${\rm SO}(3)$-invariants for 4-manifolds with $b\sp +\sb 2=1$. II

Kotschick, D.; Morgan, John

doi:10.4310/jdg/1214454879

Cited by 41 publications

(80 citation statements)

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“…In the case b C 2 > 1, the Donaldson invariants are independent of the metric as long as it is generic. In the case b C 2 D 1, Kotschick and Morgan [27] showed that the Donaldson invariants depended on the metric but only through the chamber of its period point in the positive cone in H 2 .X; R/. Crossing through a wall in H 2 .X; R/ defined by a class 2 H 2 .X; Z/ adds a certain wall-crossing term ı X ;k to the Donaldson invariants.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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SO(3)–Donaldson invariants of ℂP²and mock theta functions

Malmendier

Ono

2012

Geom. Topol.

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We compute the Moore-Witten regularized u-plane integral on CP 2 , and we confirm the conjecture that it is the generating function for the SO.3/-Donaldson invariants of CP 2 . We also derive generating functions for the SO.3/-Donaldson invariants with 2N f massless monopoles using the geometry of certain rational elliptic surfaces (N f 2 f0; 2; 3; 4g), and we show that the partition function for N f D 4 is nearly modular. Our results rely heavily on the theory of mock theta functions and harmonic Maass forms (for example, see Ono [40]). 57R57

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…(18) is a topdimensional form, ie, a form of degree 8.k 1/ on the moduli space. The first four coefficients of (27) for r D 1 were determined by Gorodenzev [18]. (18) is a form on B of degree 8k when k is even and 8.k 1/ when k is odd.…”

Section: The Vanishing Locus Of the Obstructionmentioning

confidence: 99%

SO(3)–Donaldson invariants of ℂP²and mock theta functions

Malmendier

Ono

2012

Geom. Topol.

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“…The Donaldson invariants are topological invariants of X and do not depend on the metric g if b + 2 > 1. For b + 2 = 1 the Donaldson invariants are no longer independent of the metric [11]. A metric g on X determines a ray within the set of self-dual (with respect to g) harmonic two-forms H 2 (X, R)…”

Section: Donaldson Theory Of Simply Connected Four-manifoldsmentioning

confidence: 99%

Donaldson invariants of $\mathbb{C}\mathrm{P}^1 \times \mathbb{C}\mathrm{P}^1$ and mock theta functions

Malmendier

2011

Communications in Number Theory and Physics

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“…The wallcrossing of Donaldson invariants has also been studied in gauge theory (e.g. [35], [36]). There a conjecture about the structure of the wallcrossing formulas is made.…”

Section: Moduli Of Vector Bundlesmentioning

confidence: 99%

On the Motive of the Hilbert scheme of points on a surface

Göttsche

2001

Mathematical Research Letters

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The Hilbert scheme S[n] of points on an algebraic surface S is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power S (n) . For many phenomena expected for moduli spaces and nice resolutions of singular varieties it is a model case. Hilbert schemes of points have connections to several fields of mathematics, including moduli spaces of sheaves, Donaldson invariants, enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras and also to theoretical physics. This talk will try to give an overview over these connections. IntroductionThe Hilbert scheme S[n] of points on a complex projective algebraic surface S is a a parameter variety for finite subschemes of length n on S. It is a nice (crepant) resolution of singularities of the n-fold symmetric power S (n) of S. If S is a K3 surface or an abelian surface, then S[n] is a compact, holomorphic symplectic (thus hyperkähler) manifold. Thus S [n] is at the same time a basic example of a moduli space and an example of a nice resolution of singularities of a singular variety. There are a number of conjectures and general phenomena, many of which originating from theoretical physics, both about moduli spaces for objects on surfaces and about nice resolutions of singularities. In all of these the Hilbert scheme of points can be viewed as a model case and sometimes as the main motivating example. Hilbert schemes of points on a surface have connections to many topics in mathematics, including moduli spaces of sheaves and vector bundles, Donaldson invariants, Gromov-Witten invariants and enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras, noncommutative geometry and also theoretical physics.

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${\rm SO}(3)$-invariants for 4-manifolds with $b\sp +\sb 2=1$. II

Cited by 41 publications

References 5 publications

SO(3)–Donaldson invariants of ℂP²and mock theta functions

SO(3)–Donaldson invariants of ℂP²and mock theta functions

Donaldson invariants of $\mathbb{C}\mathrm{P}^1 \times \mathbb{C}\mathrm{P}^1$ and mock theta functions

On the Motive of the Hilbert scheme of points on a surface

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${\rm SO}(3)$-invariants for 4-manifolds with $b\sp +\sb 2=1$. II

Cited by 41 publications

References 5 publications

SO(3)–Donaldson invariants of ℂP2and mock theta functions

SO(3)–Donaldson invariants of ℂP2and mock theta functions

Donaldson invariants of $\mathbb{C}\mathrm{P}^1 \times \mathbb{C}\mathrm{P}^1$ and mock theta functions

On the Motive of the Hilbert scheme of points on a surface

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SO(3)–Donaldson invariants of ℂP²and mock theta functions

SO(3)–Donaldson invariants of ℂP²and mock theta functions