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Tohoku Mathematical Journal
2016
December
SECOND SERIES VOL. 68, NO. 4
Tohoku Math. J.
68 (2016), 515558

Title
THE EQUIVARIANT $K$THEORY AND COBORDISM RINGS OF DIVISIVE WEIGHTED PROJECTIVE SPACES
Author
Andrea Cattaneo and Alice Garbagnati
(Received April 9, 2014, revised February 17, 2015) 
Abstract.
We consider CalabiYau 3folds of BorceaVoisin type, i.e. CalabiYau 3folds obtained as crepant resolutions of a quotient $(S\times E)/(\alpha_S\times \alpha_E)$, where $S$ is a K3 surface, $E$ is an elliptic curve, $\alpha_S\in \operatorname{Aut}(S)$ and $\alpha_E\in \operatorname{Aut}(E)$ act on the period of $S$ and $E$ respectively with order $n=2,3,4,6$. The case $n=2$ is very classical, the case $n=3$ was recently studied by Rohde, the other cases are less known. First, we construct explicitly a crepant resolution, $X$, of $(S\times E)/(\alpha_S\times \alpha_E)$ and we compute its Hodge numbers; some pairs of Hodge numbers we found are new. Then, we discuss the presence of maximal automorphisms and of a point with maximal unipotent monodromy for the family of $X$. Finally, we describe the map $\mathcal{E}_n: X \rightarrow S/\alpha_S$ whose generic fiber is isomorphic to $E$.
Mathematics Subject Classification.
Primary 14J32; Secondary 14J28, 14J50.
Key words and phrases.
CalabiYau 3folds, automorphisms, K3 surfaces, elliptic fibrations, isotrivial fibrations.


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