arXiv subjects
Publications and source records attributed to Diego Matessi.
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.
We study the mod $2$ cohomology of real Calabi-Yau threefolds given by real structures which preserve the torus fibrations constructed by Gross. We extend the results of Castaño-Bernard-Matessi and Arguz-Prince to the case of real structures twisted by a Lagrangian section. In particular we find exact sequences linking the cohomology of the real Calabi-Yau with the cohomology of the complex one. Applying SYZ mirror symmetry, we show that the connecting homomorphism is determined by a ``twisted squaring of divisors'' in the mirror Calabi-Yau, i.e. by $D \mapsto D^2 + DL$ where $D$ is a divisor in the mirror and $L$ is the divisor mirror to the twisting section. We use this to find an example of a connected $(M-2)$-real quintic threefold.
We use Lagrangian torus fibrations on the mirror $X$ of a toric Calabi-Yau threefold $\check X$ to construct Lagrangian sections and various Lagrangian spheres on $X$. We then propose an explicit correspondence between the sections and line bundles on $\check X$ and between spheres and sheaves supported on the toric divisors of $\check X$. We conjecture that these correspondences induce an embedding of the relevant derived Fukaya category of $X$ inside the derived category of coherent sheaves on $\check X$.
We prove that a smooth tropical hypersurface in $\mathbb{R}^3$ can be lifted to a smooth embedded Lagrangian submanifold in $(\mathbb{C}^*)^3$. This completes the proof of the result announced in the article "Lagrangian pairs pants" arXiv:1802.02993. The idea of the proof is to use Lagrangian pairs of pants as the main building blocks.
We construct a Lagrangian submanifold, inside the cotangent bundle of a real torus, which we call a Lagrangian pair of pants. It is given as the graph of the differential of a smooth function defined on the real blow up of a Lagrangian coamoeba. Lagrangian pairs of pants are the main building blocks in a construction of smooth Lagrangian submanifolds of $(\mathbb{C}^*)^n$ which lift tropical subvarieties in $\mathbb{R}^n$. As an example we explain how to lift tropical curves in $\mathbb{R}^2$ to Lagrangian submanifolds of $(\mathbb{C}^*)^2$. We also give several new examples of Lagrangian submanifolds inside toric varieties, some of which are monotone.
Mirror symmetry of Calabi-Yau manifolds can be understood via a Legendre duality between a pair of certain affine manifolds with singularities called tropical manifolds. In this article, we study conifold transitions from the point of view of Gross and Siebert. We introduce the notions of tropical nodal singularity, tropical conifolds, tropical resolutions and smoothings. We interpret known global obstructions to the complex smoothing and symplectic small resolution of compact nodal Calabi-Yaus in terms of certain tropical $2$-cycles containing the nodes in their associated tropical conifolds. We prove that the existence of such cycles implies the simultaneous vanishing of the obstruction to smoothing the original Calabi-Yau \emph{and} to resolving its mirror. We formulate a conjecture suggesting that the existence of these cycles should imply that the tropical conifold can be resolved and its mirror can be smoothed, thus showing that the mirror of the resolution is a smoothing. We partially prove the conjecture for certain configurations of nodes and for some interesting examples.
We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the quintic threefold. We interpret our results as corroboration of the view that in homological mirror symmetry, an anti-symplectic involution is the mirror of duality. In the same setting, we construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle.
We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi-Yau manifolds. For example we prove that given any real-analytic one parameter family of Riemannian metrics $g_t$ on a 3-dimensional manifold $Y$ with volume form independent of $t$ and with a real-analytic family of nowhere vanishing harmonic one forms $θ_t$, then $(Y, g_t)$ can be realized as a family of special Lagrangian submanifolds of a Calabi-Yau manifold $X$. We also prove that certain principal torus bundles can be equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus action. We use this to construct examples of $n$-parameter families of special Lagrangian tori inside $n+k$-dimensional Calabi-Yau manifolds with torus symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian fibrations with $T^2$-symmetry.
We give a simple proof of the local version of a result of R. Bryant, stating that any 3-dimensional Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We refine the theorem proving that a certain class of one-parameter families of metrics on a 3-torus can be isometrically embedded in a Calabi-Yau manifold as a one-parameter family of special Lagrangian submanifolds. We use our examples of one-parameter families to show that the semi-flat metric on the mirror manifold proposed be N. Hitchin is not necessarily Ricci-flat in dimension 3.