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1. CJM 2009 (vol 61 pp. 1050)
Examples of Calabi--Yau 3-Folds of $\mathbb{P}^{7}$ with $\rho=1$ We give some examples of Calabi--Yau $3$-folds with $\rho=1$ and
$\rho=2$, defined over $\mathbb{Q}$ and constructed as
$4$-codimensional subvarieties of $\mathbb{P}^7$ via commutative
algebra methods. We explain how to deduce their Hodge diamond and
top Chern classes from computer based computations over some
finite field $\mathbb{F}_{p}$. Three of our examples (of degree
$17$ and $20$) are new. The two others (degree $15$ and $18$) are
known, and we recover their well-known invariants with our
method. These examples are built out of Gulliksen--Neg{\aa}rd and
Kustin--Miller complexes of locally free sheaves.
Finally, we give two new examples of Calabi--Yau $3$-folds of
$\mathbb{P}^6$ of degree $14$ and $15$ (defined over
$\mathbb{Q}$). We show that they are not deformation equivalent to
Tonoli's examples of the same degree, despite the fact that they
have the same invariants $(H^3,c_2\cdot H, c_3)$ and $\rho=1$.
Categories:14J32, 14Q15 |