# On a Lefschetz-type phenomenon for elliptic Calabi--Yaus

## TL;DR

This paper investigates a Lefschetz-type phenomenon in elliptic Calabi--Yaus, showing that their Hodge structures match those of embedded projective bundles, suggesting a broader underlying principle.

## Contribution

It demonstrates a Lefschetz-type phenomenon for 18 families of elliptic Calabi--Yaus, revealing unexpected Hodge structure coincidences beyond classical hypotheses.

## Key findings

- Hodge diamonds of crepant resolutions match those of embedded projective bundles
- Results hold despite crepant resolutions not satisfying Lefschetz hyperplane theorem hypotheses
- Suggests all elliptic Calabi--Yaus may exhibit this Lefschetz-type phenomenon

## Abstract

We consider 18 families of elliptic Calabi--Yaus which arise in constructing $F$-theory compactifications of string vacua, and show in each case that the upper Hodge diamond of a crepant resolution of the associated Weierstrass model coincides with the upper Hodge diamond of the (blown up) projective bundle in which the crepant resolution is naturally embedded. Such results are unexpected, as each crepant resolution we consider does \emph{not} satisfy the hypotheses of the Lefschetz hyperplane theorem. In light of such findings, we suspect that all ellipitic Calabi--Yaus satisfy such a `Lefschetz-type phenomenon'.