International Mathematics Research Notices

Lehmann, Brian; Riedl, Eric

doi: 10.1093/imrn/rnac011pmid: N/A

Suppose that $X$ is a smooth projective variety and that $C$ is a general member of a family of free rational curves on $X$. We prove several statements showing that the Harder–Narasimhan filtration of $T_{X}|_{C}$ is approximately the same as the restriction of the Harder–Narasimhan filtration of $T_{X}$ with respect to the class of $C$. When $X$ is a Fano variety, we prove that the set of all restricted tangent bundles for general free rational curves is controlled by a finite set of data. We then apply our work to analyze Peyre’s “freeness” formulation of Manin’s Conjecture in the setting of rational curves.

Ashcroft, Calum J

doi: 10.1093/imrn/rnac111pmid: N/A

We provide a condition on the links of polygonal complexes that is sufficient to ensure groups acting properly discontinuously and cocompactly on such complexes contain a virtually free codimension-$1$ subgroup. We provide stronger conditions on the links of polygonal complexes, which are sufficient to ensure groups acting properly discontinuously and cocompactly on such complexes act properly discontinuously on a $CAT(0)$ cube complex. If the group is hyperbolic, then this action is also cocompact; hence, by Agol’s Theorem the group is virtually special (in the sense of Haglund–Wise). In particular, it is linear over $\mathbb {Z}$, virtually torsion free, and has the Haagerup property. We consider some applications of this work. Firstly, we consider the groups classified by Kangaslampi–Vdovina and Carbone–Kangaslampli–Vdovina, which act simply transitively on the vertices of $CAT(0)$ triangular complexes with the generalized quadrangle of order $2$ as their links, proving that these groups are virtually special. We further apply this theorem by considering generalized triangle groups, in particular a subset of those considered by Caprace–Conder–Kaluba–Witzel.

Chen, Xiaojuan; Tu, Qiang; Xiang, Ni

doi: 10.1093/imrn/rnac127pmid: N/A

In this paper, we consider the Dirichlet problem for a class of Hessian quotient equations involving a gradient term on the right-hand sides on Riemannian manifolds. Under the assumption of an admissible subsolution, we solve the existence and the uniqueness for the Dirichlet problem on compact Riemannian manifold, based on the a priori estimates for the solutions to the Hessian quotient type equations. Compared with the classical results for Hessian type equations, our results do not depend on the convexity assumption for the right-hand side of the equation.

Feigin, Evgeny; Makhlin, Igor; Popkovich, Alexander

doi: 10.1093/imrn/rnac116pmid: N/A

We study toric degenerations of semi-infinite Grassmannians (a.k.a. quantum Grassmannians). While the toric degenerations of the classical Grassmannians are well studied, the only known example in the semi-infinite case is due to Sottile and Sturmfels. We start by providing a new interpretation of the Sottile–Sturmfels construction by finding a poset such that their degeneration is the toric variety of the order polytope of the poset. We then use our poset to construct and study a new toric degeneration in the semi-infinite case. Our construction is based on the notion of poset polytopes introduced by Fang–Fourier–Litza–Pegel. As an application, we introduce semi-infinite PBW-semistandard tableaux, giving a basis in the homogeneous coordinate ring of a semi-infinite Grassmannian.

Fradelizi, Matthieu; Nakhle, Elie

doi: 10.1093/imrn/rnac120pmid: N/A

Let $\varphi :{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}\cup \{+\infty \}$ be an even convex function and ${\mathcal {L}}{\varphi }$ be its Legendre transform. We prove the functional form of Mahler’s conjecture concerning the functional volume product $P(\varphi )=\int e^{-\varphi }\int e^{-{\mathcal {L}}\varphi }$ in dimension 2: we give the sharp lower bound of this quantity and characterize the equality case. The proof uses the computation of the derivative in $t$ of $P(t\varphi )$ and ideas due to Meyer [16] for unconditional convex bodies, adapted to the functional case by Fradelizi and Meyer [6] and extended for symmetric convex bodies in dimension 3 by Iriyeh and Shibata [11] (see also [4]).

Baldauf, Julius; Kim, Dain

doi: 10.1093/imrn/rnac128pmid: N/A

This paper defines a parabolic frequency for solutions of the heat equation on a Ricci flow and proves its monotonicity along the flow. Frequency monotonicity is known to have many useful consequences; here it is shown to provide a simple proof of backwards uniqueness. For solutions of more general parabolic equations on a Ricci flow, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.

Åhag, Per; Czyż, Rafał

doi: 10.1093/imrn/rnac129pmid: N/A

We raise our cups to Urban Cegrell, gone but not forgotten, gone but ever here. Until we meet again in Valhalla!With inspiration from the Kähler geometry, we introduce a metric structure on the energy class, $\mathcal {E}_{1,m}$, of $m$-subharmonic functions with bounded energy and show that it is complete. After studying how the metric convergence relates to the accepted convergences in this Caffarelli–Nirenberg–Spruck model, we end by constructing geodesics in a subspace of our complete metric space.

Yao, Zijian

doi: 10.1093/imrn/rnac068pmid: N/A

We use the Nygaard filtration on Ainf cohomology to give a new proof of the following result: for a smooth proper (formal) scheme over a mixed characteristic valuation ring, the Newton polygon of its special fiber lies on or above the Hodge polygon of its generic fiber.

Bud, Andrei

doi: 10.1093/imrn/rnac135pmid: N/A

For genus $g = \frac {r(r+1)}{2}+1$, we prove that via the forgetful map, the universal Prym–Brill–Noether locus $\mathcal {V}^r_g$ has a unique irreducible component dominating the moduli space $\mathcal {R}_g$ of Prym curves.

Cui, Peiyi

doi: 10.1093/imrn/rnac133pmid: N/A

Let $p$ be a prime number and $k$ an algebraically closed field with characteristic $\ell \neq p$. We show that the supercuspidal support of irreducible smooth $k$-representations of Levi subgroups $\textrm {M}^{\prime}$ of $\textrm {SL}_n(F)$ is unique up to $\textrm {M}^{\prime}$-conjugation, where $F$ is either a finite field of characteristic $p$ or a non-Archimedean locally compact field of residual characteristic $p$.