Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.34562

Correlation inequalities for volume polynomials

June Huh·Mateusz Michałek·Botong Wang·Jian Xiao
View original source ↗Read source document ↗Export metadata ↓

Abstract

We establish sharp correlation inequalities for volume polynomials associated with covers of multisets. These inequalities simultaneously generalize the Khovanskii--Teissier and reverse Khovanskii--Teissier inequalities for intersections of nef divisor classes on projective varieties. They also hold for volume polynomials arising from convex bodies and Hermitian matrices, recovering and extending classical results such as the Alexandrov--Fenchel inequality for mixed volumes of convex bodies and the Hadamard--Fischer inequality for determinants of positive semidefinite matrices. As a combinatorial application, we prove a negative correlation inequality for partitions of bases of matroids that are linearly realizable over a field.

Explore related subjects

math.AGmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

June Huh, Mateusz Michałek, Botong Wang, Jian Xiao. 2026-10-25. Correlation inequalities for volume polynomials. https://arxiv.org/abs/2610.34562

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Low Genus Curves on Compact Shimura Surfaces

We investigate and compare the behaviors of complex curves on two major types of compact Shimura surfaces: compact variants of Picard and Hilbert modular surfaces. They are quotients of complex hyperbolic balls corresponding to moduli problems for abelian varieties with certain endomorphism types. We prove that if we fix a genus bound, then when the defining discriminant is large enough, low genus curves on the compact variant of Picard modular surfaces do not exist, while low genus curves may persist on the compact variant of Hilbert modular surfaces only if they are special curves, which parametrize those with extra endomorphisms. These results are special cases of a complex function field analogue of uniform boundedness on endomorphism types of abelian varieties. The proof requires understanding the repulsion of special subvarieties and relies on volume estimates in hyperbolic geometry.

math.AG↗
arXiv2026

Euclidean distance degree defect of singular projective varieties

The unit Euclidean distance degree and the generic Euclidean distance degree are two well-studied invariants of projective varieties. These quantities measure the algebraic complexity of nearest-point problems on a variety, and in many examples arising in optimization, engineering, statistics, and data science, there is a significant gap between them. We refer to this difference as the defect of the Euclidean distance (ED) degree. In this paper, we provide a constructible enhancement and a topological formula for the defect of the ED degree of an arbitrary complex projective variety, extending our previous results from the smooth setting. Since the generic Euclidean distance degree is typically more tractable, our approach offers a new method for computing ED degrees in broad generality.

math.AG↗
arXiv2026

Involution-equivariant topological recursion and mirror symmetry for the affine binary dihedral Calabi--Yau threefold

We prove a closed-string remodeling statement for the affine binary dihedral Calabi--Yau orbifold threefold $\mathcal X=[\mathbb C^2/Γ]\times\mathbb C$, where $Γ$ is a binary dihedral subgroup of $SU(2)$. This target lies outside the toric setting of the Bouchard--Klemm--Mariño--Pasquetti remodeling conjecture: the toric mirror curve is replaced by the type-$D_l$ relativistic Toda curve of Brini--Ma--Strachan, and the Chekhov--Eynard--Orantin topological recursion is replaced by the $\mathbb Z_2$-equivariant topological recursion of Giacchetto--Kramer--Lewański, run in the sign sector of the Toda-curve involution with the Prym kernel as its two-point input. We identify the equivariant orbifold quantum cohomology Frobenius manifold of $\mathcal X$ with the invariant Jacobian Frobenius structure of the Toda curve, and we prove that the B-model $R$-matrix, defined by regularized stationary phase, equals the A-side normalized canonical Givental--Teleman $R$-matrix on a chosen simply connected domain; this equality is anchored at the orbifold point through a semistable degeneration of the Toda curve. Comparing the resulting Givental--Teleman and Dunin-Barkowski--Orantin--Shadrin--Spitz graph sums then identifies, after a parity-twisted leaf substitution, the sign-sector recursion with the descendant Gromov--Witten generating functions of $\mathcal X$ in the stable range ($2g-2+n>0$ with $n>0$), and identifies the recursion free energies with the equivariant Gromov--Witten free energies of $\mathcal X$ for $g\geq2$. We also match the genus-one potential defined by the reduced calibrated tau function with the Gromov--Witten potential up to an additive constant.

math.AG↗