Nonsingular Algebraic Variety Research Articles

Icard groups for line bundles with connections

We study analogues of the usual Picard group for complex manifolds or non-singular complex algebraic varieties but instead of line bundles we study line bundles with connections. We choose an approach which works for both cases. We identify obstructions for the existence of a connection, or a connection which is even integrable or regular (integrable), and point out where one should be careful when passing from the analytic to the algebraic case.

Equivariant Gromov-Witten Invariants of Algebraic GKM Manifolds

An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use virtual localization to express equivariant Gromov-Witten invariants of any algebraic GKM manifold (which is not necessarily compact) in terms of Hodge integrals over moduli stacks of stable curves and the GKM graph of the GKM manifold.

Intersection products for tensor triangular Chow groups

We show that under favorable circumstances, one can construct an intersection product on the Chow groups of a tensor triangulated category T (as defined in [5]) which generalizes the usual intersection product on a non-singular algebraic variety. Our construction depends on the choice of an algebraic model for T (a tensor Frobenius pair), which has to satisfy a K-theoretic regularity condition analogous to the Gersten conjecture from algebraic geometry. In this situation, we are able to prove an analogue of the Bloch formula and use it to define an intersection product similar to Grayson's construction from [14]. We then recover the usual intersection product on a non-singular algebraic variety assuming a K-theoretic compatibility condition.

The Global Sections of the Chiral de Rham Complex on a Kummer Surface

The chiral de Rham complex is a sheaf of vertex algebras {\Omega}^ch_M on any nonsingular algebraic variety or complex manifold M, which contains the ordinary de Rham complex as the weight zero subspace. We show that when M is a Kummer surface, the algebra of global sections is isomorphic to the N = 4 superconformal vertex algebra with central charge 6. Previously, CP^n was the only manifold where a complete description of the global section algebra was known.

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.

Chains of points in the Semple tower

We introduce a proximity relation in the Semple tower over a nonsingular algebraic variety Z , and we prove a formula, relating the multiplicity of a curve C in Z at a point z with the multiplicities of its iterated Nash blowing-ups at the proximate points of z , parallel to a classical formula of Enriques. We also show that the Semple tower of a surface parametrizes its infinitely near points.

Vanishing of principal value integrals on surfaces

Principal value integrals are associated to multi-valued rational differential forms with normal crossings support on a non-singular algebraic variety. We prove their vanishing on rational surfaces in the context of a conjecture of Denef-Jacobs. As an application we obtain a strong vanishing result for candidate poles of p-adic and motivic Igusa zeta functions.

Variétés Galois-maximales et éclatements

Let X and X′ be irreducible projective nonsingular algebraic varieties over R. We suppose that X is obtained from X′ by a finite sequence of blow-ups and blow-downs along nonsingular centers and that all these centers are GM-varieties. Then we show that X is a GM-variety if and only if X′ is a GM-variety.

Asymptotic Growth of the Number of Classes of Real Plane Algebraic Curves as the Degree Grows

The work presents some results on the asymptotics of the number of real plane algebraic curves as the degree grows. In particular, we obtain the asymptotics of the number of curves considered up to the isotopy and rigid isotopy, as well as the number of isotopic classes of maximal curves realizable by T-curves. Some results are generalized to hypersurfaces in nonsingular algebraic varieties of arbitrary dimension. Bibliography: 18 titles.

Zeta Functions and ‘Kontsevich Invariants’ on Singular Varieties

AbstractLet X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Igusa) zeta function, associated to a regular function on X, which specializes to both the classical p-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant.This paper treats a generalization to singular varieties. Batyrev already considered such a ‘Kontsevich invariant’ for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any ℚ-Gorenstein variety X we associate a motivic zeta function and a ‘Kontsevich invariant’ to effective ℚ-Cartier divisors on X whose support contains the singular locus of X.

