332 research outputs found
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 to the case where
the normal bundle is nontrivial and 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
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
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