Modular Invariants for Lattice Polarized K3 Surfaces

We study the class of complex algebraic K3 surfaces admitting an embedding of H+E8+E8 inside the Neron-Severi lattice. These special K3 surfaces are classified by a pair of modular invariants, in the same manner that elliptic curves over the field of complex numbers are classified by the J-invariant. Via the canonical Shioda-Inose structure we construct a geometric correspondence relating K3 surfaces of the above type with abelian surfaces realized as cartesian products of two elliptic curves. We then use this correspondence to determine explicit formulas for the modular invariants.Comment: 29 pages, LaTe

Lattice Polarized K3 Surfaces and Siegel Modular Forms

The goal of the present paper is two-fold. First, we present a classification of algebraic K3 surfaces polarized by the lattice H+E_8+E_7. Key ingredients for this classification are: a normal form for these lattice polarized K3 surfaces, a coarse moduli space and an explicit description of the inverse period map in terms of Siegel modular forms. Second, we give explicit formulas for a Hodge correspondence that relates these K3 surfaces to principally polarized abelian surfaces. The Hodge correspondence in question underlies a geometric two-isogeny of K3 surfaces

Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds

We investigate a potential relationship between mirror symmetry for Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and Landau-Ginzburg models. More precisely, we show that if a Calabi-Yau admits a so-called Tyurin degeneration to a union of two Fano varieties, then one should be able to construct a mirror to that Calabi-Yau by gluing together the Landau-Ginzburg models of those two Fano varieties. We provide evidence for this correspondence in a number of different settings, including Batyrev-Borisov mirror symmetry for K3 surfaces and Calabi-Yau threefolds, Dolgachev-Nikulin mirror symmetry for K3 surfaces, and an explicit family of threefolds that are not realized as complete intersections in toric varieties.Comment: v2: Section 5 has been completely rewritten to accommodate results removed from Section 5 of arxiv:1501.04019. v3: Final version, to appear in String-Math 2015, forthcoming volume in the Proceedings of Symposia in Pure Mathematics serie

Les relations canado-américaines dans une ère d'incertitude

The following treatment of U.S.-Canada relations begins with the Ottawa-Quebec nexus and its impact upon the connection with the United States. Then the analysis proceeds through bilateral relations. The essay concludes with a look at multilateral interactions from the focus of both Canada and the United States. Thus the analysis proceeds from the most specific to the most general, and from the most internalized to the most external. Concluding with a paradox, the argument of the essay is that despite the end of the Cold War and the disappearance of imminent external threat, uncertainty has never loomed larger in the relation of Canada to its southern neighbor, for all parts of Canada including Quebec, and for the Canadian polity as a whole