Hypergraph Learning with Line Expansion

Previous hypergraph expansions are solely carried out on either vertex level or hyperedge level, thereby missing the symmetric nature of data co-occurrence, and resulting in information loss. To address the problem, this paper treats vertices and hyperedges equally and proposes a new hypergraph formulation named the \emph{line expansion (LE)} for hypergraphs learning. The new expansion bijectively induces a homogeneous structure from the hypergraph by treating vertex-hyperedge pairs as "line nodes". By reducing the hypergraph to a simple graph, the proposed \emph{line expansion} makes existing graph learning algorithms compatible with the higher-order structure and has been proven as a unifying framework for various hypergraph expansions. We evaluate the proposed line expansion on five hypergraph datasets, the results show that our method beats SOTA baselines by a significant margin

Higher multiplier ideals

We associate a family of ideal sheaves to any Q-effective divisor on a complex manifold, called the higher multiplier ideals, using the theory of mixed Hodge modules and V-filtrations. This family is indexed by two parameters, an integer indicating the Hodge level and a rational number, and these ideals admit a weight filtration. When the Hodge level is zero, they recover the usual multiplier ideals. We study the local and global properties of higher multiplier ideals systematically. In particular, we prove vanishing theorems and restriction theorems, and provide criteria for the nontriviality of the new ideals. The main idea is to exploit the global structure of the V-filtration along an effective divisor using the notion of twisted Hodge modules. In the local theory, we introduce the notion of the center of minimal exponent, which generalizes the notion of minimal log canonical center. As applications, we prove some cases of conjectures by Debarre, Casalaina-Martin and Grushevsky on singularities of theta divisors on principally polarized abelian varieties and the geometric Riemann-Schottky problem.Comment: 95 pages, comments are welcome

Hodge modules and Singular Hermitian Metrics

The purpose of this paper is to study certain notions of metric positivity for the lowest nonzero piece in the Hodge filtration of a Hodge module. We show that the Hodge metric satisfies the minimal extension property. In particular, this singular Hermitian metric has semi-positive curvature.Comment: 14 pages, comments are very welcome

Homological criterion for higher Du Bois and higher rational singularities

In this note, we give a homological characterization of higher Du Bois and higher rational hypersurface singularities, which were recently introduced as a natural generalization of Du Bois and rational singularities. As a key input, we characterize these singularities using the Hodge filtration on the vanishing cycle mixed Hodge module.Comment: 18 pages, comments are welcome

A log resolution for the theta divisor of a hyperelliptic curve

In this paper, we prove that the theta divisor of a smooth hyperelliptic curve has a natural and explicit embedded resolution of singularities using iterated blowups of Brill-Noether subvarieties. We also show that the Brill-Noether stratification of the hyperelliptic Jacobian is a Whitney stratification.Comment: 32 pages, comments are welcome. v2: a new section about open problems is added; v3: some typos are corrected; v4: We add an appendix with the proof of the reducedness of Brill-Noether varieties of hyperelliptic curve

Hirzebruch-Milnor classes of hypersurfaces with nontrivial normal bundles and applications to higher du Bois and rational singularities

We extend the Hirzebruch-Milnor class of a hypersurface $X$ to the case where the normal bundle is nontrivial and $X$ cannot be defined by a global function, using the associated line bundle and the graded quotients of the monodromy filtration. The earlier definition requiring a global defining function of $X$ can be applied rarely to projective hypersurfaces with non-isolated singularities. Indeed, it is surprisingly difficult to get a one-parameter smoothing with total space smooth without destroying the singularities by blowing-ups (except certain quite special cases). As an application, assuming the singular locus is a projective variety, we show that the minimal exponent of a hypersurface can be captured by the spectral Hirzebruch-Milnor class, and higher du Bois and rational singularities of a hypersurface are detectable by the unnormalized Hirzebruch-Milnor class. Here the unnormalized class can be replaced by the normalized one in the higher du Bois case, but for the higher rational case, we must use also the decomposition of the Hirzebruch-Milnor class by the action of the semisimple part of the monodromy (which is equivalent to the spectral Hirzebruch-Milnor class).Comment: this paper supersedes the earlier 2-authored pape