September 2017, 24: 87-99. doi: 10.3934/era.2017.24.010

Real orientations, real Gromov-Witten theory, and real enumerative geometry

1. 

Institut de Mathématiques de Jussieu – Paris Rive Gauche, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 5, France

2. 

Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794, USA

Supported by ERC grant STEIN-259118.
Partially supported by NSF grant DMS 1500875 and MPIM.

Received  April 04, 2017 Published  September 2017

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.

Citation: Penka Georgieva, Aleksey Zinger. Real orientations, real Gromov-Witten theory, and real enumerative geometry. Electronic Research Announcements, 2017, 24: 87-99. doi: 10.3934/era.2017.24.010
References:
[1]

I. BiswasJ. Huisman and J. Hurtubise, The moduli space of stable vector bundles over a real algebraic curve, Math. Ann., 347 (2010), 201-233. doi: 10.1007/s00208-009-0442-5.

[2]

C.-H. Cho, Counting real J-holomorphic discs and spheres in dimension four and six, J. Korean Math. Soc., 45 (2008), 1427-1442. doi: 10.4134/JKMS.2008.45.5.1427.

[3]

R. Crétois, Automorphismes réels d'un fibré et opérateurs de Cauchy-Riemann, Math. Z., 275 (2013), 453-497. doi: 10.1007/s00209-013-1143-z.

[4]

R. Crétois, Déterminant des opérateurs de Cauchy-Riemann réels et application á l'orientabilité d'espaces de modules de courbes réelles, arXiv: 1207.4771.

[5]

K. Fukaya, Y. -G. Oh, H. Ohta and K. Ono, Lagrangian Intersection Theory: Anomaly and Obstruction. Part I, AMS/IP Studies in Advanced Mathematics, 46. 1, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009.

[6]

M. Farajzadeh Tehrani, Counting genus zero real curves in symplectic manifolds, Geom. Topol., 20 (2016), 629-695. doi: 10.2140/gt.2016.20.629.

[7]

P. Georgieva, Open Gromov-Witten invariants in the presence of an anti-symplectic involution, Adv. Math., 301 (2016), 116-160. doi: 10.1016/j.aim.2016.06.009.

[8]

P. Georgieva and A. Zinger, The moduli space of maps with crosscaps: The relative signs of the natural automorphisms, J. Symplectic Geom., 14 (2016), 359-430. doi: 10.4310/JSG.2016.v14.n2.a2.

[9]

P. Georgieva and A. Zinger, Enumeration of real curves in $\mathbb{C}\mathbb{P}^{2n-1}$ and a WDVV relation for real Gromov-Witten invariants, arXiv: 1309.4079.

[10]

P. Georgieva and A. Zinger, Real Gromov-Witten theory in all genera and real enumerative geometry: Construction, arXiv: 1504.06617.

[11]

P. Georgieva and A. Zinger, Real Gromov-Witten theory in all genera and real enumerative geometry: Properties, arXiv: 1507.06633.

[12]

P. Georgieva and A. Zinger, Real Gromov-Witten theory in all genera and real enumerative geometry: Computation, arXiv: 1510.07568.

[13]

P. Georgieva and A. Zinger, On the topology of real bundle pairs over nodal symmetric surfaces, Topology Appl., 214 (2016), 109-126. doi: 10.1016/j.topol.2016.10.002.

[14]

M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.

[15]

I. ItenbergV. Kharlamov and E. Shustin, Welschinger invariant and enumeration of real rational curves, IMRN, (2003), 2639-2653. doi: 10.1155/S1073792803131352.

[16]

J. Kollár, Examples of vanishing Gromov-Witten-Welschinger invariants, J. Math. Sci. Univ. Tokyo, 22 (2015), 261-278.

[17]

M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., 164 (1994), 525-562. doi: 10.1007/BF02101490.

