Searcharxiv⌕ Search

arXiv subjects

Hülya Argüz

Publications and source records attributed to Hülya Argüz.

24 records · Page 2Linked to original sources
arXiv2021

Real loci in (log-) Calabi-Yau manifolds via Kato-Nakayama spaces of toric degenerations

We study the real loci of toric degenerations of complex varieties with reducible central fibre. We show that the topology of such degenerations can be explicitly described via the Kato-Nakayama space of the central fibre as a log space. We furthermore provide generalities of real structures in log geometry and their lift to Kato-Nakayama spaces. A key point of this paper is a description of the Kato-Nakayama space of a toric degeneration and its real locus, both as bundles determined by combinatorial data. We provide several examples including real toric degenerations of K3-surfaces and a toric degeneration of local $\textbb{P}^2$.

math.AG↗
arXiv2020

On the cohomology groups of real Lagrangians in Calabi-Yau threefolds

The quintic threefold $X$ is the most studied Calabi-Yau $3$-fold in the mathematics literature. In this paper, using Čech-to-derived spectral sequences, we investigate the mod $2$ and integral cohomology groups of a real Lagrangian $\breve{L}_{\mathbb{R}}$, obtained as the fixed locus of an anti-symplectic involution in the mirror to $X$. We show that $\breve{L}_{\mathbb{R}}$ is the disjoint union of a $3$-sphere and a rational homology sphere. Analysing the mod $2$ cohomology further, we deduce a correspondence between the mod $2$ Betti numbers of $\breve{L}_{\mathbb{R}}$ and certain counts of integral points on the base of a singular torus fibration on $X$. By work of Batyrev, this identifies the mod $2$ Betti numbers of $\breve{L}_{\mathbb{R}}$ with certain Hodge numbers of $X$. Furthermore, we show that the integral cohomology groups $H^j(\breve{L}_{\mathbb{R}},\mathbb{Z})$ of $\breve{L}_{\mathbb{R}}$ are $2$-primary for $j \neq 0,3$; we conjecture that this holds in much greater generality.

math.AG↗
arXiv2019

Real Lagrangians in Calabi-Yau Threefolds

We compute the mod $2$ cohomology groups of real Lagrangians in Calabi-Yau threefolds using well-behaved torus fibrations constructed by Gross. To do this we study a long exact sequence introduced by Castaño-Bernard and Matessi, which relates the cohomology of the Lagrangians to the cohomology of the Calabi-Yau. We show that the connecting homomorphism in this sequence is given by the map squaring divisor classes in the mirror Calabi-Yau.

math.AG↗
arXiv2013

Self-Dual metrics on non-simply connected 4-manifolds

We construct self-dual(SD) but not locally conformally flat(LCF) metrics on families of non-simply connected 4-manifolds with small signature. We construct various sequences with bounded or unbounded Betti numbers and Euler characteristic. These metrics have negative scalar curvature. As an application, this addresses the Remark 4.79 of Besse.

math.DG↗