arXiv subjects
Publications and source records attributed to Uli Walther.
We discuss the connections between tautological systems and the local cohomology of cones over homogeneous spaces. We study a derived version of tautological systems, related to the Chevalley--Eilenberg complex, and show that in many cases it underlies a complex of mixed Hodge modules.
Configuration polynomials generalize the Kirchhoff polynomial of a graph, as well as the Symanzik polynomials that appear in the denominators of Feynman integrands. The configuration hypersurfaces cut out by such polynomials are typically highly singular, which poses a challenge for the evaluation of Feynman integrals even in simplified settings. In this paper, we provide a two-step recipe for a resolution of singularities of any irreducible configuration hypersurface. We first consider the normalization of the Nash blow-up, which we identify with an incidence variety introduced by Bloch. This variety is typically still not smooth, but it is the closure of a smooth subvariety of a torus. The latter then a smooth, tropical compactification, using work of Tevelev. We construct explicitly such a compactification and a morphism to the normalized Nash blow-up for every configuration, described in terms of bipermutohedral matroid combinatorics introduced by Ardila, Denham and Huh. Along the way, we find that the normalized Nash blow-up of the configuration hypersurface has strongly $F$-regular singularities in positive characteristic. We deduce this by certifying $F$-rationality of its biprojective cone, and infer from it that the normalized Nash blow-up has rational singularities over the complex numbers.
We compute the arithmetic rank as well as the local/\'etale cohomological dimension of nullcone ideals arising from the classical actions of the symplectic group, the general linear group, and the orthogonal group. We use these calculations to establish striking vanishing results for local cohomology modules supported at these nullcone ideals; this is achieved via a careful analysis of the critical local cohomology modules. The vanishing theorems that we prove are sharp in various respects.
We prove that hypersurfaces defined by irreducible square-free polynomials have rational singularities. As an easy consequence, we deduce that certain (possibly non-square-free) polynomials associated to pairs of square-free polynomials define hypersurfaces with rational singularities. This extends results on certain classes of polynomials associated to matroids and Feynman diagrams in [BW].
Given a matroid or flag of matroids we introduce several broad classes of polynomials satisfying Deletion-Contraction identities, and study their singularities. There are three main families of polynomials captured by our approach: matroidal polynomials on a matroid (including matroid basis polynomials, configuration polynomials, Tutte polynomials); flag matroidal polynomials on a flag matroid; and Feynman integrands. The last class includes under general kinematics the inhomogeneous Feynman diagram polynomials which naturally arise in the Lee--Pomeransky form of the Feynman integral attached to a Feynman diagram. Assuming that the primary underlying matroid is connected and of positive rank (and in the flag case, has rank at least two), we show: a) in positive characteristic, homogeneous matroidal polynomials are strongly $F$-regular; b) over an algebraically closed field of characteristic zero, the associated jet schemes of (flag) matroidal polynomials as well as those of Feynman integrands are irreducible. Consequently, all these polynomials have rational singularities (or are smooth).
We consider $A$-hypergeometric (or GKZ-)systems in the case where the grading (character) group is an arbitrary finitely generated Abelian group. Emulating the approach taken for classical GKZ-systems in arXiv:math/0406383 that allows for a coefficient module, we show that these $D$-modules are holonomic systems. For this purpose we formulate an Euler--Koszul complex in this context, built on an extension of the category of $A$-toric modules. We derive that these new systems are regular holonomic under circumstances that are similar to those that lead to regular holonomic classical GKZ-systems. For the appropriate coefficient module, our $D$-modules specialize to the "better behaved GKZ-systems" introduced by Borisov and Horja. We certify the corresponding $D$-modules as regular holonomic, and establish a holonomic duality on the level of $D$-modules that was suggested on the level of solutions by Borisov and Horja and later shown by Borisov and Han in a special situation (arXiv:1308.2238, arXiv:2301.01374).
Consider a group acting on a polynomial ring S over a field K by degree-preserving K-algebra automorphisms. Several key properties of the invariant ring can be deduced by studying the nullcone of the action, that is, the vanishing locus of all non-constant homogeneous invariant polynomials. These properties include the finite generation of the invariant ring and the purity of its embedding in S. In this article, we study the nullcones arising from the natural actions of the symplectic and general linear groups. For the natural representation of the symplectic group (via copies of the regular representation), the invariant ring is a generic Pfaffian ring. We show that the nullcone of this embedding is F-regular in positive characteristic. Independent of characteristic, we give a complete description of the divisor class group of the nullcone and determine precisely when it is Gorenstein. For the natural representation of the general linear group (via copies of the regular representation and copies of its dual), the invariant ring is a generic determinantal ring. The nullcone of this embedding is typically non-equidimensional; its irreducible components are the varieties of complexes introduced by Buchsbaum and Eisenbud. We show that each of these irreducible components are F-regular in positive characteristic. We also show that the Frobenius splittings of the varieties of complexes may be chosen compatibly so that the nullcone is F-pure.
