229,173 research outputs found
On the expected number of perfect matchings in cubic planar graphs
A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that
a bridgeless cubic graph has exponentially many perfect matchings. It was
solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other
hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the
special case of cubic planar graphs. In our work we consider random bridgeless
cubic planar graphs with the uniform distribution on graphs with vertices.
Under this model we show that the expected number of perfect matchings in
labeled bridgeless cubic planar graphs is asymptotically , where
and is an explicit algebraic number. We also
compute the expected number of perfect matchings in (non necessarily
bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs.
Our starting point is a correspondence between counting perfect matchings in
rooted cubic planar maps and the partition function of the Ising model in
rooted triangulations.Comment: 19 pages, 4 figure
Curves on K3 surfaces and modular forms
We study the virtual geometry of the moduli spaces of curves and sheaves on
K3 surfaces in primitive classes. Equivalences relating the reduced
Gromov-Witten invariants of K3 surfaces to characteristic numbers of stable
pairs moduli spaces are proven. As a consequence, we prove the Katz-Klemm-Vafa
conjecture evaluating integrals (in all genera) in terms of
explicit modular forms. Indeed, all K3 invariants in primitive classes are
shown to be governed by modular forms.
The method of proof is by degeneration to elliptically fibered rational
surfaces. New formulas relating reduced virtual classes on K3 surfaces to
standard virtual classes after degeneration are needed for both maps and
sheaves. We also prove a Gromov-Witten/Pairs correspondence for toric 3-folds.
Our approach uses a result of Kiem and Li to produce reduced classes. In
Appendix A, we answer a number of questions about the relationship between the
Kiem-Li approach, traditional virtual cycles, and symmetric obstruction
theories.
The interplay between the boundary geometry of the moduli spaces of curves,
K3 surfaces, and modular forms is explored in Appendix B by A. Pixton.Comment: An incorrect example in Appendix A, pointed out to us by Dominic
Joyce, has been replaced by a reference to a new paper arXiv:1204.3958
containing a corrected exampl
Gromov-Witten invariants of blow-ups along submanifolds with convex normal bundles
Given a submanifold Z inside X, let Y be the blow-up of X along Z. When the
normal bundle of Z in X is convex with a minor assumption, we prove that
genus-zero GW-invariants of Y with cohomology insertions from X, are identical
to GW-invariants of X. Under the same hypothesis, a vanishing theorem is also
proved. An example to which these two theorems apply is when the normal bundle
is generated by global sections. These two main theorems do not hold for
arbitrary blow-ups, and counter-examples are included.Comment: 34 page
Virtual push-forwards
Let be a morphism of stacks of positive \emph{virtual} relative
dimension and let . We give sufficient conditions for
to be a multiple of . We apply this
result to show an analogue of the conservation of number for virtually smooth
families. We show implications to Gromov-Witten invariants and give a new proof
of a theorem of Marian, Oprea and Pandharipande which compares the virtual
classes of moduli spaces of stable maps and moduli spaces of stable quotients
Witt groups of sheaves on topological spaces
This paper investigates the Witt groups of triangulated categories of sheaves
(of modules over a ring R in which 2 is invertible) equipped with
Poincare-Verdier duality. We consider two main cases, that of perfect complexes
of sheaves on locally compact Hausdorff spaces and that of cohomologically
constructible complexes of sheaves on polyhedra. We show that the Witt groups
of the latter form a generalised homology theory for polyhedra and continuous
maps. Under certain restrictions on the ring R, we identify the constructible
Witt groups of a finite simplicial complex with Ranicki's free symmetric
L-groups. Witt spaces are the natural class of spaces for which the rational
intersection homology groups have Poincare duality. When the ring R is the
rationals we show that every Witt space has a natural L-theory, or Witt,
orientation and we identify the constructible Witt groups with the 4-periodic
colimit of the bordism groups of Witt spaces. This allows us to interpret
Goresky and Macpherson's L-classes of singular spaces as stable homology
operations from the constructible Witt groups to rational homology.Comment: 38 pages, reformatted, minor corrections and changes as suggested by
referee. To appear in Commentarii Mathematici Helvetici no. 8
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