A New Cohomology Theory of Orbifold

Maschke's Calabi-Yau three-fold is the double cover of projective three space branched along Maschke's octic surface. This surface is defined by the lowest degree invariant of a certain finite group acting on a four-dimensional (4D) vector space. Using this group, we show that the middle Betti cohomology group of the three-fold decomposes into the direct sum of 150 2D Hodge substructures. We exhibit 1D families of rational curves on the three-fold and verify that the associated Abel-Jacobi map is non-trivial. By counting the number of points over finite fields, we determine the rank of the Néron-Severi group of Maschke's surface and the Galois representation on the transcendental lattice of some of its quotients.We also formulate precise conjectures on the modularity of the Galois representations associated to Maschke's three-fold (these have now been proven by M. Schütt) and to a genus 33 curve, which parametrizes rational curves in the three-fold.The Hodge structure on the middle dimensional Betti cohomology group of a Calabi-Yau (CY) three-fold carries important information on the moduli and the one-dimensional (1D) algebraic cycles on the three-fold. However, if the three-fold is easy to define, say by one equation in a (weighted) projective space, the dimension h 3 of this vector space tends to be large. For example, a smooth quintic three-fold in P 4 has h 3 = 204 and a double octic, i.e., a double cover of P 3 branched along a smooth surface of degree 8, has h 3 = 300. Using automorphisms of the three-folds, one can decompose the cohomology into subrepresentations, which give rise to Hodge substructures. In this paper, we consider a double octic with a particularly large automorphism group G of order 16 ·

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“…Thus the Hodge numbers of Y and Y are the same (see [2]) and we determined them using orbifold cohomology (see [10]):…”

Section: The Heisenberg Quotient Y Of Xmentioning

confidence: 99%

Geometry and arithmetic of Maschke’s Calabi–Yau three-fold

Bini

Geemen

2011

Communications in Number Theory and Physics

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“…, a n )/S 1 . We can compute the orbifold cohomology of X a following Chen and Ruan, see [3]. This is done by taking the direct sum of the homologies of the singular subspaces with appropriate degree shifts.…”

Section: A Generic Contact Form Formentioning

confidence: 99%

Contact homology of Brieskorn manifolds

Koert¹

2008

Forum Mathematicum

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Abstract. We give an algorithm for computing the contact homology of some Brieskorn manifolds. Brieskorn manifolds can be regarded as circle-bundles over orbifolds and our algorithm expresses cylindrical contact homology of the Brieskorn manifold in terms of the homology of the underlying orbifold. As an application, we construct infinitely many contact structures on the class of simply connected contact manifolds that admit nice contact forms (i.e. no Reeb orbits of degree -1, 0 or 1) and have index positivity with trivial first Chern class. IntroductionFor a long time it has been known that contact manifolds may carry many nonisomorphic contact structures. A first way to distinguish these structures from each other is by considering their Chern class or their formal homotopy class. In dimension 3 we can, in addition, sometimes distinguish contact structures by showing tightness or overtwistedness as was shown first by Bennequin. At present, the latter two notions do not have generalizations to higher dimensions.Eliashberg was the first to show that the classical invariants (i.e. the Chern class and formal homotopy class) are not always enough for a full classification of contact structures on a given manifold in dimension larger than 3. He showed that spheres in dimensions 4n + 1 admit a non-standard, yet homotopically trivial contact structure in [4]. Giroux gave an interesting example in dimension 3 in [7]. He exhibited infinitely many tight contact structures on T 3 in the same homotopy class of plane fields. Since then more general techniques to distinguish contact structures were introduced. Eliashberg and Hofer's contact homology is such a technique (for a survey, see [5], and for a more recent description, [6]). It works, roughly speaking, as follows. The Reeb orbits on a given contact manifold can be given a grading. The chain complexes are freely generated by the closed Reeb orbits. Then a differential can be defined by counting certain holomorphic curves asymptotic to Reeb orbits in the symplectization of the contact manifold. It can be shown that the homology of this differential is independent of the contact form under suitable conditions and it is hence an invariant of the contact structure. Since its introduction, several examples of non-isomorphic contact structures in the same homotopy class have been found.One of these examples is a family of Brieskorn spheres in dimensions 4n + 1. These were found by Ustilovsky in [14]. In his thesis he used a few other tools that can easily be applied to construct more contact structures on the same manifold. This is done by connect summing. It is well known that the connected sum of two contact manifolds carries again a contact structure, see [11] and [16]. In his thesis 1991 Mathematics Subject Classification. Primary 53D10 .

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“…Recall that for any almost complex orbifold X, we can associate a Chen-Ruan orbifold cohomology ring H * CR (X, C) [CR1] as the summation of ordinary cohomology over all the sectors with appropriate degree shifting. There are two important factors of this ring.…”

Section: Introductionmentioning

confidence: 99%