We show that the sum of the local cohomological dimension and the rectified $\mathbb Q$-homological depth of a closed analytic subspace of a complex manifold coincide with the dimension of the ambient manifold. The local cohomological dimension is then calculated using the cohomology of the links of the analytic space. In the algebraic case the first assertion is equivalent to the coincidence of the rectified $\mathbb Q$-homological depth with the de Rham depth studied by Ogus, and follows essentially from his work. As a corollary we show that the local cohomological dimension of a quasi-projective variety is determined by that of its general hyperplane section together with the link cohomology at 0-dimensional strata of a complex analytic Whitney stratification.
Many hypergeometric differential systems that arise from a geometric setting can be endowed with the structure of mixed Hodge modules. We generalize this fundamental result to the tautological systems associated to homogeneous spaces by giving a functorial construction for them. As an application, we solve the holonomic rank problem for such tautological systems in full generality.
We show in several important cases that the $A$-hypergeometric system attached to a Feynman diagram in Lee--Pomeransky form, obtained by viewing the momenta and the nonzero masses as indeterminates, has a normal underlying semigroup. This continues a quest initiated by Klausen, and studied by Helmer and Tellander. In the process we identify several relevant matroids related to the situation and explore their relationships.
If $β\in\CC^d$ is integral but not a strongly resonant parameter for the homogeneous matrix $A\in\ZZ^{d\times n}$ with $\ZZ A=\ZZ^d$, then the associated GKZ-system carries a naturally defined mixed Hodge module structure. We study here in the normal case the corresponding weight filtration by computing the intersection complexes with respective multiplicities on the associated graded parts. We do this by computing the weight filtration of a Gauss-Manin system with respect to a locally closed embedding of a torus inside an affine space. We then produce a result, based on a Fourier-Sato transforms, that allows to port an MHS structure on a monodromic module through a Fourier-Laplace transform, from the Gauss-Manin system to the GKZ-system. Our results show that these data, which we express in terms of intersection cohomology groups on induced toric varieties, are purely combinatorial, and not arithmetic, in the sense that they only depend on the polytopal structure of the cone of $A$ but not on the semigroup itself. As a corollary we get a purely combinatorial formula for the length of the underlying (regular) holonomic GKZ-system, irrespective of homogeneity. In dimension up to three, and for simplicial semigroups, we give explicit generators of the weight filtration.
This article is part introduction and part survey to the mathematical area centered around local cohomology.
Let $\bf{x}$ be a random variable with density $ρ(x)$ taking values in ${\mathbb R}^d$. We are interested in finding a representation for the shape of $ρ(x)$, i.e. for the orbit $\{ ρ(g\cdot x) | g\in E(d) \}$ of $ρ$ under the Euclidean group. Let $x_1$ and $x_2$ be two random samples picked, independently, following $ρ(x)$, and let $Δ$ be the squared Euclidean distance between $x_1$ and $x_2$. We show, if $ρ(x)$ is a mixture of Gaussians whose covariance matrix is the identity, and if the means of the Gaussians are in generic position, then the density $ρ(x)$ is reconstructible, up to a rigid motion in $E(d)$, from the density of $\bfΔ$. In other words, any two such Gaussian mixtures $ρ(x)$ and $\barρ (x)$ with the same distribution of distances are guaranteed to be related by a rigid motion $g\in E(d)$ as $ρ(x)=\barρ (g\cdot x)$. We also show that a similar result holds when the distance is defined by a symmetric bilinear form.
Generalizing the star graphs of Müller-Stach and Westrich, we describe a class of graphs whose associated graph hypersurface is equipped with a non-trivial torus action. For such graphs, we show that the Euler characteristic of the corresponding projective graph hypersurface complement is zero. In contrast, we also show that the Euler characteristic in question can take any integer value for a suitable graph. This disproves a conjecture of Aluffi in a strong sense.
We discuss for an affine variety $Y$ embedded in affine space $X$ two sets of integers attached to $Y\subseteq X$ via local and de Rham cohomology spectral sequences. We give topological interpretations, study them in small dimension, and investigate to what extent one can attach them to projective varieties.
Configuration polynomials generalize the classical Kirchhoff polynomial defined by a graph. Their study sheds light on certain polynomials appearing in Feynman integrands. Contact equivalence provides a way to study the associated configuration hypersurface. In the contact equivalence class of any configuration polynomial we identify a polynomial with minimal number of variables; it is a configuration polynomial. This minimal number is bounded by $r+1\choose 2$, where $r$ is the rank of the underlying matroid. We show that the number of equivalence classes is finite exactly up to rank $3$ and list explicit normal forms for these classes.
Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen-Macaulay of codimension 3. If the matroid is 3-connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions.
For a local ring $(A,\mathfrak{m})$ of dimension $n$, we study the natural map from the Koszul cohomology module $H^n(\mathfrak{m}; A)$ to the local cohomology module $H^n_\mathfrak{m}(A)$. We prove that the injectivity of this map characterizes the Cohen-Macaulay property of the ring $A$. We also answer a question of Dutta by constructing normal rings $A$ for which this map is zero.