Real Seifert Forms, Hodge Numbers and Blanchfield Pairings

Abstract We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The $l$-dimensional variety $Y_{(n)}^{(l)}$, which is the quotient of the product of a certain curve $C_{(n)}$ by itself $l$ times by a group $G\simeq \left ({\mathbb{Z}}/n{\mathbb{Z}}\right )^{l-1}$ of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If $n=3^c$ Schreieder constructed an explicit smooth birational model of it, and Flapan proved that the Kodaira dimension of this smooth model is 1, if $c>1$; if $l=2$ it is a modular elliptic surface; if $l=3$ it admits a fibration in K3 surfaces. In this paper we generalize these results: without any assumption on $n$ and $l$ we prove that $Y_{(n)}^{(l)}$ admits many elliptic fibrations and its Kodaira dimension is at most 1. Moreover, if $l=2$, its minimal resolution is a modular elliptic surface, obtained by a base change of order $n$ on a specific extremal rational elliptic surface; if $l\geq 3$ it has a birational model that admits a fibration in K3 surfaces and a fibration in $(l-1)$-dimensional varieties of Kodaira dimension at most 0.

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Hodge numbers of Landau–Ginzburg models

Advances in Mathematics ◽

10.1016/j.aim.2020.107436 ◽

2021 ◽

Vol 378 ◽

pp. 107436

Author(s):

Andrew Harder

Keyword(s):

Hodge Numbers

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Topological mirror symmetry for rank two character varieties of surface groups

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg ◽

10.1007/s12188-021-00246-y ◽

2021 ◽

Cited By ~ 1

Author(s):

Mirko Mauri

Keyword(s):

Moduli Spaces ◽

Mirror Symmetry ◽

Surface Groups ◽

Character Varieties ◽

Hitchin System ◽

Global Topology ◽

Hodge Numbers

AbstractThe moduli spaces of flat $${\text{SL}}_2$$ SL 2 - and $${\text{PGL}}_2$$ PGL 2 -connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tamás Hausel in Remark 3.30 of “Global topology of the Hitchin system”.

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Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers

Journal of High Energy Physics ◽

10.1007/jhep02(2019)087 ◽

2019 ◽

Vol 2019 (2) ◽

Cited By ~ 4

Author(s):

Yu-Chien Huang ◽

Washington Taylor

Keyword(s):

Hodge Numbers

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NEW EXAMPLES OF CALABI–YAU 3-FOLDS AND GENUS ZERO SURFACES

Communications in Contemporary Mathematics ◽

10.1142/s0219199713500107 ◽

2014 ◽

Vol 16 (02) ◽

pp. 1350010 ◽

Cited By ~ 5

Author(s):

GILBERTO BINI ◽

FILIPPO F. FAVALE ◽

JORGE NEVES ◽

ROBERTO PIGNATELLI

Keyword(s):

Fundamental Group ◽

Moduli Space ◽

General Type ◽

Picard Number ◽

Surfaces Of General Type ◽

Geometric Genus ◽

Genus Zero ◽

New Family ◽

Hodge Numbers ◽

Projective Lines

We classify the subgroups of the automorphism group of the product of four projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi–Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is nontrivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

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