[18]

C. -C. Liu, Moduli of $J$-Holomorphic Curves with Lagrangian Boundary Condition and Open Gromov-Witten Invariants for an $S^1$ -Pair, arXiv: math/0210257v2, Thesis (Ph. D.), Harvard University, 2002.

[19]

J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, in Topics in Symplectic 4-Manifolds, First Int. Press Lect. Ser., I, Internat. Press, 1998, 47–83.

[20]

D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology Second edition, American Mathematical Society Colloquium Publications, 52, American Mathematical Society, Providence, RI, 2012.

[21]

S. Natanzon, Moduli of real algebraic curves and their superanalogues: Spinors and Jacobians of real curves, Russian Math. Surveys, 54 (1999), 1091-1147. doi: 10.1070/rm1999v054n06ABEH000229.

[22]

J. Niu and A. Zinger, Lower bounds for the enumerative geometry of positive-genus real curves, arXiv: 1511.02206.

[23]

J. Niu and A. Zinger, Lower bounds for the enumerative geometry of positive-genus real curves, appendix, available from the authors' websites.

[24]

R. Pandharipande, Three questions in Gromov-Witten theory, Proceedings of ICM, Beijing, 2 (2002), 503–512.

[25]

Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Differential Geom., 42 (1995), 259-367.

[26]

J. Solomon, Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions, arXiv: math/0606429.

[27]

J.-Y. Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math., 162 (2005), 195-234. doi: 10.1007/s00222-005-0445-0.

[28]

J.-Y. Welschinger, Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants, Duke Math. J., 127 (2005), 89-121. doi: 10.1215/S0012-7094-04-12713-7.

[29]

A. Zinger, A comparison theorem for Gromov-Witten invariants in the symplectic category, Adv. Math., 228 (2011), 535-574. doi: 10.1016/j.aim.2011.05.021.

show all references

References:
[1]

I. BiswasJ. Huisman and J. Hurtubise, The moduli space of stable vector bundles over a real algebraic curve, Math. Ann., 347 (2010), 201-233. doi: 10.1007/s00208-009-0442-5.

[2]

C.-H. Cho, Counting real J-holomorphic discs and spheres in dimension four and six, J. Korean Math. Soc., 45 (2008), 1427-1442. doi: 10.4134/JKMS.2008.45.5.1427.

[3]

R. Crétois, Automorphismes réels d'un fibré et opérateurs de Cauchy-Riemann, Math. Z., 275 (2013), 453-497. doi: 10.1007/s00209-013-1143-z.

[4]

R. Crétois, Déterminant des opérateurs de Cauchy-Riemann réels et application á l'orientabilité d'espaces de modules de courbes réelles, arXiv: 1207.4771.

[5]

K. Fukaya, Y. -G. Oh, H. Ohta and K. Ono, Lagrangian Intersection Theory: Anomaly and Obstruction. Part I, AMS/IP Studies in Advanced Mathematics, 46. 1, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009.

[6]

M. Farajzadeh Tehrani, Counting genus zero real curves in symplectic manifolds, Geom. Topol., 20 (2016), 629-695. doi: 10.2140/gt.2016.20.629.

[7]

P. Georgieva, Open Gromov-Witten invariants in the presence of an anti-symplectic involution, Adv. Math., 301 (2016), 116-160. doi: 10.1016/j.aim.2016.06.009.

[8]

P. Georgieva and A. Zinger, The moduli space of maps with crosscaps: The relative signs of the natural automorphisms, J. Symplectic Geom., 14 (2016), 359-430. doi: 10.4310/JSG.2016.v14.n2.a2.

[9]

P. Georgieva and A. Zinger, Enumeration of real curves in $\mathbb{C}\mathbb{P}^{2n-1}$ and a WDVV relation for real Gromov-Witten invariants, arXiv: 1309.4079.

[10]

P. Georgieva and A. Zinger, Real Gromov-Witten theory in all genera and real enumerative geometry: Construction, arXiv: 1504.06617.

[11]

P. Georgieva and A. Zinger, Real Gromov-Witten theory in all genera and real enumerative geometry: Properties, arXiv: 1507.06633.

