mean convex flow, i.e., smooth, multiplicity 1 , and convex. Our results generalize the well-known fact that the level set flow of a mean convex initial hypersurface M does not fatten. They also provide the first instance where nonfattening is concluded from local information around the singular set or from information about the singularity profiles of a flow. © 2019 Wiley Periodicals, Inc.     

Standard P3-bundles of exponent two on algebraic surfaces

Let V_K be a twisted form of P³ over the function field of an algebraic surface with Pic V_K generated by the half of the canonical line bundle. We construct an algebraic fibre space V → X projective flat over a smooth projective surface X with the generic fibre V _K → Spac K satisfying some properties.     

On holonomic D-modules attached to Reiffen’s hypersurface isolated singularities

A family of hypersurface isolated singularities, called Reiffen’s examples, is considered in the context of holonomic D-modules. Algebraic local cohomology classes attached to Reiffen’s singularities and their annihilating differential operators are studied. The algebraic local cohomology solution space to the first-order holonomic D-module is determined explicitly. As an application, it is shown that the multiplicity of the holonomic D-module can be described in terms of classical invariants of singularities. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

The Aluffi algebra of hypersurfaces with isolated singularities

The Alu? algebra is an algebraic definition of a characteristic cycle of a hypersurface in intersection theory. In this paper, we study the Alu? algebra of quasi-homogeneous and locally Eulerian hypersurfaces with only isolated singularities. We prove that the Jacobian ideal of an a?ne hypersurface with isolated singularities is of linear type if and only if it is locally Eulerian. We show that the gradient ideal of a projective hypersurface with only isolated singularities is of linear type if and only if the a?ne curve in each a?ne chart associated to singular points is locally Eulerian. We show that the gradient ideal of Nodal and Cuspidal projective plane curves are of linear type.     

Algebraicity of cycles on smooth manifolds

According to the Nash–Tognoli theorem, each compact smooth manifold M is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of M. It is interesting to investigate to what extent algebraic and differential topology of compact smooth manifolds can be transferred into the algebraic-geometric setting. Many results, examples and counterexamples depend on the detailed study of the homology classes represented by algebraic subsets of X, as X runs through the class of all algebraic models of M. The present paper contains several new results concerning such algebraic homology classes. In particular, a complete solution in codimension 2 and strong results in codimensions 3 and 4.     

On power series rings over valuation domains

Let V be a valuation domain and V[[X]] be the power series ring over V. In this paper, we show that if V[[X]] is a locally finite intersection of valuation domains, then V is an SFT domain and hence a discrete valuation domain. As a consequence, it is shown that the power series ring V[[X]] is a Krull domain if and only if V[[X]] is a generalized Krull domain if and only if V[[X]] is an integral domain of Krull type (or equivalently, a PvMD of finite t-character) if and only if V is a discrete valuation domain with Krull dimension at most one.     

Smooth three-dimensional canonical thresholds

If X is an algebraic variety with at most canonical singularities and S is a ?-Cartier hypersurface in X, then the canonical threshold of the pair (X, S) is defined as the least upper bound of the reals c for which the pair (X, cS) is canonical. We show that the set of all possible canonical thresholds of the pairs (X, S), where X is smooth and three-dimensional, satisfies the ascending chain condition. We also derive a formula for the canonical threshold of the pair (?³, S), where S is a Brieskorn singularity.     

Is Every Toric Variety an M-Variety?

A complex algebraic variety X defined over the real numbers is called an M-variety if the sum of its Betti numbers (for homology with closed supports and coefficients in ) coincides with the corresponding sum for the real part of X. It has been known for a long time that any nonsingular complete toric variety is an M-variety. In this paper we consider whether this remains true for toric varieties that are singular or not complete, and we give a positive answer when the dimension of X is less than or equal to 3 or when X is complete with isolated singularities.An erratum to this article can be found at     

Affine normal surfaces with simply-connected smooth locus

Given a normal affine surface V defined over \mathbbC{\mathbb{C}}, we look for algebraic and topological conditions on V which imply that V is smooth or has at most rational singularities. The surfaces under consideration are algebraic quotients \mathbbCⁿ/G{\mathbb{C}^n/G} with an algebraic group action of G and topologically contractible surfaces. Theorem 3.6 can be considered as a global version of the well-known result of Mumford giving a smoothness criterion for a germ of a normal surface in terms of the local fundamental group.     

Rational surfaces in algebraic models of smooth manifolds

Every compact smooth manifold M is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of M. We study modulo 2 homology classes represented by rational algebraic surfaces in X, as X runs through the class of all algebraic models of M. Received: 16 June 2007     

Characterization of isolated homogeneous hypersurface singularities in ℂ4

Let V be a hypersurface with an isolated singularity at the origin in ? ⁿ⁺¹. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface signularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau’s theorem remains true for singularities with geometric genus equal to zero.     

Hodge genera of algebraic varieties I

The aim of this paper is to study the behavior of Hodge‐theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a complex algebraic variety X to such invariants of singularities of proper algebraic maps defined on X. Such formulae severely constrain, both topologically and analytically, the singularities of complex maps, even between smooth varieties. Similar results were announced by the first and third author in [13, 32]. © 2007 Wiley Periodicals, Inc.