arXiv subjects
Publications and source records attributed to Arthur Renaudineau.
Continuing the investigation of real Calabi-Yau hypersurfaces in toric varieties obtained by patchworking, we present a new theorem concerning the computation of their first Betti number using mirror symmetry. Although the proof of this result will appear elsewhere, we focus here on its consequences and applications to the topology of real Calabi-Yau hypersurfaces.
In the first part of the paper, we prove a mirror symmetry isomorphism between integral tropical homology groups of a pair of mirror tropical Calabi-Yau hypersurfaces. We then apply this isomorphism to prove that a primitive patchworking of a central triangulation of a reflexive polytope gives a connected real Calabi-Yau hypersurface if and only if the corresponding divisor class on the mirror is not zero.
This paper generalises the homeomorphism theorem behind Viro's combinatorial patchworking of hypersurfaces in toric varieties to arbitrary codimension using tropical geometry. We first define the patchwork of a polyhedral space equipped with a real phase structure. When the polyhedral subspace is tropically non-singular, we show that the patchwork is a topological manifold. When a non-singular tropical variety appears as a tropical limit of a real analytic family, we show that the real part of a fibre of the family near the tropical limit is homeomorphic to the patchwork. Finally we extend the spectral sequence introduced by the last two authors in the case of hypersurfaces to non-singular tropical varieties with real phase structures. As a corollary, we obtain bounds on the Betti numbers of the patchwork in terms of the dimensions of the tropical homology groups with coefficients modulo two.
We introduce the notion of real phase structure on rational polyhedral fans in Euclidean space. Such a structure consists of an assignment of affine spaces over $\mathbb{Z}/2\mathbb{Z}$ to each top dimensional face of the fan subject to two conditions. Given an oriented matroid we can construct a real phase structure on the fan of the underlying matroid. Conversely, we show that from a real phase structure on a matroid fan we can produce an orientation of the underlying matroid. Thus real phase structures are cryptomorphic to matroid orientations. The topes of the orientated matroid are recovered immediately from the real phase structure. We also provide a direct way to recover the signed circuits of the oriented matroid from the real phase structure.
We establish a patchworking theorem à la Viro for the Log-critical locus of algebraic curves in $(\mathbb{C}^*)^2$. As an application, we prove the existence of projective curves of arbitrary degree with smooth connected Log-critical locus. To prove our patchworking theorem, we study the behaviour of Log-inflection points along families of curves defined by Viro polynomials. In particular, we prove a generalisation of a theorem of Mikhalkin and the second author on the tropical limit of Log-inflection points.
We prove a bound conjectured by Itenberg on the Betti numbers of real algebraic hypersurfaces near non-singular tropical limits. These bounds are given in terms of the Hodge numbers of the complexification. To prove the conjecture we introduce a real variant of tropical homology and define a filtration on the corresponding chain complex inspired by Kalinin's filtration. The spectral sequence associated to this filtration converges to the homology groups of the real algebraic variety and we show that the terms of the first page are tropical homology groups with $\mathbb{Z}_2$-coefficients. The dimensions of these homology groups correspond to the Hodge numbers of complex projective hypersurfaces. The bounds on the Betti numbers of the real part follow, as well as a criterion to obtain a maximal variety. We also generalise a known formula relating the signature of the complex hypersurface and the Euler characteristic of the real algebraic hypersurface, as well as Haas' combinatorial criterion for the maximality of plane curves near the tropical limit.
We establish variants of the Lefschetz hyperplane section theorem for the integral tropical homology groups of tropical hypersurfaces of toric varieties. It follows from these theorems that the integral tropical homology groups of non-singular tropical hypersurfaces which are compact or contained in $\mathbb{R}^n$ are torsion free. We prove a relationship between the coefficients of the $χ_y$ genera of complex hypersurfaces in toric varieties and Euler characteristics of the integral tropical cellular chain complexes of their tropical counterparts. It follows that the integral tropical homology groups give the Hodge numbers of compact non-singular hypersurfaces of complex toric varieties. Finally for tropical hypersurfaces in certain affine toric varieties, we relate the ranks of their tropical homology groups to the Hodge-Deligne numbers of their complex counterparts.
The paper describes behavior of log-inflection points of curves in $(\mathbb{C}^*)^2$ under passing to the tropical limit. We show that such points accumulate by pairs at the midpoints of bounded edges in the limiting tropical curve. Log-inflection points are points of inflection with respect to the parallelization of $(\mathbb{C}^*)^2$ given by the multiplicative group law.
Haas' theorem describes all partchworkings of a given non-singular plane tropical curve $C$ giving rise to a maximal real algebraic curve. The space of such patchworkings is naturally a linear subspace $W_C$ of the $\mathbb{Z}/2\mathbb{Z}$-vector space $\overrightarrow Π_C$ generated by the bounded edges of $C$, and whose origin is the Harnack patchworking. The aim of this note is to provide an interpretation of affine subspaces of $\overrightarrow Π_C $ parallel to $W_C$. To this purpose, we work in the setting of abstract graphs rather than plane tropical curves. We introduce a topological surface $S_Γ$ above a trivalent graph $Γ$, and consider a suitable affine space $Π_Γ$ of real structures on $S_Γ$ compatible with $Γ$. We characterise $W_Γ$ as the vector subspace of $\overrightarrow Π_Γ$ whose associated involutions induce the same action on $H_1(S_Γ,\mathbb{Z}/2\mathbb{Z})$. We then deduce from this statement another proof of Haas' original result.
In this text, we study Viro's conjecture and related problems for real algebraic surfaces in $(\mathbb{CP}^1)^3$. We construct a counter-example to Viro's conjecture in tridegree $(4,4,2)$ and a family of real algebraic surfaces of tridegree $(2k,2l,2)$ in $(\mathbb{CP}^1)^3$ with asymptotically maximal first Betti number of the real part. To perform such constructions, we consider double covers of blow-ups of $(\mathbb{CP}^1)^2$ and we glue singular curves with special position of the singularities adapting the proof of Shustin's theorem for gluing singular hypersurfaces.
We give a constructive proof using tropical modifications of the existence of a family of real algebraic plane curves with asymptotically maximal numbers of even ovals.
It follows from classical restrictions on the topology of real algebraic varieties that the first Betti number of the real part of a real nonsingular sextic in $\mathbb{CP}^3$ can not exceed $94$. We construct a real nonsingular sextic $X$ in $\mathbb{CP}^3$ satisfying $b_1(\mathbb{R}X)=90$, improving a result of F.Bihan. The construction uses Viro's patchworking and an equivariant version of a deformation due to E.Horikawa.