[12]

P. Georgieva and A. Zinger, Real Gromov-Witten theory in all genera and real enumerative geometry: Computation, arXiv: 1510.07568.

[13]

P. Georgieva and A. Zinger, On the topology of real bundle pairs over nodal symmetric surfaces, Topology Appl., 214 (2016), 109-126. doi: 10.1016/j.topol.2016.10.002.

[14]

M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.

[15]

I. ItenbergV. Kharlamov and E. Shustin, Welschinger invariant and enumeration of real rational curves, IMRN, (2003), 2639-2653. doi: 10.1155/S1073792803131352.

[16]

J. Kollár, Examples of vanishing Gromov-Witten-Welschinger invariants, J. Math. Sci. Univ. Tokyo, 22 (2015), 261-278.

[17]

M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., 164 (1994), 525-562. doi: 10.1007/BF02101490.

[18]

C. -C. Liu, Moduli of $J$-Holomorphic Curves with Lagrangian Boundary Condition and Open Gromov-Witten Invariants for an $S^1$ -Pair, arXiv: math/0210257v2, Thesis (Ph. D.), Harvard University, 2002.

[19]

J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, in Topics in Symplectic 4-Manifolds, First Int. Press Lect. Ser., I, Internat. Press, 1998, 47–83.

[20]

D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology Second edition, American Mathematical Society Colloquium Publications, 52, American Mathematical Society, Providence, RI, 2012.

[21]

S. Natanzon, Moduli of real algebraic curves and their superanalogues: Spinors and Jacobians of real curves, Russian Math. Surveys, 54 (1999), 1091-1147. doi: 10.1070/rm1999v054n06ABEH000229.

[22]

J. Niu and A. Zinger, Lower bounds for the enumerative geometry of positive-genus real curves, arXiv: 1511.02206.

[23]

J. Niu and A. Zinger, Lower bounds for the enumerative geometry of positive-genus real curves, appendix, available from the authors' websites.

[24]

R. Pandharipande, Three questions in Gromov-Witten theory, Proceedings of ICM, Beijing, 2 (2002), 503–512.

[25]

Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Differential Geom., 42 (1995), 259-367.

[26]

J. Solomon, Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions, arXiv: math/0606429.

[27]

J.-Y. Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math., 162 (2005), 195-234. doi: 10.1007/s00222-005-0445-0.

[28]

J.-Y. Welschinger, Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants, Duke Math. J., 127 (2005), 89-121. doi: 10.1215/S0012-7094-04-12713-7.

[29]

A. Zinger, A comparison theorem for Gromov-Witten invariants in the symplectic category, Adv. Math., 228 (2011), 535-574. doi: 10.1016/j.aim.2011.05.021.

Table .  The extendability of the canonical orientations factoring into (13) and of the parity of the number of components of $\sum ^{\sigma }$ across the codimension 1 strata: $+$ extends, $-$ flips
(E)/(H1)(H2)/(H3)
orientation on (11) with $(V, \varphi ) = u^*(TX, \text{d}\phi)$ $+$ $+$
orientation induced by KS isomorphism $-$ $-$
orientation induced by SD isomorphism $+$ $+$
orientation on (11) with $(V, \varphi ) = (T^*\sum , (\text{d}\sigma )^*)^{\otimes2}$ $+$ $-$
parity of $|\pi_0(\sum ^{\sigma })|$ $-$ $+$
(E)/(H1)(H2)/(H3)
orientation on (11) with $(V, \varphi ) = u^*(TX, \text{d}\phi)$ $+$ $+$
orientation induced by KS isomorphism $-$ $-$
orientation induced by SD isomorphism $+$ $+$
orientation on (11) with $(V, \varphi ) = (T^*\sum , (\text{d}\sigma )^*)^{\otimes2}$ $+$ $-$
parity of $|\pi_0(\sum ^{\sigma })|$ $-$ $+$
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