Smooth Compactification Research Articles

On the standard conjecture for complex 4-dimensional elliptic varieties and compactifications of Néron minimal models

We prove that the Grothendieck standard conjecture of Lefschetz type on the algebraicity of operators and of Hodge theory holds for every smooth complex projective model of the fibre product , where is an elliptic surface over a smooth projective curve and is a family of K3 surfaces with semistable degenerations of rational type such that for a generic geometric fibre . We also show that holds for any smooth projective compactification of the Néron minimal model of an Abelian scheme of relative dimension 3 over an affine curve provided that the generic scheme fibre is an absolutely simple Abelian variety with reductions of multiplicative type at all infinite places.

Groupe de Brauer non ramifié d’espaces homogènes de tores

Let k be a field, X a smooth, projective k-variety. If X is geometrically rational, there is an injective map from the quotient of Brauer groups Br(X)/Br(k) into the first Galois cohomology group of the lattice given by the geometric Picard group. In this note, where the main attention is on smooth compactifications of homogeneous spaces of algebraic k-tori, we show how under some hypotheses the map is onto, and how one may in some special cases exhibit concrete generators in Br(X). This is applied to the analysis of counterexamples to the local-global principle for norms in biquadratic extensions of number fields.

Gauge threshold corrections for $ \mathcal{N}=2 $ heterotic local models with flux, and mock modular forms

AbstractWe determine threshold corrections to the gauge couplings in local models of$ \mathcal{N}=2 $smooth heterotic compactifications with torsion, given by the direct product of a warped Eguchi-Hanson space and a two-torus, together with a line bundle. Using the worldsheet cft description previously found and by suitably regularising the infinite target space volume divergence, we show that threshold corrections to the various gauge factors are governed by the non-holomorphic completion of the Appell-Lerch sum. While its holomorphic Mock-modular component captures the contribution of states that localise on the blown-up two-cycle, the non-holomorphic correction originates from non-localised bulk states. We infer from this analysis universality properties for$ \mathcal{N}=2 $heterotic local models with flux, based on target space modular invariance and the presence of such non-localised states. We finally determine the explicit dependence of these one-loop gauge threshold corrections on the moduli of the two-torus, and by S-duality we extract the corresponding string-loop and E1-instanton corrections to the Kähler potential and gauge kinetic functions of the dual type i model. In both cases, the presence of non-localised bulk states brings about novel perturbative and non-perturbative corrections, some features of which can be interpreted in the light of analogous corrections to the effective theory in compact models.

Arithmetic of 0-cycles on varieties defined over number fields

Let $X$ be a rationally connected algebraic variety, defined over a number field $k$. We find a relation between the arithmetic of rational points on $X$ and the arithmetic of zero-cycles. More precisely, we consider the following statements: (1) the Brauer-Manin obstruction is the only obstruction to weak approximation for $K$-rational points on $X_K$ for all finite extensions $K/k$; (2) the Brauer-Manin obstruction is the only obstruction to weak approximation in some sense that we define for zero-cycles of degree 1 on $X_K$ for all finite extensions $K/k$; (3) a certain sequence of local-global type for Chow groups of 0-cycles on $X_K$ is exact for all finite extensions $K/k$. We prove that (1) implies (2), and that (2) and (3) are equivalent. We also prove a similar implication for the Hasse principle. As an application, we prove the exactness of the sequence mentioned above for smooth compactifications of certain homogeneous spaces of linear algebraic groups.

Normality and smoothness of simple linear group compactifications

If G is a complex semisimple algebraic group, we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant GxG-compactifications which possess a unique closed orbit and which arise in a projective space of the shape P(End(V)), where V is finite dimensional rational G-module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of V. In particular, we show that Sp(2r) (with r > 0) is the unique non-adjoint simple group which admits a simple smooth compactification.

Finite-volume complex-hyperbolic surfaces, their toroidal compactifications, and geometric applications

We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. Several results concerning the Riemannian and complex algebraic geometry of these spaces are given. In particular we show that there are compact complex surfaces which admit Riemannian metrics of nonpositive curvature, but which do not admit Kahler metrics of nonpositive curvature. An infinite class of such examples arise as smooth toroidal compactifications of ball quotients.

