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1. Blown-up toric surfaces with non-polyhedral effective cone

   https://doi.org/10.1515/crelle-2023-0022  

   Castravet, Ana-Maria; Laface, Antonio; Tevelev, Jenia; Ugaglia, Luca (May 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We construct projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone.As a consequence, we prove that the pseudo-effective cone of the Grothendieck–Knudsen moduli space ${\overline{M}}_{0,n}$\overline{M}_{0,n}of stable rational curves is not polyhedral for $n\ge 10$n\geq 10.These results hold both in characteristic 0 and in characteristic 𝑝, for all primes 𝑝.Many of these toric surfaces are related to an interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order.Our analysis relies on tools of arithmetic geometry and Galois representations in the spirit of the Lang–Trotter conjecture, producing toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone in characteristic 0 and in characteristic 𝑝, for an infinite set of primes 𝑝 of positive density. 

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2. Staircase patterns in Hirzebruch surfaces

   https://doi.org/10.4171/cmh/571  

   Magill, Nicki; McDuff, Dusa; Weiler, Morgan (July 2024, Commentarii Mathematici Helvetici)

   The ellipsoidal capacity function of a symplectic four manifoldXmeasures how much the form onXmust be dilated in order for it to admit an embedded ellipsoid of eccentricityz. In most cases there are just finitely many obstructions to such an embedding besides the volume. If there are infinitely many obstructions,Xis said to have a staircase. This paper gives an almost complete description of the staircases in the ellipsoidal capacity functions of the family of symplectic Hirzebruch surfacesH_{b}formed by blowing up the projective plane with weightb. We describe an interweaving, recursively defined, family of obstructions to symplectic embeddings of ellipsoids that show there is an open dense set of shape parametersbthat are blocked, i.e. have no staircase, and an uncountable number of other values ofbthat do admit staircases. The remainingb-values form a countable sequence of special rational numbers that are closely related to the symmetries discussed in Magill–McDuff (arXiv:2106.09143). We show that none of them admit ascending staircases. Conjecturally, none admit descending staircases. Finally, we show that, as long asbis not one of these special rational values, any staircase inH_{b}has irrational accumulation point. A crucial ingredient of our proofs is the new, more indirect approach to using almost toric fibrations in the analysis of staircases by Magill (arXiv:2204.12460). In particular, the structure of the relevant mutations of the set of almost toric fibrations onH_{b}is echoed in the structure of the set of blockedb-intervals. 

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3. The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers

   https://doi.org/10.1090/cams/23  

   Parisi, Matteo; Sherman-Bennett, Melissa; Williams, Lauren (July 2023, Communications of the American Mathematical Society)

   The hypersimplex ${\mathrm{\Delta <\#comment/>}}_{k+1,n}$is the image of the positive Grassmannian $G{r}_{k+1,n}^{\ge <\#comment/>0}$under the moment map. It is a polytope of dimension $n-<\#comment/>1$in ${\mathbb{R}}^n$. Meanwhile, the amplituhedron ${\mathcal{A}}_{n,k,2}\left(Z\right)$is the projection of the positive Grassmannian $G{r}_{k,n}^{\ge <\#comment/>0}$into the Grassmannian $G{r}_{k,k+2}$under a map $\stackrel{~<\#comment/>}{Z}$induced by a positive matrix $Z\in <\#comment/>Ma{t}_{n,k+2}^{>0}$. Introduced in the context ofscattering amplitudes, it is not a polytope, and has full dimension $2k$inside $G{r}_{k,k+2}$. Nevertheless, there seem to be remarkable connections between these two objects viaT-duality, as conjectured by Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]. In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting outpositroid polytopes—images of positroid cells of $G{r}_{k+1,n}^{\ge <\#comment/>0}$under the moment map—translate into sign conditions characterizing the T-dualGrasstopes—images of positroid cells of $G{r}_{k,n}^{\ge <\#comment/>0}$under $\stackrel{~<\#comment/>}{Z}$. Moreover, we subdivide the amplituhedron intochambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We use these properties to prove the main conjecture of Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]: a collection of positroid polytopes is a tiling of the hypersimplex if and only if the collection of T-dual Grasstopes is a tiling of the amplituhedron ${\mathcal{A}}_{n,k,2}\left(Z\right)$for all $Z$. Moreover, we prove Arkani-Hamed–Thomas–Trnka’s conjectural sign-flip characterization of ${\mathcal{A}}_{n,k,2}$, and Łukowski–Parisi–Spradlin–Volovich’s conjectures on $m=2$cluster adjacencyand onpositroid tilesfor ${\mathcal{A}}_{n,k,2}$(images of $2k$-dimensional positroid cells which map injectively into ${\mathcal{A}}_{n,k,2}$). Finally, we introduce new cluster structures in the amplituhedron. 

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4. The Riemannian and symplectic geometry of the space of generalized Kähler structures

   https://doi.org/10.4171/CMH/585  

   Apostolov, Vestislav; Streets, Jeffrey; Ustinovskiy, Yury (February 2025, Commentarii Mathematici Helvetici)

   On a compact complex manifold(M, J)endowed with a holomorphic Poisson tensor \pi_{J}and a de Rham class\alpha\in H^{2}(M, \mathbb{R}), we study the space of generalized Kähler (GK) structures defined by a symplectic formF\in \alphaand whose holomorphic Poisson tensor is\pi_{J}. We define a notion of generalized Kähler class of such structures, and use the moment map framework of Boulanger (2019) and Goto (2020) to extend the Calabi program to GK geometry. We obtain generalizations of the Futaki–Mabuchi extremal vector field (1995) and the Calabi–Lichnerowicz–Matsushima result (1982, 1958, 1957) for the Lie algebra of the group of automorphisms of(M, J, \pi_{J}). We define a closed1-form on a GK class, which yields a generalization of the Mabuchi energy and thus a variational characterization of GK structures of constant scalar curvature. Next we introduce a formal Riemannian metric on a given GK class, generalizing the fundamental construction of Mabuchi–Semmes–Donaldson (1987, 1992, 1997) We show that this metric has nonpositive sectional curvature, and that the Mabuchi energy is convex along geodesics, leading to a conditional uniqueness result for constant scalar curvature GK structures. We finally examine the toric case, proving the uniqueness of extremal generalized Kähler structures and showing that their existence is obstructed by the uniform relative K-stability of the corresponding Delzant polytope. Using the resolution of the Yau–Tian–Donaldson conjecture in the toric case by Chen–Cheng (2021) and He (2019), we show in some settings that this condition suffices for existence and thus construct new examples. 

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5. Rigidity of nonpositively curved manifolds with convex boundary

   https://doi.org/10.1090/proc/16475  

   Ghomi, Mohammad; Spruck, Joel (November 2023, Proceedings of the American Mathematical Society)

   We show that a compact Riemannian $3$-manifold $M$with strictly convex simply connected boundary and sectional curvature $K\le <\#comment/>a\le <\#comment/>0$is isometric to a convex domain in a complete simply connected space of constant curvature $a$, provided that $K\equiv <\#comment/>a$on planes tangent to the boundary of $M$. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard $3$-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for $K\ge <\#comment/>a\ge <\#comment/>0$. 

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