Abelian Fibrations Research Articles

Mathematical foundation of foliation-based quantization

We have recently proposed in \cite{Park:2014tia} the quantization of pure 4D Einstein gravity through hypersurface foliation, and observed that the 4D Einstein gravity becomes renormalizable once all (or most) of the unphysical degrees of freedom are removed. In this work, we confirm this observation from a more mathematical angle. In particular, we show that the physical state condition arising from the shift vector constraint connects with the requirement that the manifold admit "totally geodesic (TG) foliation". The TG foliation, in turn, makes it possible to view the 4D manifold as abelian fibration over a 3D base. Associating the abelian fibration with the 4D diffeomorphism leads to reduction of the 4D manifold to 3D, thereby realizing and generalizing the holography of 't Hooft.

A support theorem for the Hitchin fibration: the case of

We prove that the direct image complex for the $D$-twisted $\text{SL}_{n}$ Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for $\text{GL}_{n}$ is due to Chaudouard and Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove $\unicode[STIX]{x1D6FF}$-regularity results for some auxiliary weak abelian fibrations.

Torsion sections of Abelian fibrations

Torsion sections of Abelian fibrations

Collapsing of abelian fibered Calabi–Yau manifolds

We study the collapsing behavior of Ricci-flat Kähler metrics on a projective Calabi–Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov–Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkähler manifolds in the large complex structure limit. In this setting, we prove an analogue of a result of Gross and Wilson for K3 surfaces, which is motivated by the Strominger–Yau–Zaslow picture of mirror symmetry.

Abelian fibrations and SYZ mirror conjecture

SYZ mirror conjecture predicts that a Calabi–Yau manifold X consists of a family of tori which are dual to a family of special Lagrangian tori on the mirror dual manifold Xˆ. Here we consider a fibration of polarized abelian varieties and we construct a dual one. Moreover we prove that they are equivalent at the level of derived categories.

Abelian fibrations, string junctions, and flux/geometry duality

In previous work, it was argued that the type IIB T^6/Z_2 orientifold with a choice of flux preserving N=2 supersymmetry is dual to a class of purely geometric type IIA compactifications on abelian surface (T^4) fibered Calabi-Yau threefolds. We provide two explicit constructions of the resulting Calabi-Yau duals. The first is a monodromy based description, analogous to F-theory encoding of Calabi-Yau geometry via 7-branes and string junctions, except for T^4 rather than T^2 fibers. The second is an explicit algebro-geometric construction in which the T^4 fibers arise as the Jacobian tori of a family of genus-2 curves. This improved description of the duality map will be a useful tool to extend our understanding of warped compactifications. We sketch applications to related work to define warped Kaluza-Klein reduction in toroidal orientifolds, and to check the modified rules for D-brane instanton zero mode counting due to the presence of flux and other D-branes. The nontrivial fundamental groups of the Calabi-Yau manifolds constructed also have potential applications to heterotic model building.

Survey on the fundamental lemma

This is a survey on the recent proof of the fundamental lemma. The fundamental lemma and the related transfer conjecture were formulated by R. Langlands in the context of endoscopy theory for automorphic representations in [26]. Important arithmetic applications follow from the endoscopy theory, including the transfer of automorphic representations from classical groups to linear groups and the construction of Galois representations attached to automorphic forms via Shimura varieties. Independent of applications, endoscopy theory is instrumental in building a stable trace formula that seems necessary to any decisive progress toward Langlands’ conjecture on functoriality of automorphic representations. There are already several expository texts on endoscopy theory and in particular on the fundamental lemma. The original text [26] and articles of Kottwitz [19], [20] are always the best places to learn the theory. The two introductory articles to endoscopy, one by Labesse [24], the other [14] written by Harris for the Book project are highly recommended. So are the reports on the proof of the fundamental lemma in the unitary case written by Dat for Bourbaki [7] and in general written by Dat and Ngo Dac for the Book project [8]. I have also written three expository notes on Hitchin fibration and the fundamental lemma : [Fibration de Hitchin et structure endoscopique de la formule des traces, International Congress of Mathematicians. Vol. II, 1213–1225, Eur. Math. Soc., Zurich, 2006] reports on endoscopic structure of the cohomology of the Hitchin fibration, [Vietnamese congress of mathematicians (2008)] is a more gentle introduction to the fundamental lemma, and [Decomposition theorem and abelian fibration] reports on the support theorem, a key point in the proof of the fundamental lemma written for the Book project. The survey follows the same plan as [Vietnamese congress of mathematicians (2008)] but more details have been added. This report is written when its author enjoyed the hospitality of the Institute for Advanced Study in Princeton. He acknowledges the support of the Simonyi foundation and the Monell Foundation during his stay in the Institute.

