Elliptic Calabi-Yau threefolds over a del Pezzo surface (original) (raw)

Modular Calabi–Yau Threefolds of Level Eight

International Journal of Mathematics, 2007

In the studies on the modularity conjecture for rigid Calabi–Yau threefolds several examples with the unique level 8 cusp form were constructed. According to the Tate conjecture, correspondences inducing isomorphisms on the middle cohomologies should exist between these varieties. In the paper, we construct several examples of such correspondences. In the constructions elliptic fibrations play a crucial role. In fact we show that all but three examples are in some sense built upon two modular curves from the Beauville list.

3 New Examples of Calabi–Yau Threefolds and Genus Zero Surfaces

2016

We classify the subgroups of the automorphism group of the product of 4 projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is non-trivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K 2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds

Contemporary Mathematics, 2006

Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds are compared. In certain situations, the Donaldson-Thomas invariants are very easy to handle, sometimes easier than the other invariants. This point is illustrated in several ways, especially by revisiting computations of Gopakumar-Vafa invariants by Katz, Klemm, and Vafa in a rigorous mathematical framework. This note is based on my talk at the 2004 Snowbird Conference on String Geometry.

New examples of Calabi-Yau threefolds and genus zero surfaces

2012

We classify the subgroups of the automorphism group of the product of 4 projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is non-trivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K^2=3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

New realizations of modular forms in Calabi-Yau threefolds arising from ϕ 4 theory

Journal of Number Theory, 2017

Brown and Schnetz found that the number of points over Fp of a graph hypersurface is often related to the coefficients of a modular form. We set some of the reduction techniques used to discover such relation in a general geometric context. We also prove the relation for one example of a modular form of weight 4 and two of weight 3, refine the statement and suggest a method of proving it for four more of weight 4, and use the one proved example to construct two new rigid Calabi-Yau threefolds that realize Hecke eigenforms of weight 4 (one provably and one conjecturally).

On the global moduli of Calabi-Yau threefolds

arXiv (Cornell University), 2017

In this note we initiate a program to obtain global descriptions of Calabi-Yau moduli spaces, to calculate their Picard group, and to identify within that group the Hodge line bundle. We do this here for several Calabi-Yau's obtained in [DW09] as crepant resolutions of the orbifold quotient of the product of three elliptic curves. In particular we verify in these cases a recent claim of [GHKSST16] by noting that a power of the Hodge line bundle is trivial-even though in most of these cases the Picard group is infinite.

New Examples of Calabi–Yau 3-FOLDS and Genus Zero Surfaces

Communications in Contemporary Mathematics, 2014

We classify the subgroups of the automorphism group of the product of 4 projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is non-trivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K 2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

The Euler number of certain primitive Calabi–Yau threefolds

Mathematical Proceedings of the Cambridge Philosophical Society, 2000

Recently Calabi–Yau threefolds have been studied intensively by physicists and mathematicians. They are used as physical models of superstring theory [Y] and they are one of the building blocks in the classification of complex threefolds [KMM]. These are three dimensional analogues of K3 surfaces. However, there is a fundamental difference as is to be expected. For K3 surfaces, the moduli space N of K3 surfaces is irreducible of dimension 20, inside which a countable number of families Ng with g [ges ] 2 of algebraic K3 surfaces of dimension 19 lie as a dense subset. More explicitly, an element in Ng is (S, H), where S is a K3 surface and H is a primitive ample divisor on S with H2 = 2g − 2. For a generic (S, H), Pic (S) is generated by H, so that the rank of the Picard group of S is 1. A generic surface S in N is not algebraic and it has Pic (S) = 0, but dim N = h1(S, TS) = 20 [BPV]. It is quite an interesting problem whether or not the moduli space M of all Calabi–Yau threefolds i...