On a class of non-simply connected Calabi-Yau 3-folds with positive Euler characteristic

Date

2022-01-01

Author

Karayayla, Tolga

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

208
views

0
downloads

In this work we obtain a class of non-simply connected Calabi-Yau 3-folds with positive Euler characteristic as the quotient of projective small resolutions of singular Schoen 3-folds under the free action of finite groups. A Schoen 3-fold is a fiber product X = B-1 x(P1) B-2 of two relatively minimal rational elliptic surfaces with section beta(i) : B-i -> P-1, i = 1, 2. Schoen has shown that if X is smooth, then X is a simply connected Calabi-Yau 3-fold, and if the only singularities of X are on I-r x I-s type fibers with r > 1 and s > 1, then there exists a projective small resolution (X) over cap of X, and (X) over cap is a simply connected Calabi-Yau 3-fold [7]. If G is a finite group which acts freely on a smooth Schoen 3-fold X, then the quotient X/G is a non-simply connected Calabi-Yau 3-fold with fundamental group G, and all such group actions have been classified by Bouchard and Donagi [2]. Bouchard and Donagi have proposed the open problem of classifying all finite groups G which act freely on projective small resolutions (X) over cap of singular Schoen 3-folds X. In this case the quotient (X) over cap /G is again a Calabi-Yau 3-fold with fundamental group G. In this paper we first classify the finite groups G which act freely on singular Schoen 3-folds X where the only singularities of X are on I-r x I-s type fibers with r > 1 and s > 1 and the elements of G act on X as an automorphism tau(1) x tau(2) where each tau(i) is an automorphism of the elliptic surface B-i. A projective small resolution (X) over cap of X is obtained by blowing up some components of the I-r x I-s fibers on X. We determine which of the free actions on the singular 3-fold X lift to free actions on the Calabi-Yau 3-fold (X) over cap. For the non-simply connected Calabi-Yau 3-folds (X) over cap /G obtained with this construction, the distinct fundamental groups are Z(3) x Z(3), Z(4) x Z(2), Z(2) x Z(2), and Z(n) for n = 6, 5, 4, 3, 2. These are the same groups obtained by Bouchard and Donagi by working on free actions on smooth Schoen 3-folds. While the Euler characteristic of each (X) over cap /G obtained by Bouchard and Donagi is 0, the Euler characteristics of all non-simply connected Calabi-Yau 3-folds X/G we obtain in this paper are positive and they range in: 64, 54, 48, 40 and 2k for 2 1 and s > 1, the classification of such group actions which lift to free actions on projective small resolutions (X) over cap of X, and the fundamental groups and Euler characteristic values of the non-simply connected Calabi-Yau 3-folds (C) over cap /G are displayed in several tables. The study of the group actions on X which induce a non-trivial action on the base curve P-1 and which induce a trivial action on P-1 is carried out separately.

Subject Keywords

Calabi-Yau 3-folds, Schoen 3-folds, fiber product of relatively minimal rational elliptic surfaces with section, non-simply connected Calabi-Yau 3-folds, group actionss, automorphisms of rational elliptic surfaces, RATIONAL ELLIPTIC-SURFACES

URI

https://hdl.handle.net/11511/96197

Journal

COMMUNICATIONS IN NUMBER THEORY AND PHYSICS

DOI

https://doi.org/10.4310/CNTP.2022.v16.n1.a-karayayla

Collections

Department of Mathematics, Article

Suggestions

OpenMETU
Core

On the arithmetic operations over finite fields of characteristic three with low complexity AKLEYLEK, SEDAT; Özbudak, Ferruh; Özel, Claire Susanna (2014-03-15) In this paper, the Hermite polynomial representation is adapted as a new way to represent certain finite fields of characteristic three. We give the multiplication method to multiply two elements of F-3n in the Hermite polynomial representation with subquadratic computational complexity by using a divide-and-conquer idea. We show that in some cases there is a set of irreducible binomials in the Hermite polynomial representation to obtain modular reduction with a lower addition complexity than the standard p...
On finite groups admitting a fixed point free abelian operator group whose order is a product of three primes Mut Sağdıçoğlu, Öznur; Ercan, Gülin; Department of Mathematics (2009) A long-standing conjecture states that if A is a finite group acting fixed point freely on a finite solvable group G of order coprime to jAj, then the Fitting length of G is bounded by the length of the longest chain of subgroups of A. If A is nilpotent, it is expected that the conjecture is true without the coprimeness condition. We prove that the conjecture without the coprimeness condition is true when A is a cyclic group whose order is a product of three primes which are coprime to 6 and the Sylow 2-sub...
ON THE IDEAL TRIANGULATION GRAPH OF A PUNCTURED SURFACE Korkmaz, Mustafa (2012-01-01) We study the ideal triangulation graph T(S) of an oriented punctured surface S of finite type. We show that if S is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of S into the simplicial automorphism group of T(S) is an isomorphism. We also show that, the graph T(S) of such a surface S. equipped with its natural simplicial metric is not Gromov hyperbolic. We also show that if the triangulation graph of two oriented punc...
Non-simply connected Calabi-Yau threefolds constructed as quotients of Schoen threefolds Karayayla, Tolga (2017-07-01) The aim of this paper is to complete the classification of all Calabi-Yau threefolds which are constructed as the quotient of a smooth Schoen threefold X = B-1 x p1 B-2 (fiber product over P-1 of two relatively minimal rational elliptic surfaces B-1 and B-2 with section) under a finite group action acting freely on the Schoen threefold X. The abelian group actions on smooth Schoen threefolds which induce cyclic group actions on the base curve P-1 were studied by Bouchard and Donagi (2008), and all such acti...
On a Fitting length conjecture without the coprimeness condition Ercan, Gülin (Springer Science and Business Media LLC, 2012-08-01) Let A be a finite nilpotent group acting fixed point freely by automorphisms on the finite solvable group G. It is conjectured that the Fitting length of G is bounded by the number of primes dividing the order of A, counted with multiplicities. The main result of this paper shows that the conjecture is true in the case where A is cyclic of order p (n) q, for prime numbers p and q coprime to 6 and G has abelian Sylow 2-subgroups.

Citation Formats

T. Karayayla, “On a class of non-simply connected Calabi-Yau 3-folds with positive Euler characteristic,” COMMUNICATIONS IN NUMBER THEORY AND PHYSICS, vol. 16, no. 1, pp. 159–213, 2022, Accessed: 00, 2022. [Online]. Available: https://hdl.handle.net/11511/96197.