Hodge numbers and deformations of Fano 3-folds. (English) Zbl 1505.14089
Summary: We show that index 1 Fano 3-folds which lie in weighted Grassmannians in their total anticanonical embedding have finite automorphism group, and we relate the deformation theory of any Fano 3-fold that has a \(K3\) elephant to its Hodge theory. Combining these results with standard Gorenstein projection techniques calculates both the number of deformations and the Hodge numbers of most quasi-smooth Fano 3-folds in low codimension. This provides detailed new information for hundreds of families of Fano 3-folds.
MSC:
| 14J30 | \(3\)-folds |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
| 14E30 | Minimal model program (Mori theory, extremal rays) |
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