Volume 2025, Issue 821

In this paper, we prove the optimal volume growth for complete Riemannian manifolds ( M n , g ) (M^{n},g) with nonnegative Ricci curvature everywhere and bi-Ricci curvature bounded from below by n − 2 n-2 outside a compact set when the dimension is less than eight. This answers a question proposed by Antonelli–Xu in dimensions six and seven. As a by-product, we also prove an analogy of Gromov’s volume bound conjecture under the condition of positive bi-Ricci curvature.

We prove that the Chow class κ ∞ ⁢ ( C t ) \kappa_{\infty}(C_{t}) of the Ceresa cycle of the genus-three curve C t : y 3 = x 4 + 2 ⁢ t ⁢ x 2 + 1 C_{t}\colon y^{3}=x^{4}+2tx^{2}+1 is torsion if and only if Q t = ( t 2 − 1 3 , t ) Q_{t}=(\sqrt[3]{t^{2}-1},t) is a torsion point on the elliptic curve y 2 = x 3 + 1 y^{2}=x^{3}+1 . In particular, there are infinitely many plane quartic curves over ℂ with torsion Ceresa cycle. Over Q ̄ \overline{\mathbb{Q}} , we show that the Beilinson–Bloch height of κ ∞ ⁢ ( C t ) \kappa_{\infty}(C_{t}) is proportional to the Néron–Tate height of Q t Q_{t} . Thus the height of κ ∞ ⁢ ( C t ) \kappa_{\infty}(C_{t}) is nondegenerate and satisfies a Northcott property. To prove all this, we show that the Chow motive that controls κ ∞ ⁢ ( C t ) \kappa_{\infty}(C_{t}) is isomorphic to h 1 \mathfrak{h}^{1} of an appropriate elliptic curve.

To any element of a connected, simply connected, semisimple complex algebraic group 𝐺 and a choice of an element of the corresponding Weyl group, there is an associated Lusztig variety. When the element of 𝐺 is regular semisimple, the corresponding variety carries an action of the Weyl group on its (equivariant) intersection cohomology. From this action, we recover the induced characters of an element of the Kazhdan–Lusztig basis of the corresponding Hecke algebra. In type 𝐴, we prove a more precise statement: that the Frobenius character of this action is precisely the symmetric function given by the characters of a Kazhdan–Lusztig basis element. The main idea is to find cellular decompositions of desingularizations of these varieties and apply the Brosnan–Chow palindromicity criterion for determining when the local invariant cycle map is an isomorphism. This recovers some results of Lusztig about character sheaves and gives a generalization of the Brosnan–Chow [P. Brosnan and T. Y. Chow, Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties, Adv. Math. 329 (2018), 955–1001] solution to the Shareshian–Wachs [J. Shareshian and M. L. Wachs, Chromatic quasisymmetric functions, Adv. Math. 295 (2016), 497–551] conjecture to non-codominant permutations, where singularities are involved.

Let p : X → S p\colon X\to S be a flat (proper) and regular scheme of finite type over a strictly henselian discrete valuation ring. We prove that the singularity category of the special fiber with its natural two-periodic structure allows to recover the ℓ-adic vanishing cohomology of 𝑝. Along the way, we compute homotopy-invariant non-connective algebraic K-theory with compact support of certain embeddings X t ↪ X T X_{t}\hookrightarrow X_{T} in terms of the motivic realization of the dg-category of relatively perfect complexes.

Using tropical geometry, one can translate problems in enumerative geometry to combinatorial problems. Thus, tropical geometry is a powerful tool in enumerative geometry over the complex and real numbers. Results from A 1 \mathbb{A}^{1} -homotopy theory allow to enrich classical enumerative geometry questions and get answers over an arbitrary field. In the resulting area, A 1 \mathbb{A}^{1} -enumerative geometry, the answer to these questions lives in the Grothendieck–Witt ring of the base field 𝑘. In this paper, we use tropical methods in this enriched set up by showing Bézout’s theorem and a generalization, namely the Bernstein–Kushnirenko theorem, for tropical hypersurfaces enriched in GW ⁡ ( k ) \operatorname{GW}(k) .

A nonsingular real algebraic variety 𝑌 is said to have the approximation property if, for every nonsingular real algebraic variety 𝑋, the following holds: if f : X → Y f\colon X\rightarrow Y is a C ∞ \mathcal{C}^{\infty} map that is homotopic to a regular map, then 𝑓 can be approximated in the C ∞ \mathcal{C}^{\infty} topology by regular maps. In this paper, we characterize the varieties 𝑌 with the approximation property. We also characterize the varieties 𝑌 with the approximation property combined with a suitable interpolation condition. Some of our results have variants concerning the regular approximation of continuous maps defined on (possibly singular) real algebraic varieties.

Building on some classical techniques in approximation theory of functions of one variable, we obtain new results concerning C p \mathcal{C}^{p} functions and definable (e.g., semialgebraic) C p \mathcal{C}^{p} functions of several variables. The starting point for our work is the following vague problem: can C p \mathcal{C}^{p} smoothness be tested on lines in R N \mathbb{R}^{N} parallel to the coordinate axes? Specific aspects of this problem are investigated both in general setting and in o-minimal setting. Of course, the behaviour of a function is not examined separately on each line, but it should be uniformly controlled (for otherwise, we cannot expect any positive answers to the problem). All the main results of the paper are accompanied by numerous examples and counterexamples.

A salami is a connected, locally finite, weighted graph with non-negative Ollivier Ricci curvature and at least two ends of infinite volume. We show that every salami has exactly two ends and no vertices with positive curvature. We moreover show that every salami is recurrent and admits harmonic functions with constant gradient. The proofs are based on extremal Lipschitz extensions, a variational principle and the study of harmonic functions. Assuming a lower bound on the edge weight, we prove that salamis are quasi-isometric to the line, that the space of all harmonic functions has finite dimension and that the space of subexponentially growing harmonic functions is two-dimensional. Moreover, we give a Cheng–Yau gradient estimate for harmonic functions on balls.