80 research outputs found
Log BPS numbers of log Calabi-Yau surfaces
Let be a log Calabi-Yau surface pair with a smooth divisor. We
define new conjecturally integer-valued counts of -curves in
. These log BPS numbers are derived from genus 0 log Gromov-Witten
invariants of maximal tangency along via a formula analogous to the
multiple cover formula for disk counts. A conjectural relationship to genus 0
local BPS numbers is described and verified for del Pezzo surfaces and curve
classes of arithmetic genus up to 2. We state a number of conjectures and
provide computational evidence.Comment: 49 pages, 2 figure
Birational cobordism invariance of uniruled symplectic manifolds
A symplectic manifold is called {\em (symplectically) uniruled}
if there is a nonzero genus zero GW invariant involving a point constraint. We
prove that symplectic uniruledness is invariant under symplectic blow-up and
blow-down. This theorem follows from a general Relative/Absolute correspondence
for a symplectic manifold together with a symplectic submanifold. A direct
consequence is that symplectic uniruledness is a symplectic birational
invariant. Here we use Guillemin and Sternberg's notion of cobordism as the
symplectic analogue of the birational equivalence.Comment: To appear in Invent. Mat
Log BPS numbers of log Calabi-Yau surfaces
Let be a log Calabi-Yau surface pair with a smooth divisor. We define new conjecturally integer-valued counts of -curves in . These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence
Pencils of quadrics and Gromov-Witten-Welschinger invariants of
We establish a formula for the Gromov-Witten-Welschinger invariants of
with mixed real and conjugate point constraints. The method is
based on a suggestion by J. Koll\'ar that, considering pencils of quadrics,
some real and complex enumerative invariants of could be
computed in terms of enumerative invariants of
and of elliptic curves.Comment: 14 pages, 4 figures, minor corrections following referee's
suggestion
Reconstruction theorems for genus 2 Gromov-Witten invariants
We use Pixton's relations to prove a reconstruction theorem for genus 2
Gromov-Witten invariants in the style of Kontsevich-Manin (genus 0) and Getzler
(genus 1). We also calculate genus 2 (descendant) Gromov-Witten invariants of
blown up at a finite number of points in general position.Comment: 28 page
Surgery of real symplectic fourfolds and Welschinger invariants
A surgery of a real symplectic manifold along a real
Lagrangian sphere is a modification of the symplectic and real structure on
in a neigborhood of . Genus 0 Welschinger invariants of two
real symplectic -manifolds differing by such a surgery have been related in
a previous work in collaboration with N. Puignau. In the present paper, we
explore some particular situations where these general formulas greatly
simplify. As an application, we complete the computation of genus 0 Welschinger
invariants of all del~Pezzo surfaces, and of all -minimal real conic
bundles. As a by-product, we establish the existence of some new relative
Welschinger invariants. We also generalize our results to the enumeration of
curves of higher genus, and give relations between hypothetical invariants
defined in the same vein as a previous work by Shustin.Comment: 28 pages, 2 figures. V2: Major edition (hopefully simplifications) of
the first version, references precised. V3: Minor edition
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