Hirzebruch genera of manifolds with torus action

A quasitoric manifold is a smooth -manifold with an action of the compact torus such that the action is locally isomorphic to the standard action of on and the orbit space is diffeomorphic, as a manifold with corners, to a simple polytope . The name refers to the fact that topological and combinatorial properties of quasitoric manifolds are similar to those of non-singular algebraic toric varieties (or toric manifolds). Unlike toric varieties, quasitoric manifolds may fail to be complex. However, they always admit a stably (or weakly almost) complex structure, and their cobordism classes generate the complex cobordism ring. Buchstaber and Ray have recently shown that the stably complex structure on a quasitoric manifold is determined in purely combinatorial terms, namely, by an orientation of the polytope and a function from the set of codimension-one faces of the polytope to primitive vectors of the integer lattice. We calculate the -genus of a quasitoric manifold with a fixed stably complex structure in terms of the corresponding combinatorial data. In particular, this gives explicit formulae for the classical Todd genus and the signature. We also compare our results with well-known facts in the theory of toric varieties.

The push-forward and Todd class of flag bundles

Consider a connected reductive algebraic group G over an algebraically closed field k, and a principal G-bundle π : X → Y , where X and Y are non-singular algebraic varieties over k. For any parabolic subgroup P ⊂ G, the map π factors through the flag bundle h : X/P → Y . In this note, we describe the push-forward (or Gysin homomorphism) h∗ : A∗(X/P ) → A∗(Y ) where A∗ denotes the Chow group. Moreover, we compute the Todd class of the tangent bundle to h in A∗(X/P )Q. In the case when k is the field of complex numbers, our results hold when the Chow ring is replaced by the rational cohomology ring, and the proofs are the same. The push-forward is described in [P] when G is the general linear group, and in [AC] for the canonical map G/B → G/P where G is arbitrary and B is a Borel subgroup of P . Note that this map is a flag bundle associated with the principal P/R(P )-bundle G/R(P ) → G/P , where R(P ) denotes the radical of P . Our formula for the Todd class seems to be new.

RESIDUE FREE DIFFERENTIALS AND DIFFERENTIALS OF THE SECOND KIND

Let k be a field of characteristic 0, V an n-dimensional non-singular algebraic variety over k,K the function field of V and ΩK/k1 the module of differentials of K over k. A closed differential ω∈ΩK/k1 is called residue free if resW (ω)=0 for any prime divisor W of V and a differential ω is called second kind if for any prime divisor W , there exists an element θW∈K such that νW(ω−dθW)≥0, where νW is the canonical valuation with respect to W. In this paper, we prove the following theorem: Let ω be a closed element of ΩK/k1. Then ω is residue free if and only if ω is of second kind.

The algebraic de Rham theorem for toric varieties

On an arbitrary toric variety, we introduce the logarithmic double complex, which is essentially the same as the algebraic de Rham complex in the nonsingular case, but which behaves much better in the singular case. Over the field of complex numbers, we prove the toric analog of the algebraic de Rham theorem which Grothendieck formulated and proved for general nonsingular algebraic varieties re-interpreting an earlier work of Hodge-Atiyah. Namely, for a finite simplicial fan which need not be complete, the complex cohomology groups of the corresponding toric variety as an analytic space coincide with the hypercohomology groups of the single complex associated to the logarithmic double complex. They can then be described combinatorially as Ishida's cohomology groups for the fan. We also prove vanishing theorems for Ishida's cohomology groups. As a consequence, we deduce directly that the complex cohomology groups vanish in odd degrees for toric varieties which correspond to finite simplicial fans with full-dimensional convex support. In the particular case of complete simplicial fans, we thus have a direct proof for an earlier result of Danilov and the author.

