9,493 research outputs found
Lipschitz geometry of complex surfaces: analytic invariants and equisingularity
We prove that the outer Lipschitz geometry of a germ of a normal
complex surface singularity determines a large amount of its analytic
structure. In particular, it follows that any analytic family of normal surface
singularities with constant Lipschitz geometry is Zariski equisingular. We also
prove a strong converse for families of normal complex hypersurface
singularities in : Zariski equisingularity implies Lipschitz
triviality. So for such a family Lipschitz triviality, constant Lipschitz
geometry and Zariski equisingularity are equivalent to each other.Comment: Added a new section 10 to correct a minor gap and simplify some
argument
Global Euler obstruction and polar invariants
For an affine complex algebraic singular space Y, we define a global Euler
obstruction Eu(Y) which extends the Euler-Poincare characteristic of a
nonsingular Y. Using Lefschetz pencils, we express Eu(Y) as alternating sum of
global polar invariants.Comment: 9 pages, 1 figur
Complements of hypersurfaces, variation maps and minimal models of arrangements
We prove the minimality of the CW-complex structure for complements of
hyperplane arrangements in by using the theory of Lefschetz
pencils and results on the variation maps within a pencil of hyperplanes. This
also provides a method to compute the Betti numbers of complements of
arrangements via global polar invariants
The L\^e numbers of the square of a function and their applications
L\^e numbers were introduced by Massey with the purpose of numerically
controlling the topological properties of families of non-isolated hypersurface
singularities and describing the topology associated with a function with
non-isolated singularities. They are a generalization of the Milnor number for
isolated hypersurface singularities.
In this note the authors investigate the composite of an arbitrary
square-free f and . They get a formula for the L\^e numbers of the
composite, and consider two applications of these numbers. The first
application is concerned with the extent to which the L\^e numbers are
invariant in a family of functions which satisfy some equisingularity
condition, the second is a quick proof of a new formula for the Euler
obstruction of a hypersurface singularity. Several examples are computed using
this formula including any X defined by a function which only has transverse
D(q,p) singularities off the origin.Comment: 14 page
The elliptic genus from split flows and Donaldson-Thomas invariants
We analyze a mixed ensemble of low charge D4-D2-D0 brane states on the
quintic and show that these can be successfully enumerated using attractor flow
tree techniques and Donaldson-Thomas invariants. In this low charge regime one
needs to take into account worldsheet instanton corrections to the central
charges, which is accomplished by making use of mirror symmetry. All the
charges considered can be realized as fluxed D6-D2-D0 and anti-D6-D2-D0 pairs
which we enumerate using DT invariants. Our procedure uses the low charge
counterpart of the picture developed Denef and Moore. By establishing the
existence of flow trees numerically and refining the index factorization
scheme, we reproduce and improve some results obtained by Gaiotto, Strominger
and Yin. Our results provide appealing evidence that the strong split flow tree
conjecture holds and allows to compute exact results for an important sector of
the theory. Our refined scheme for computing indices might shed some light on
how to improve index computations for systems with larger charges.Comment: 37 pages, 12 figure
BPS Spectra, Barcodes and Walls
BPS spectra give important insights into the non-perturbative regimes of
supersymmetric theories. Often from the study of BPS states one can infer
properties of the geometrical or algebraic structures underlying such theories.
In this paper we approach this problem from the perspective of persistent
homology. Persistent homology is at the base of topological data analysis,
which aims at extracting topological features out of a set of points. We use
these techniques to investigate the topological properties which characterize
the spectra of several supersymmetric models in field and string theory. We
discuss how such features change upon crossing walls of marginal stability in a
few examples. Then we look at the topological properties of the distributions
of BPS invariants in string compactifications on compact threefolds, used to
engineer black hole microstates. Finally we discuss the interplay between
persistent homology and modularity by considering certain number theoretical
functions used to count dyons in string compactifications and by studying
equivariant elliptic genera in the context of the Mathieu moonshine
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