Related papers: Newton-Okounkov bodies and complexity functions

We compute the generic infinitesimal Newton-Okounkov body at any point for some box-product polarizations on products of curves. This appears to be the first nontrivial description of such a body in arbitrary dimension.

We construct families of Newton-Okounkov bodies for the free group character varieties and configuration spaces of any connected reductive group. We use this construction to give a proof that these spaces are Cohen-Macaulay.

Nekrasov-Okounkov identity gives a product representation of the sum over partitions of a certain function of partition hook length. In this paper we give several generalizations of the Nekrasov-Okounkov identity using the cyclic symmetry…

We show that given a face of Hironaka's characteristic polyhedron, it does only depend on the singularity and a flag defined by the linear form determining the face. As a consequence we get that certain numerical data obtained from the…

We study the dual complexes of boundary divisors in log resolutions of compactifications of algebraic varieties and show that the homotopy types of these complexes are independent of all choices. Inspired by recent developments in…

We give a new proof that bounded non-commutative functions on polynomial polyhedra can be represented by a realization formula, a generalization of the transfer function realization formula for bounded analytic functions on the unit disk.

We prove that the Newton product of efficient polynomial projectors is still efficient. Various polynomial approximation theorems are established involving Newton product projectors on spaces of holomorphic functions on a neighborhood of a…

We classify the holomorphic parabolic geometries on compact complex manifolds of general type. We accomplish this by bounding the numerical dimension of any smooth projective variety in terms of geometric invariants of the flag variety…

We give a uniform proof for the conjectured Gromov width of coadjoint orbits of all compact connected simple Lie groups, by analyzing simplices in Newton-Okounkov bodies.

We prove inequalities relating the degrees of holomorphic distributions and of holomorphic foliations forming a flag on $\mathbb{P}^n$. Such inequalities are inspired by the so called Poincar\'e problem for foliations.

Starting from the data of a big line bundle $L$ on a projective manifold $X$ with a choice of $N\geq 1$ different points on $X$ we provide a new construction of $N$ Okounkov bodies which encodes important geometric features of ($L\to…

We explain that the Pl\"ucker relations provide the defining equations of the thick flag manifold associated to a Kac-Moody algebra. This naturally transplant the result of Kumar-Mathieu-Schwede about the Frobenius splitting of thin flag…

We disprove a conjecture of A. Koldobsky asking whether it is enough to compare $(n-2)$-derivatives of the projection functions of two symmetric convex bodies in the Shephard problem in order to get a positive answer in all dimensions.

Newton-Okounkov bodies serve as a bridge between algebraic geometry and convex geometry, enabling the application of combinatorial and geometric methods to the study of linear systems on algebraic varieties. This paper contributes to…

We compute the Newton--Okounkov bodies of line bundles on a Bott--Samelson resolution of the complete flag variety of $GL_n$ for a geometric valuation coming from a flag of translated Schubert subvarieties. The Bott--Samelson resolution…

Using an intuition from metric geometry, we prove that any flag and normal simplicial complex satisfies the non-revisiting path conjecture. As a consequence, the diameter of its facet-ridge graph is smaller than the number of vertices minus…

Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincar\'e…

Estimating the coefficient functionals on various classes of holomorphic functions traditionally forms an important field of geometric complex analysis and its mathematical and physical applications. These coefficients reflect fundamental…

We establish basic facts about the varieties of homogeneous polynomials divisible by powers of linear forms, and explain consequences for geometric complexity theory. This includes quadratic set-theoretic equations, a description of the…

We compare the $T$-polynomial convexity of Guedj with holomorphic convexity away from the support of $T$. In particular we show an Oka--Weil theorem for $T$-polynomial convexity, as well as present a situation when the notions of…