29 research outputs found
Non-liftable Calabi-Yau spaces
We construct many new non-liftable three-dimensional Calabi-Yau spaces in
positive characteristic. The technique relies on lifting a nodal model to a
smooth rigid Calabi-Yau space over some number field as introduced by the first
author and D. van Straten.Comment: 16 pages, 5 tables; v2: minor corrections and addition
Generalized Borcea-Voisin Construction
C. Voisin and C. Borcea have constructed mirror pairs of families of
Calabi-Yau threefolds by taking the quotient of the product of an elliptic
curve with a K3 surface endowed with a non-symplectic involution. In this
paper, we generalize the construction of Borcea and Voisin to any prime order
and build three and four dimensional Calabi-Yau orbifolds. We classify the
topological types that are obtained and show that, in dimension 4, orbifolds
built with an involution admit a crepant resolution and come in topological
mirror pairs. We show that for odd primes, there are generically no minimal
resolutions and the mirror pairing is lost.Comment: 15 pages, 2 figures. v2: typos corrected & references adde
On the monodromy of the moduli space of Calabi-Yau threefolds coming from eight planes in
It is a fundamental problem in geometry to decide which moduli spaces of
polarized algebraic varieties are embedded by their period maps as Zariski open
subsets of locally Hermitian symmetric domains. In the present work we prove
that the moduli space of Calabi-Yau threefolds coming from eight planes in
does {\em not} have this property. We show furthermore that the
monodromy group of a good family is Zariski dense in the corresponding
symplectic group. Moreover, we study a natural sublocus which we call
hyperelliptic locus, over which the variation of Hodge structures is naturally
isomorphic to wedge product of a variation of Hodge structures of weight one.
It turns out the hyperelliptic locus does not extend to a Shimura subvariety of
type III (Siegel space) within the moduli space. Besides general Hodge theory,
representation theory and computational commutative algebra, one of the proofs
depends on a new result on the tensor product decomposition of complex
polarized variations of Hodge structures.Comment: 26 page