From integer polynomials to complex and hypercomplex manifolds

Abstract: We say that a polynomial $p\in\mathbb{Z}[x]$ belongs to $\Delta_n$ if it is monic of degree $n$, all its roots are positive and distinct, and the product of its roots is 1. For $p\in \Delta_n$, we show how to construct a $(2n+2)$-dimensional solvmanifold equipped with an invariant complex structure, and a $(4n+4)$-dimensional solvmanifold equipped with an invariant hypercomplex structure. These solvmanifolds are completely solvable, therefore their de Rham cohomology can be computed using invariant differential forms. Imposing certain restrictions on $p$ (namely, the "full rank" or "quasi-full rank" conditions), we obtain explicit expressions for their Betti numbers. In the complex case, we show that these solvmanifolds are generalized Nakamura manifolds (as defined by Cattaneo and Tomassini) and, under the same conditions on $p$ plus an extra condition on the lattice, we determine their Hodge numbers.

This talk is based on joint work with María Laura Barberis and Valentina Chaves (Córdoba).

differential geometry

Audience: researchers in the topic

( paper )

---

Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Differential geometry.

The seminar meets every other Wednesday. To accommodate most time zones, the time rotates. The Zoom link is sent to the mailing list around 24 hours before each talk. To subscribe to the mailing list, fill the following form: docs.google.com/forms/d/e/1FAIpQLSdKrJ-nivgjr7ZVJmIY0qkN-VbzTl5NHHNyg6nNsCqjhB-4WA/viewform?usp=sf_link.

---