On Real Structures of Rigid Surfaces

V. Kharlamov; Vik. S. Kulikov

Home/Archive/arXiv

arXiv · math/0101098

On Real Structures of Rigid Surfaces

V. Kharlamov·Vik. S. Kulikov

View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct several rigid (i.e., unique in their deformation class) surfaces which have particular behavior with respect to real structures: in one example the surface has no any real structure, in the other one it has a unique real structure and this structure is not maximal with respect to the Smith-Thom inequality. So, it answers in negative to the following problems: existence of real surfaces in each complex deformation class and existence of maximal surfaces in each complex deformation class containing real surfaces. Besides, we prove that there is no real surfaces among the surfaces of general type with $p_g=q=0$ and $K^2=9$. As a by-product, the surfaces constructed give one more counterexample to "Dif=Def" problem.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

V. Kharlamov, Vik. S. Kulikov. 2001-06-07. On Real Structures of Rigid Surfaces. https://arxiv.org/abs/math/0101098

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Good minimal models with nowhere vanishing holomorphic $1$-forms

Popa and Schnell show that any holomorphic 1-form on a smooth projective variety of general type has zeros. In this article, we show that a smooth good minimal model has a holomorphic 1-form without zero if and only if it admits an analytic fiber bundle structure over a positive dimensional abelian variety.

math.AG↗

arXiv2026

The automorphism groups of Kummer surfaces in characteristic two and their complex analogues

We calculate the automorphism group of the Kummer surface associated with a curve of genus 2 or the product of two elliptic curves in characteristic two under the assumption that the Kummer surface is a $K3$ surface. Moreover we discuss the complex $K3$ surfaces with the same Picard lattice as these Kummer surfaces. The paper has two appendices.

math.AG↗

arXiv2026

Algebraic Zhou valuations

In this paper, we generalize Zhou valuations, originally defined on complex domains, to the framework of general schemes. We demonstrate that an algebraic version of the Jonsson--Mustaţă conjecture is equivalent to the statement that every Zhou valuation is quasi-monomial. By introducing a mixed version of jumping numbers and Tian functions associated with valuations, we obtain characterizations of a valuation being a Zhou valuation or computing some jumping number using the Tian functions. Furthermore, we establish the correspondence between Zhou valuations in algebraic settings and their counterparts in analytic settings.

math.AG↗