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1. Special Lagrangian submanifolds of log Calabi-Yau manifolds

   Collins, Tristan C.; Jacob, A.; Lin, Y.-S. (January 2021, Duke mathematical journal)

   null (Ed.)

   We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D is a smooth divisor in the linear system of K_Y with D^2=d, then X=Y/D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y=P^2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type I_d fiber appearing as a singular fiber in a rational elliptic surface. 

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2. Non-toric Brane Webs, Calabi–Yau 3-Folds, and 5d SCFTs

   https://doi.org/10.1007/s00220-025-05461-9  

   Alexeev, Valery; Argüz, Hülya; Bousseau, Pierrick (October 2025, Communications in Mathematical Physics)

   Abstract We study webs of 5-branes with 7-branes in Type IIB string theory from a geometric perspective. Mathematically, a web of 5-branes with 7-branes is a tropical curve in$$\mathbb {R}^2$$ $R^2$with focus-focus singularities introduced. To any such a webW, we attach a log Calabi–Yau surface (Y, D) with a line bundleL. We then describe supersymmetric webs, which are webs defining 5d superconformal field theories (SCFTs), in terms of the geometry of (Y, D, L). We also introduce particular supersymmetric webs called “consistent webs, and show that any 5d SCFT defined by a supersymmetric web can be obtained from a consistent web by adding free hypermultiplets. Using birational geometry of degenerations of log Calabi–Yau surfaces, we provide an algorithm to test the consistency of a web in terms of its dual polygon. Moreover, for a consistent webW, we provide an algebro-geometric construction of the mirror$$\mathcal {X}^{\textrm{can}}$$ $X^{\text{can}}$to (Y, D, L), as a non-toric canonical 3-fold singularity, and show that M-theory on$$\mathcal {X}^{\textrm{can}}$$ $X^{\text{can}}$engineers the same 5d SCFT asW. We also explain how to derive explicit equations for$$\mathcal {X}^{\textrm{can}}$$ $X^{\text{can}}$using scattering diagrams, encoding disk worldsheet instantons in the A-model, or equivalently the BPS states of an auxiliary rank one 4d$$\mathcal {N}=2$$ $N=2$theory. 

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3. Deformations of Calabi–Yau Varieties With Isolated Log Canonical Singularities

   https://doi.org/10.1093/imrn/rnaf112  

   Friedman, Robert; Laza, Radu (May 2025, International Mathematics Research Notices)

   Abstract Recent progress in the deformation theory of Calabi–Yau varieties $$Y$$ with canonical singularities has highlighted the key role played by the higher Du Bois and higher rational singularities, and especially by the so-called $$k$$-liminal singularities for $$k\ge 1$$. The goal of this paper is to show that certain aspects of this study extend naturally to the $$0$$-liminal case as well, that is, to Calabi–Yau varieties $$Y$$ with Gorenstein log canonical, but not canonical, singularities. In particular, we show the existence of first order smoothings of $$Y$$ in the case of isolated $$0$$-liminal hypersurface singularities, and extend Namikawa’s unobstructedness theorem for deformations of singular Calabi–Yau three-folds $$Y$$ with canonical singularities to the case where $$Y$$ has an isolated $$0$$-liminal lci singularity under suitable hypotheses. Finally, we describe an interesting series of examples. 

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4. Brane brick models for the Sasaki-Einstein 7-manifolds Yp,k(ℂℙ1 × ℂℙ1) and Yp,k(ℂℙ2)

   https://doi.org/10.1007/JHEP03(2023)050  

   Franco, Sebastián; Ghim, Dongwook; Seong, Rak-Kyeong (March 2023, Journal of High Energy Physics)

   A bstract The 2 d (0 , 2) supersymmetric gauge theories corresponding to the classes of Y p,k (ℂℙ 1 × ℂℙ 1 ) and Y p,k (ℂℙ 2 ) manifolds are identified. The complex cones over these Sasaki-Einstein 7-manifolds are non-compact toric Calabi-Yau 4-folds. These infinite families of geometries are the largest ones for Sasaki-Einstein 7-manifolds whose metrics, toric diagrams, and volume functions are known explicitly. This work therefore presents the largest list of 2 d (0 , 2) supersymmetric gauge theories corresponding to Calabi-Yau 4-folds with known metrics. 

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5. Bourgeois contact structures: Tightness, fillability and applications

   https://doi.org/10.1007/s00222-022-01131-y  

   Bowden, Jonathan; Gironella, Fabio; Moreno, Agustin (July 2022, Inventiones mathematicae)

   Abstract Given a contact structure on a manifold V together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $$V \times {\mathbb {T}}^2$$ V × T 2 . We prove that all such structures are universally tight in dimension 5, independent of whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact 5-manifolds, as well as the first examples of weakly but not strongly fillable contact structures in all odd dimensions. These obstructions are particular instances of more general obstructions for $${\mathbb {S}}^1$$ S 1 -invariant contact manifolds. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the n -torus has a unique symplectically aspherical strong filling up to diffeomorphism. 

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