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.

Asymptotic Simplicity and Static Data

The present article considers time symmetric initial data sets for the vacuum Einstein field equations which in a neighbourhood of infinity have the same massless part as that of some static initial data set. It is shown that the solutions to the regular finite initial value problem at spatial infinity for this class of initial data sets extend smoothly through the critical sets where null infinity touches spatial infinity if and only if the initial data sets coincide with static data in a neighbourhood of infinity. This result highlights the special role played by static data among the class of initial data sets for the Einstein field equations whose development gives rise to a spacetime with a smooth conformal compactification at null infinity.

Stabilizing all geometric moduli in heterotic Calabi-Yau vacua

We propose a scenario to stabilize all geometric moduli - that is, the complex structure, Kahler moduli and the dilaton - in smooth heterotic Calabi-Yau compactifications without Neveu-Schwarz three-form flux. This is accomplished using the gauge bundle required in any heterotic compactification, whose perturbative effects on the moduli are combined with non-perturbative corrections. We argue that, for appropriate gauge bundles, all complex structure and a large number of other moduli can be perturbatively stabilized - in the most restrictive case, leaving only one combination of Kahler moduli and the dilaton as a flat direction. At this stage, the remaining moduli space consists of Minkowski vacua. That is, the perturbative superpotential vanishes in the vacuum without the necessity to fine-tune flux. Finally, we incorporate non-perturbative effects such as gaugino condensation and/or instantons. These are strongly constrained by the anomalous U(1) symmetries which arise from the required bundle constructions. We present a specific example, with a consistent choice of non-perturbative effects, where all remaining flat directions are stabilized in an AdS vacuum.

Constant curvature foliations in asymptotically hyperbolic spaces

Let (M,g) be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on \partial M and Weingarten foliations in some neighbourhood of infinity in M . We focus mostly on foliations where each leaf has constant mean curvature, though our results apply equally well to foliations where the leaves have constant \sigma_k -curvature. In particular, we prove the existence of a unique foliation near infinity in any quasi-Fuchsian 3-manifold by surfaces with constant Gauss curvature. There is a subtle interplay between the precise terms in the expansion for g and various properties of the foliation. Unlike other recent works in this area, by Rigger ([The foliation of asymptotically hyperbolic manifolds by surfaces of constant mean curvature (including the evolution equations and estimates). Manuscripta Math. 113 (2004), 403-421]) and Neves-Tian ([Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. Geom. Funct. Anal. 19 (2009), no.3, 910-942], [Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. II. J. Reine Angew. Math. 641 (2010), 69-93]), we work in the context of conformally compact spaces, which are more general than perturbations of the AdS-Schwarzschild space, but we do assume a nondegeneracy condition.

On the rational cohomology of moduli spaces of curves with level structures

We investigate low degree rational cohomology groups of smooth compactifications of moduli spaces of curves with level structures. In particular, we determine \({H^k\left({\bar S}_{g}, {\mathbb Q}\right)}\) for g ≥ 2 and k ≤ 3, where \({{\bar S}_{g}}\) denotes the moduli space of spin curves of genus g.

Remarks on $\aone$-homotopy groups of smooth toric models

We extend previous results on A 1 -homotopy groups of smooth proper toric varieties to the case of smooth proper toric models, i.e., smooth proper equivariant compactifications of possibly non-split tori, in characteristic 0.

Positive sheaves of differentials coming from coarse moduli spaces

Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base Y ∘ , and suppose the family is non-isotrivial. If Y is a smooth compactification of Y ∘ , such that D:=Y∖Y ∘ is a simple normal crossing divisor, then we can consider the sheaf of differentials with logarithmic poles along D. Viehweg and Zuo have shown that for some m>0, the m th symmetric power of this sheaf admits many sections. More precisely, the m th symmetric power contains an invertible sheaf whose Kodaira-Iitaka dimension is at least the variation of the family. We refine this result and show that this “Viehweg-Zuo sheaf” comes from the coarse moduli space associated to the given family, at least generically.

