Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCalabi--Yau Operators
135
0
0.0
(
0
)
Authors
Duco van Straten
Ask ChatGPT about the research
No Arabic abstract
Motivated by mirror symmetry of one-parameter models, an interesting class of Fuchsian differential operators can be singled out, the so-called Calabi--Yau operators, introduced by Almkvist and Zudilin. They conjecturally determine $Sp(4)$-local systems that underly a $mathbb{Q}$-VHS with Hodge numbers [h^{3 0}=h^{2 1}=h^{1 2}=h^{0 3}=1] and in the best cases they make their appearance as Picard--Fuchs operators of families of Calabi--Yau threefolds with $h^{12}=1$ and encode the numbers of rational curves on a mirror manifold with $h^{11}=1$. We review some of the striking properties of this rich class of operators.
rate research
Read More
Let X be an n-dimensional Calabi-Yau with ordinary double points, where n is odd. Friedman showed that for n=3 the existence of a smoothing of X implies a specific type of relation between homology classes on a resolution of X. (The converse is also true, due to work of Friedman, Kawamata and Tian.) We sketch a more topological proof of this result, and then extend it to higher dimensions. For n>3 the Yukawa product on the middle dimensional (co)homology plays an unexpected role. We also discuss a converse, proving it for nodal Calabi-Yau hypersurfaces in projective space.
Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 5-folds. We find recursions for meeting numbers of genus 0 curves, and we determine the contributions of moving multiple covers of genus 0 curves to the genus 1 Gromov-Witten invariants. The resulting invariants, conjectured to be integral, are analogous to the previously defined BPS counts for Calabi-Yau 3 and 4-folds. We comment on the situation in higher dimensions where new issues arise. Two main examples are considered: the local Calabi-Yau P^2 with balanced normal bundle 3O(-1) and the compact Calabi-Yau hypersurface X_7 in P^6. In the former case, a closed form for our integer invariants has been conjectured by G. Martin. In the latter case, we recover in low degrees the classical enumeration of elliptic curves by Ellingsrud and Stromme.
We generalize the cohomological mirror duality of Borcea and Voisin in any dimension and for any number of factors. Our proof applies to all examples which can be constructed through Berglund-H{u}bsch duality. Our method is a variant of the so-called Landau-Ginzburg/Calabi-Yau correspondence of Calabi-Yau orbifolds with an involution that does not preserve the volume form. We deduce a version of mirror duality for the fixed loci of the involution, which are beyond the Calabi-Yau category and feature hypersurfaces of general type.
We compute numerical approximations of the period integrals for eleven rigid double octic Calabi--Yau threefolds and compare them with the periods of corresponding weight our cusp forms and find, as to be expected, commensurabilities. These give information on character of the correspondences of these varieties with the associated Kuga-Sato modular threefolds.
We prove a structure theorem for the Albanese maps of varieties with Q-linearly trivial log canonical divisors. Our start point is the action of a nonlinear algebraic group on a projective variety.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
