Journal title
Virtual Fundamental Cycles in Symplectic Topology
Last updated
2024-04-10T08:35:05.89+01:00
Abstract
This is a survey of the author's in-progress book arXiv:1409.6908. 'Kuranishi
spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic
geometry (see e.g. arXiv:1503.07631), as the geometric structure on moduli
spaces of $J$-holomorphic curves. We propose a new definition of Kuranishi
space, which has the nice property that they form a 2-category $\bf Kur$. Thus
the homotopy category Ho$({\bf Kur})$ is an ordinary category of Kuranishi
spaces.
Any Fukaya-Oh-Ohta-Ono (FOOO) Kuranishi space $\bf X$ can be made into a
compact Kuranishi space $\bf X'$ uniquely up to equivalence in $\bf Kur$ (that
is, up to isomorphism in Ho$({\bf Kur})$), and conversely any compact Kuranishi
space $\bf X'$ comes from some (nonunique) FOOO Kuranishi space $\bf X$. So
FOOO Kuranishi spaces are equivalent to ours at one level, but our definition
has better categorical properties. The same holds for McDuff and Wehrheim's
'Kuranishi atlases' in arXiv:1508.01556.
Using results of Yang on polyfolds and Kuranishi spaces surveyed in
arXiv:1510.06849, a compact topological space $X$ with a 'polyfold Fredholm
structure' in the sense of Hofer, Wysocki and Zehnder (see e.g.
arXiv:1407.3185) can be made into a Kuranishi space $\bf X$ uniquely up to
equivalence in $\bf Kur$.
Our Kuranishi spaces are based on the author's theory of Derived Differential
Geometry (see e.g. arXiv:1206.4207), the study of classes of derived manifolds
and orbifolds that we call 'd-manifolds' and 'd-orbifolds'. There is an
equivalence of 2-categories ${\bf Kur}\simeq{\bf dOrb}$, where $\bf dOrb$ is
the 2-category of d-orbifolds. So Kuranishi spaces are really a form of derived
orbifold.
We discuss the differential geometry of Kuranishi spaces, and the author's
programme for applying these ideas in symplectic geometry.
spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic
geometry (see e.g. arXiv:1503.07631), as the geometric structure on moduli
spaces of $J$-holomorphic curves. We propose a new definition of Kuranishi
space, which has the nice property that they form a 2-category $\bf Kur$. Thus
the homotopy category Ho$({\bf Kur})$ is an ordinary category of Kuranishi
spaces.
Any Fukaya-Oh-Ohta-Ono (FOOO) Kuranishi space $\bf X$ can be made into a
compact Kuranishi space $\bf X'$ uniquely up to equivalence in $\bf Kur$ (that
is, up to isomorphism in Ho$({\bf Kur})$), and conversely any compact Kuranishi
space $\bf X'$ comes from some (nonunique) FOOO Kuranishi space $\bf X$. So
FOOO Kuranishi spaces are equivalent to ours at one level, but our definition
has better categorical properties. The same holds for McDuff and Wehrheim's
'Kuranishi atlases' in arXiv:1508.01556.
Using results of Yang on polyfolds and Kuranishi spaces surveyed in
arXiv:1510.06849, a compact topological space $X$ with a 'polyfold Fredholm
structure' in the sense of Hofer, Wysocki and Zehnder (see e.g.
arXiv:1407.3185) can be made into a Kuranishi space $\bf X$ uniquely up to
equivalence in $\bf Kur$.
Our Kuranishi spaces are based on the author's theory of Derived Differential
Geometry (see e.g. arXiv:1206.4207), the study of classes of derived manifolds
and orbifolds that we call 'd-manifolds' and 'd-orbifolds'. There is an
equivalence of 2-categories ${\bf Kur}\simeq{\bf dOrb}$, where $\bf dOrb$ is
the 2-category of d-orbifolds. So Kuranishi spaces are really a form of derived
orbifold.
We discuss the differential geometry of Kuranishi spaces, and the author's
programme for applying these ideas in symplectic geometry.
Symplectic ID
572431
Download URL
http://arxiv.org/abs/1510.07444v1
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Publication type
Chapter
Publication date
01 Mar 2019