Abstract
The present note overviews our recent construction of real Gromov-Witten theory in arbitrary genera for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold, its properties, and its connections with real enumerative geometry. Our construction introduces the principle of orienting the determinant of a differential operator relative to a suitable base operator and a real setting analogue of the (relative) spin structure of open Gromov-Witten theory. Orienting the relative determinant, which in the now-standard cases is canonically equivalent to orienting the usual determinant, is naturally related to the topology of vector bundles in the relevant category. This principle and its applications allow us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces, thus implementing a far-reaching proposal from C.-C. Liu's thesis.
Highlights
The present note overviews our recent construction of real GromovWitten theory in arbitrary genera for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold, its properties, and its connections with real enumerative geometry
The progress in establishing the foundations of real GW-theory, i.e., of counts of J-holomorphic curves in symplectic manifolds preserved by anti-symplectic involutions, has been much slower: it did not exist in positive genera until [10]
C 2016 American Institute of Mathematical Sciences form, the coordinate conjugation τn : Pn1 ÝÑ Pn1, τnrZ1, . . . , Zns “ “Z1, . . . , Zn‰, and the standard complex structure. Another important example is a real quintic threefold X5, i.e., a smooth hypersurface in P4 cut out by a real equation; it plays a prominent role in the interactions with string theory and algebraic geometry
Summary
The study of curves in projective varieties has been central to algebraic geometry since the nineteenth century It was reinvigorated through its introduction into symplectic topology in [14] and plays prominent roles in symplectic topology and string theory as well. Another important example is a real quintic threefold X5, i.e., a smooth hypersurface in P4 cut out by a real equation; it plays a prominent role in the interactions with string theory and algebraic geometry. (H1) x12 is a non-isolated real node and the topological component Σσ12 of Σσ containing x12 is algebraically irreducible (the normalization Σrσ1r2 of Σσ12 is connected);. (H2) x12 is a non-isolated real node and the topological component Σσ12 of Σσ containing x12 is algebraically reducible, but Σ is algebraically irreducible (the normalization Σrσ1r2 of Σ12 is disconnected, but the normalization Σr of Σ is connected);. The one-nodal symmetric surfaces can be smoothed out in one-parameter families to symmetric surfaces, typically of different involution types for the two directions of smoothings (the smoothings of (H3) are always of the same type though)
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