Extensions of Toric Varieties

In this paper, we introduce the notion of "extension" of a toric variety and study its fundamental properties. This gives rise to infinitely many toric varieties with a special property, such as being set theoretic complete intersection or arithmetically Cohen-Macaulay (Gorenstein) and having a Cohen-Macaulay tangent cone or a local ring with non-decreasing Hilbert function, from just one single example with the same property, verifying Rossi's conjecture for larger classes and extending some results appeared in literature.Comment: 8 page

NC Calabi-Yau Manifolds in Toric Varieties with NC Torus fibration

Using the algebraic geometry method of Berenstein and Leigh (BL), hep-th/0009209 and hep-th/0105229), and considering singular toric varieties ${\cal V}_{d+1}$ with NC irrational torus fibration, we construct NC extensions ${\cal M}_{d}^{(nc)}$ of complex d dimension Calabi-Yau (CY) manifolds embedded in ${\cal V}_{d+1}^{(nc)}$. We give realizations of the NC $\mathbf{C}^{\ast r}$ toric group, derive the constraint eqs for NC Calabi-Yau (NCCY) manifolds ${\cal M}^{nc}_d$ embedded in ${\cal V}_{d+1}^{nc}$ and work out solutions for their generators. We study fractional $D$ branes at singularities and show that, due to the complete reducibility property of $\mathbf{C}^{\ast r}$ group representations, there is an infinite number of non compact fractional branes at fixed points of the NC toric group.Comment: 12 pages, LaTex, no figur

Toric Calabi-Yau supermanifolds and mirror symmetry

We study mirror symmetry of supermanifolds constructed as fermionic extensions of compact toric varieties. We mainly discuss the case where the linear sigma A-model contains as many fermionic fields as there are U(1) factors in the gauge group. In the mirror super-Landau-Ginzburg B-model, focus is on the bosonic structure obtained after integrating out all the fermions. Our key observation is that there is a relation between the super-Calabi-Yau conditions of the A-model and quasi-homogeneity of the B-model, and that the degree of the associated superpotential in the B-model is given in terms of the determinant of the fermion charge matrix of the A-model.Comment: 20 pages, v2: references adde

Toric Varieties with NC Toric Actions: NC Type IIA Geometry

Extending the usual $\mathbf{C}^{\ast r}$ actions of toric manifolds by allowing asymmetries between the various $\mathbf{C}^{\ast}$ factors, we build a class of non commutative (NC) toric varieties $\mathcal{V}$. We construct NC complex $d$ dimension Calabi-Yau manifolds embedded in $\mathcal{V}_{d+1}^{(nc)}$ by using the algebraic geometry method. Realizations of NC $\mathbf{C}^{\ast r}$ toric group are given in presence and absence of quantum symmetries and for both cases of discrete or continuous spectrums. We also derive the constraint eqs for NC Calabi-Yau backgrounds $\mathcal{M}_{d}^{nc}$ embedded in $\mathcal{V}_{d+1}^{nc}$ and work out their solutions. The latters depend on the Calabi-Yau condition $\sum_{i}q_{i}^{a}=0$, $q_{i}^{a}$ being the charges of $\mathbf{C}^{\ast r}$% ; but also on the toric data ${q_{i}^{a},\nu_{i}^{A};p_{I}^{\alpha},\nu _{iA}^{\ast}}$ of the polygons associated to $\mathcal{V}$. Moreover, we study fractional $D$ branes at singularities and show that, due to the complete reducibility property of $\mathbf{C}^{\ast r}$ group representations, there is an infinite number of fractional $D$ branes. We also give the generalized Berenstein and Leigh quiver diagrams for discrete and continuous $\mathbf{C}^{\ast r}$ representation spectrums. An illustrating example is presented.Comment: 25 pages, no figure

Horizontal non-vanishing of Heegner points and toric periods

Let $F/\mathbb{Q}$ be a totally real field and $A$ a modular \GL_2-type abelian variety over $F$. Let $K/F$ be a CM quadratic extension. Let $\chi$ be a class group character over $K$ such that the Rankin-Selberg convolution $L(s,A,\chi)$ is self-dual with root number $-1$. We show that the number of class group characters $\chi$ with bounded ramification such that $L'(1, A, \chi) \neq 0$ increases with the absolute value of the discriminant of $K$. We also consider a rather general rank zero situation. Let $\pi$ be a cuspidal cohomological automorphic representation over \GL_{2}(\BA_{F}). Let $\chi$ be a Hecke character over $K$ such that the Rankin-Selberg convolution $L(s,\pi,\chi)$ is self-dual with root number $1$. We show that the number of Hecke characters $\chi$ with fixed $\infty$-type and bounded ramification such that $L(1/2, \pi, \chi) \neq 0$ increases with the absolute value of the discriminant of $K$. The Gross-Zagier formula and the Waldspurger formula relate the question to horizontal non-vanishing of Heegner points and toric periods, respectively. For both situations, the strategy is geometric relying on the Zariski density of CM points on self-products of a quaternionic Shimura variety. The recent result \cite{Ts, YZ, AGHP} on the Andr\'e-Oort conjecture is accordingly fundamental to the approach.Comment: Adv. Math., to appear. arXiv admin note: text overlap with arXiv:1712.0214

Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties

We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In view of the Raynaud parametrisation, this translates into a purely algebraic problem concerning the number of $H$-invariant points on a quotient of $C_n$-lattices $\Lambda/e\Lambda'$ for varying subgroups $H$ of $C_n$ and integers $e$. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions (corresponding to varying $e$) and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on the $p$-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties, that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the form $y^2=f(x)$, under some simplifying hypotheses.Comment: Two new lemmas are added. The first describes permutation representations, and the second describes the dependence of the B-group on the maximal fixpoint-free invariant sublattice. Contact details and bibliographic details have been update