Mordell–Weil Groups of a Hyperkähler a Manifold—A Question of F. Campana

Among other things, we show that Mordell–Weil groups of finitely many different abelian fibrations of a hyperkähler manifold have essentially no relation, as its birational transformation. Precise definition of the terms “essentially no relation” will be given in Introduction.

Stability conditions and Calabi–Yau fibrations

In this paper, we describe the spaces of stability conditions for the triangulated categories associated to three-dimensional Calabi–Yau fibrations. We deal with two cases, the flat elliptic fibrations and smooth K 3 K3 (Abelian) fibrations. In the first case, we will see that there exist chamber structures similar to those of the movable cone used in birational geometry. In the second case, we will compare the space with the space of stability conditions for the closed fiber of the fibration.

THE ABELIAN FIBRATION ON THE HILBERT CUBE OF A K3 SURFACE OF GENUS 9

In this paper we construct an abelian fibration over P3on the Hilbert cube of the primitive K3 surface of genus 9. After the abelian fibration constructed by Hassett and Tschinkel on the Hilbert square on the primitive K3 surface of genus 5, this is the second example where the abelian fibration is constructed directly on [Formula: see text]. The recent more general result of Sawon proves the existence of an abelian fibration on the Hilbert scheme [Formula: see text] of a primitive K3 surface S of degree 2g - 2 = m2(2n - 2). Our example provides an alternative proof in the case m = 2, n = 3. Furthermore we identify the general fiber with the Hilbert scheme of twisted cubic curves in a Fano 3-fold of genus 9, and interpret the addition law on this variety.

Abelian fibrations on S[ n

Let S → φ P 1 be an elliptic fibration on a K3 surface S. Then the composition S [n] → π S (n) → sym nφ P n gives an Abelian fibration on S [ n] . Let E be the exceptional divisor of π, then sym n φ∘ π( E) is of dimension n−1. We prove the inverse in this Note. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

A canonical bundle formula for certain algebraic fiber spaces and its applications

AbstractWe investigate period maps of polarized variations of Hodge structures of weight one or two. We treat the case when the period domains are bounded symmetric domains. We deal with a relationship between canonical extensions of some Hodge bundles and automorphic forms. As applications, we obtain a canonical bundle formula for certain algebraic fiber spaces, such as Abelian fibrations, K3 fibrations, and solve Iitaka’s famous conjecture Cn,m for some algebraic fiber spaces.

Star products of Green's currents and automorphic forms

In previous work, the authors computed archimedian heights of hermitian line bundles on families of polarized, n-dimensional abelian varieties. In this paper, a detailed analysis of the results obtained in the setting of abelian fibrations is given, and it is shown that the proofs can be modified in such a way that they no longer depend on the specific setting of abelian fibrations and hence extend to a quite general situation. Specifically, we let f : X→Y be any family of smooth, projective, n-dimensional complex varieties over some base, and consider a line bundle on X equipped with a smooth, hermitian metric. To this data is associated a hermitian line bundle M on Y characterized by conditions on the first Chern class. Under mild additional hypotheses, it is shown that, for generically chosen sections of L, the integral of the (n+1)-fold star product of Green's currents associated to the sections, integrated along the fibers of f, is the log-norm of a global section of M. Furthermore, it is proven that in certain general settings the global section of M can be explicitly expressed in terms of point evaluations of the original sections. A particularly interesting example of this general result appears in the setting of polarized Enriques surfaces when M is a moduli space of degree-2 polarizations. In this setting, the global section constructed via Green's currents is equal to a power of the Φ-function first studied by R. Borcherds. Additional examples and problems are presented.

ABELIAN FIBRATIONS AND RATIONAL POINTS ON SYMMETRIC PRODUCTS

Given a variety over a number field, are its rational points potentially dense, i.e. does there exist a finite extension over which rational points are Zariski dense? We study the question of potential density for symmetric products of surfaces. Contrary to the situation for curves, rational points are not necessarily potentially dense on a sufficiently high symmetric product. Our main result is that rational points are potentially dense for the Nth symmetric product of a K3 surface, where N is explicitly determined by the geometry of the surface. The basic construction is that for some N, the Nth symmetric power of a K3 surface is birational to an Abelian fibration over ℙN. It is an interesting geometric problem to find the smallest N with this property.

Theta line bundles and the determinant of the Hodge bundle

We give an expression of the determinant of the push forward of a symmetric line bundle on a complex abelian fibration, in terms of the pull back of the relative dualizing sheaf via the zero section.