9,907 research outputs found
Chern Classes of Logarithmic Vector Fields
Let be a nonsingular complex variety and a reduced effective divisor
in . In this paper we study the conditions under which the formula
is true. We prove that this
formula is equivalent to a Riemann-Roch type of formula. As a corollary, we
show that over a surface, the formula is true if and only if the Milnor number
equals the Tjurina number at each singularity of . We also show the
Rimann-Roch type of formula is true if the Jacobian scheme of is
nonsingular or a complete intersection
Chern Classes Of Logarithmic Vector Fields For Locally-Homogenous Free Divisors
Let be a nonsingular complex projective variety and a locally
quasi-homogeneous free divisor in . In this paper we study a numerical
relation between the Chern class of the sheaf of logarithmic derivations on
with respect to , and the Chern-Schwartz-MacPherson class of the complement
of in . Our result confirms a conjectural formula for these classes, at
least after push-forward to projective space; it proves the full form of the
conjecture for locally quasi-homogeneous free divisors in . The
result generalizes several previously known results. For example, it recovers a
formula of M. Mustata and H. Schenck for Chern classes for free hyperplane
arrangements. Our main tools are Riemann-Roch and the logarithmic comparison
theorem of Calderon-Moreno, Castro-Jimenez, Narvaez-Macarro, and David Mond. As
a subproduct of the main argument, we also obtain a schematic Bertini statement
for locally quasi-homogeneous divisors.Comment: To Appear in Mathematical Research Letter
Chern Classes of Logarithmic Derivations for Free Divisors with Jacobian Ideal of Linear Type
Let be a nonsingular variety defined over an algebraically closed field
of characteristic , and be a free divisor with Jacobian ideal of linear
type. We compute the Chern class of the sheaf of logarithmic derivations along
and compare it with the Chern-Schwartz-MacPherson class of the hypersurface
complement. Our result establishes a conjecture by Aluffi raised in
\cite{hyparr}.Comment: To appear in the Journal of the Mathematical Society of Japa
Stable Birational Equivalence and Geometric Chevalley-Warning
We propose a 'geometric Chevalley-Warning' conjecture, that is a motivic
extension of the Chevalley-Warning theorem in number theory. It is equivalent
to a particular case of a recent conjecture of F. Brown and O.Schnetz. In this
paper, we show the conjecture is true for linear hyperplane arrangements,
quadratic and singular cubic hypersurfaces of any dimension, and cubic surfaces
in \Pbb^3. The last section is devoted to verifying the conjecture for
certain special kinds of hypersurfaces of any dimension. As a by-product, we
obtain information on the Grothendieck classes of the affine 'Potts model'
hypersurfaces considered in \cite{aluffimarcolli1}
A cohomological interpretation of derivations on graded algebras
We trace derivations through Demazure's correspondence between a finitely
generated positively graded normal -algebras and normal projective
-varieties equipped with an ample -Cartier
-divisor . We obtain a generalized Euler sequence involving a
sheaf on whose space of global sections consists of all homogeneous
-linear derivations of and a sheaf of logarithmic derivations on .Comment: 19 page
An approach to Lagrangian specialisation through MacPherson's graph construction
Let be a holomorphic map between two complex manifolds. Assume
is flat and sans \'{e}clatement en codimension 0 (no blowup in codimension
0). We study the theory of Lagrangian specialisation for such , and prove a
Gonz\'{a}lez-Sprinberg type formula for the local Euler obstruction relative to
. With the help of this formula and MacPherson's graph construction for the
vector bundle map , we find the Lagrangian cycle of the
Milnor number constructible function . As an application, we study the
Chern class transformation of when has finite contact type
Recovery of graded index profile by cubic spline function
We present in this paper a method to recover the refractive index profile of
graded waveguide from the effective indices by cubic spline interpolation
function. It is proved by numerical analysis of several typical index
distributions that the refractive index profile can be reconstructed closely to
its exact profile with the presented interpolation model. This method can
reliably retrieve index profile of both more-mode (more than 4 guiding mode)
and fewer-mode (2-4) waveguides.Comment: 16 pages, 8 figure
Rotation-Sensitive Regression for Oriented Scene Text Detection
Text in natural images is of arbitrary orientations, requiring detection in
terms of oriented bounding boxes. Normally, a multi-oriented text detector
often involves two key tasks: 1) text presence detection, which is a
classification problem disregarding text orientation; 2) oriented bounding box
regression, which concerns about text orientation. Previous methods rely on
shared features for both tasks, resulting in degraded performance due to the
incompatibility of the two tasks. To address this issue, we propose to perform
classification and regression on features of different characteristics,
extracted by two network branches of different designs. Concretely, the
regression branch extracts rotation-sensitive features by actively rotating the
convolutional filters, while the classification branch extracts
rotation-invariant features by pooling the rotation-sensitive features. The
proposed method named Rotation-sensitive Regression Detector (RRD) achieves
state-of-the-art performance on three oriented scene text benchmark datasets,
including ICDAR 2015, MSRA-TD500, RCTW-17 and COCO-Text. Furthermore, RRD
achieves a significant improvement on a ship collection dataset, demonstrating
its generality on oriented object detection.Comment: accepted by CVPR 201
On the explicit calculation of Hirzebruch-Milnor classes of hyperplane arrangements
The Hirzebruch-Milnor class is given by the difference between the homology
Hirzebruch characteristic class and the virtual one. It is known that the
Hirzebruch-Milnor class for a certain singular hypersurface can be calculated
by using the Hodge spectrum of each stratum of singular locus. So far there is
no explicit calculation of this invariant for any non-trivial examples, and we
calculate this invariant by two different ways for low dimmensional hyperplane
arrangements.Comment: 24 page
Nonuniform dichotomy spectrum and reducibility for nonautonomous difference equations
For nonautonomous linear difference equations, we introduce the notion of the
so-called nonuniform dichotomy spectrum and prove a spectral theorem. Moreover,
we introduce the notion of weak kinematical similarity and prove a reducibility
result by the spectral theorem
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