On Isotopies and Homologies of Subvarieties of Toric Varieties

In ℂn we consider an algebraic surface Y and a finite collection of hypersurfaces Si. Froissart’s theorem states that if Y and Si are in general position in the projective compactification of ℂn together with the hyperplane at infinity then for the homologies of Y \∪ Si we have a special decomposition in terms of the homology of Y and all possible intersections of Si in Y. We prove the validity of this homological decomposition on assuming a weaker condition: there exists a smooth toric compactification of ℂn in which Y and Si are in general position with all divisors at infinity. One of the key steps of the proof is the construction of an isotopy in Y leaving invariant all hypersurfaces Y ∩ Sk with the exception of one Y ∩ Si, which is shifted away from a given compact set. Moreover, we consider a purely toric version of the decomposition theorem, taking instead of an affine surface Y the complement of a surface in a compact toric variety to a collection of hypersurfaces in it.

Cut-by-curves criterion for the log extendability of overconvergent isocrystals

In this paper, we prove a ‘cut-by-curves criterion’ for an overconvergent isocrystal on a smooth variety over a field of characteristic p > 0 to extend logarithmically to its smooth compactification whose complement is a simple normal crossing divisor, under certain assumption. This is a p-adic analogue of a version of cut-by-curves criterion for regular singularity of an integrable connection on a smooth variety over a field of characteristic 0. In the course of the proof, we also prove a kind of cut-by-curves criteria on solvability, highest ramification break and exponent of ∇-modules.

Yukawa Couplings in Heterotic Compactification

We present a practical, algebraic method for efficiently calculating the Yukawa couplings of a large class of heterotic compactifications on Calabi-Yau three-folds with non-standard embeddings. Our methodology covers all of, though is not restricted to, the recently classified positive monads over favourable complete intersection Calabi-Yau three-folds. Since the algorithm is based on manipulating polynomials it can be easily implemented on a computer. This makes the automated investigation of Yukawa couplings for large classes of smooth heterotic compactifications a viable possibility.

Cubic polynomial maps with periodic critical orbit, Part II: Escape regions

The parameter space S p \mathcal {S}_p for monic centered cubic polynomial maps with a marked critical point of period p p is a smooth affine algebraic curve whose genus increases rapidly with p p . Each S p \mathcal {S}_p consists of a compact connectedness locus together with finitely many escape regions, each of which is biholomorphic to a punctured disk and is characterized by an essentially unique Puiseux series. This note will describe the topology of S p \mathcal {S}_p , and of its smooth compactification, in terms of these escape regions. In particular, it computes the Euler characteristic. It concludes with a discussion of the real sub-locus of S p \mathcal {S}_p .

On logarithmic extension of overconvergent isocrystals

In this paper, we establish a criterion for an overconvergent isocrystal on a smooth variety over a field of characteristic p > 0 to extend logarithmically to its smooth compactification whose complement is a simple normal crossing divisor. This is a generalization of a result of Kedlaya, who treated the case of unipotent monodromy. Our result is regarded as a p-adic analogue of the theory of canonical extension of regular singular integrable connections on smooth varieties of characteristic 0.

The homology of real subspace arrangements

Associated to any subspace arrangement is a "De Concini-Procesi model", a certain smooth compactification of its complement, which in the case of the braid arrangement produces the Deligne-Mumford compactification of the moduli space of genus 0 curves with marked points. In the present work, we calculate the integral homology of real De Concini-Procesi models, extending earlier work of Etingof, Henriques, Kamnitzer and the author on the (2-adic) integral cohomology of the real locus of the moduli space. To be precise, we show that the integral homology of a real De Concini-Procesi model is isomorphic modulo its 2-torsion with a sum of cohomology groups of subposets of the intersection lattice of the arrangement. As part of the proof, we construct a large family of natural maps between De Concini-Procesi models (generalizing the operad structure of moduli space), and determine the induced action on poset cohomology. In particular, this determines the ring structure of the cohomology of De Concini-Procesi models (modulo 2-torsion).

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]