4D fermionic superstrings from thirteen dimensions: σ-model and superconformal properties

In this paper we consider in more detail the general structure of heterotic superstrings compactified on M target = M 4 ⊗ G/B where G is SU(20 3 and B⊂is a suitable discrete subgroup. We classify the nine allowed choices of B by solving the constraints of multi-loop modular invariance in the equivalent fermionic formulation, and we show that G/B is a non-singular algebraic variety. Distinguishing between solitonic and non-solitonic conformal fields, we show that the extended superconformal algebra associated with target SUSY is solitonic and therefore it is not necessary that any complex structure of the target exists. At the quantum level, the conformal field theory on G/B allows a re-interpretation in terms of asymmetric orbifolds. Utilizing harmonic expansion on G/B, we retrieve the non-solitonic massles vertices in the zeroth picture the deformations of the 3D ó-model. Furthermore we discuss the auxiliary field vertices showing how they are embedded in the 13D spin connection. From their equations of motion we obtain information on the scalar manifold of the effective theory; in particular we derive the general structure of the non-solitonic submainfold and its immersion in the complete manifold.

Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure

A variation of polarized Hodge structures on the complement of a divisor with normal crossings gives rise, locally (on the divisor) to a commuting set {Ni} of nilpotent endomorphisms of the vector space underlying the variation: the logarithms of the unipotent parts of the various Picard-Lefschetz transformations. These reflect properties of the singularities of the period map associated to the variation (cf. [5, 7, 9] and Sect. 3). For example, a variation depending on a single parameter defines asymptotically a mixed Hodge structure whose weight filtration is the monodromy weight filtration of the corresponding endomorphism N ([9, 11]). This paper is mainly concerned with the properties of the set {N/} arising in the several parameters case. In particular, Theorem 3.3 asserts that all the elements in the open polyhedral cone C spanned over R by the N,.'s, determine the same monodromy weight filtration. It also describes the relationship between this common filtration and those associated to the faces of the cone. This statement was conjectured by Deligne based on his analogous result ([4], 1.9.2) for the Q-cone associated to those variations which arise from families of polarized, non-singular algebraic varieties. In w 1 we discuss some general properties of commuting sets of nilpotent endomorphisms and their associated filtrations. Section2 focuses on the case when each element in such set defines in the sense of (2.4) a polarized mixed Hodge structure with a fixed Hodge filtration F (as is the case for the monodromy cone C of a variation). These conditions imply (cf. (2.16)) a generic uniqueness for the monodromy weight filtration. We also consider those pola-

On singular sets of flat holomorphic mappings with isolated singularities

In [4] B. Iversen studied critical points of algebraic mappings, using algebraic-geometry methods. In particular when algebraic maps have only isolated singularities, he shows the following relation; Let V and S be compact connected non-singular algebraic varieties of dimcV = n, and dimc S = 1, respectively. Suppose f is an algebraic map of V onto S with isolated singularities. Then it follows thatwhere χ denotes the Euler number, μf(p) is the Milnor number of f at the singular point p, and F is the general fiber of f : V → S.

On the semisimplicity of the algebra associated to a polarized algebraic variety

Let V be a compact nonsingular algebraic variety of dimension n with a Hodge structure ω and let Hi,i(V,C) be the subgroup of 2i-th cohomology group H2i(V,C) represented by harmonic (i,i)-forms on V with respect to ω.

Characteristic invariants of foliated bundles

This paper gives a construction of characteristic invariants of foliated principal bundles in the category of smooth and complex manifolds or non-singular algebraic varieties. It contains a generalization of the Chern-Weil theory requiring no use of global connections. This construction leads for foliated bundles automatically to secondary characteristic invariants. The generalized Weil-homomorphism induces a homomorphism of spectral sequences. On the E1-level this gives rise to further characteristic invariants (derived characteristic classes). The new invariants are geometrically interpreted and examples are discussed.

CHERN CLASSES OF AMPLE BUNDLES

In the article Ample vector bundles, Inst. Hautes Etudes Sci.Publ. Math. no. 29 (1966), 63-94, R. Hartshorne has extended the notion of ample vector bundle to vector bundles of arbitrary rank and has raised the following question. Let be an ample vector bundle over a nonsingular algebraic variety and assume that the rank of is equal to . Is it true that the th Chern class is numerically positive for ? In this paper it is proved that in the case the degree of the point-cycle is positive.Bibliography: 8 items.