1,720,988 research outputs found
Joyce-Song wall-crossing as an asymptotic expansion
Full text linkWe conjecture that the Joyce-Song wall-crossing formula for Donaldson-Thomas invariants arises naturally from an asymptotic expansion in the field-theoretic work of Gaiotto, Moore, and Neitzke. This would also give a new perspective on how the formulae of Joyce and Song and of Kontsevich and Soibelman are related. We check the conjecture in many examples
Unstable blowups
Full text linkLet (X,L) be a polarised manifold. We show that K-stability and as- ymptotic Chow stability of the blowup of X along a 0-dimensional cycle are closely related to Chow stability of the cycle itself, for polarisations making the exceptional divisors small. This can be used to give (almost) a converse to the results of Arezzo and Pacard (2004 and 2007) and to give new examples of K ̈ahler classes with no constant scalar curvature representatives
K-stability of constant scalar curvature Kähler manifolds
Full text linkAbstractWe show that a polarised manifold with a constant scalar curvature Kähler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S.K. Donaldson
D0-D6 states counting and GW invariants
Full text linkWe describe a correspondence between the Donaldson–Thomas invariants enumerating D0–D6 bound states on a Calabi–Yau 3-fold and certain Gromov–Witten invariants counting rational curves in a family of blowups of weighted projective planes. This is a variation on a correspondence found by Gross–Pandharipande, with D0–D6 bound states replacing representations of generalised Kronecker quivers. We build on a small part of the theories developed by Joyce–Song and Kontsevich–Soibelman for wall-crossing formulae and by Gross–Pandharipande–Siebert for factorisations in the tropical vertex group. Along the way we write down an explicit formula for the BPS state counts which arise up to rank 3 and prove their integrality. We also compare with previous “noncommutative DT invariants” computations in the physics literature
Twisted constant scalar curvature Kahler metrics and Kahler slope stability
No full textOn a compact Ka ̈hler manifold we introduce a cohomological obstruction to the solvability of the constant scalar curvature (cscK) equation twisted by a semipositive form, appearing in works of Fine and Song-Tian.
As a special case we find an obstruction for a manifold to be the base of a holomorphic submersion carrying a cscK metric in certain “adiabatic” classes. We apply this to find new examples of general type threefolds with classes which do not admit a cscK representative.
When the twist vanishes our obstruction extends the slope stability of Ross-Thomas to effective divisors on a Kahler manifold. Thus we find examples of non-projective slope unstable manifolds
The HcscK equations in symplectic coordinates
No full textThe Donaldson–Fujiki Kähler reduction of the space of compatible almost complex structures, leading to the interpretation of the scalar curvature of Kähler metrics as a moment map, can be lifted canonically to a hyperkähler reduction. Donaldson proposed to consider the corresponding vanishing moment map conditions as (fully nonlinear) analogues of Hitchin’s equations, for which the underlying bundle is replaced by a polarised manifold. However this construction is well understood only in the case of complex curves. In this paper we study Donaldson’s hyperkähler reduction on abelian varieties and toric manifolds. We obtain a decoupling result, a variational characterisation, a relation to K-stability in the toric case, and prove existence and uniqueness under suitable assumptions on the “Higgs tensor”. We also discuss some aspects of the analogy with Higgs bundles
Log Calabi-Yau surfaces and Jeffrey-Kirwan residues
No full textWe prove an equality, predicted in the physical literature, between the
Jeffrey-Kirwan residues of certain explicit meromorphic forms attached to a
quiver without loops or oriented cycles and its Donaldson-Thomas type
invariants. In the special case of complete bipartite quivers we also show
independently, using scattering diagrams and theta functions, that the same
Jeffrey-Kirwan residues are determined by the the Gross-Hacking-Keel mirror
family to a log Calabi-Yau surface.Comment: 43 pages, accepted versio
Universal covers and the GW/Kronecker correspondence
Full text linkThe tropical vertex is an incarnation of mirror symmetry found by Gross, Pandharipande and Siebert. It can be applied to m-Kronecker quivers K(m) (together with a result of Reineke) to compute the Euler characteristics of the moduli spaces of their (framed) representations in terms of Gromov–Witten invariants (as shown by Gross and Pandharipande). In this paper, we study a possible geometric picture behind this correspondence, in particular constructing rational tropical curves from subquivers of the universal covering quiver of K(m). Additional motivation comes from the physical interpretation of m-Kronecker quivers in the context of quiver quantum mechanics (especially, work of Denef)
A simple limit for slope instability
No full textRoss and Thomas have shown that subschemes can K-destabilise polarised varieties, yielding a notion known as slope (in)stability for varieties. Here we describe a special situation in which slope instability for varieties (for example of general type) corresponds to a slope instability type condition for certain bundles, making the computations almost trivial
Relative K-stability of extremal metrics
No full textWe show that if a polarised manifold admits an extremal metric then it is K-polystable relative to a maximal torus of automorphisms
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