## Geometry & Topology

### Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties

Alexander Ritter

#### Abstract

We define a class of noncompact Fano toric manifolds which we call admissible toric manifolds, for which Floer theory and quantum cohomology are defined. The class includes Fano toric negative line bundles, and it allows blow-ups along fixed point sets.

We prove closed-string mirror symmetry for this class of manifolds: the Jacobian ring of the superpotential is the symplectic cohomology (not the quantum cohomology). Moreover, $SH∗(M)$ is obtained from $QH∗(M)$ by localizing at the toric divisors. We give explicit presentations of $SH∗(M)$ and $QH∗(M)$, using ideas of Batyrev, McDuff and Tolman.

Assuming that the superpotential is Morse (or a milder semisimplicity assumption), we prove that the wrapped Fukaya category for this class of manifolds satisfies the toric generation criterion, ie is split-generated by the natural Lagrangian torus fibers of the moment map taken with suitable holonomies. In particular, the wrapped category is compactly generated and cohomologically finite.

We prove a generic generation theorem: a generic deformation of the monotone toric symplectic form defines a local system for which the twisted wrapped Fukaya category satisfies the toric generation criterion. This theorem, together with a limiting argument about continuity of eigenspaces, are used to prove the untwisted generation results.

We prove that for any closed Fano toric manifold, and a generic local system, the twisted Fukaya category satisfies the toric generation criterion. If the superpotential is Morse (or assuming semisimplicity), also the untwisted Fukaya category satisfies the criterion.

The key ingredients are nonvanishing results for the open-closed string map, using tools from the paper by Ritter and Smith; we also prove a conjecture from that paper that any monotone toric negative line bundle contains a nondisplaceable monotone Lagrangian torus. The above presentation results require foundational work: we extend the class of Hamiltonians for which the maximum principle holds for symplectic manifolds conical at infinity, thus extending the class of Hamiltonian circle actions for which invertible elements can be constructed in $SH∗(M)$. Computing $SH∗(M)$ is notoriously hard and there are very few known examples beyond the cases of cotangent bundles and subcritical Stein manifolds. So this computation is significant in itself, as well as being the key ingredient in proving the above results in homological mirror symmetry.

#### Article information

Source
Geom. Topol., Volume 20, Number 4 (2016), 1941-2052.

Dates
Revised: 16 July 2015
Accepted: 25 August 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859020

Digital Object Identifier
doi:10.2140/gt.2016.20.1941

Mathematical Reviews number (MathSciNet)
MR3548462

Zentralblatt MATH identifier
1348.53076

#### Citation

Ritter, Alexander. Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties. Geom. Topol. 20 (2016), no. 4, 1941--2052. doi:10.2140/gt.2016.20.1941. https://projecteuclid.org/euclid.gt/1510859020

#### References

• M Abouzaid, A geometric criterion for generating the Fukaya category, Publ. Math. Inst. Hautes Études Sci. 112 (2010) 191–240
• P Albers, A Lagrangian Piunikhin–Salamon–Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not. 2008 (2008) Art. rnm134
• D Auroux, Mirror symmetry and $T$–duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT 1 (2007) 51–91
• D Auroux, Special Lagrangian fibrations, wall-crossing, and mirror symmetry, from: “Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry”, (H-D Cao, S-T Yau, editors), Surv. Differ. Geom. 13, Int. Press, Somerville, MA (2009) 1–47
• V V Batyrev, Quantum cohomology rings of toric manifolds, from: “Journées de Géométrie Algébrique d'Orsay”, Astérisque 218, Société Mathématique de France, Paris (1993) 9–34
• V V Batyrev, On the classification of toric Fano $4$–folds, J. Math. Sci. $($New York$)$ 94 (1999) 1021–1050
• P Biran, O Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009) 2881–2989
• C-H Cho, Products of Floer cohomology of torus fibers in toric Fano manifolds, Comm. Math. Phys. 260 (2005) 613–640
• C-H Cho, Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle, J. Geom. Phys. 58 (2008) 1465–1476
• C-H Cho, Y-G Oh, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 (2006) 773–814
• K Cieliebak, D Salamon, Wall crossing for symplectic vortices and quantum cohomology, Math. Ann. 335 (2006) 133–192
• D A Cox, S Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs 68, Amer. Math. Soc., Providence, RI (1999)
• T Dyckerhoff, Compact generators in categories of matrix factorizations, Duke Math. J. 159 (2011) 223–274
• L C Evans, Partial differential equations, Graduate Studies in Mathematics 19, Amer. Math. Soc., Providence, RI (1998)
• K Fukaya, Y-G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory: anomaly and obstruction, I, II, AMS/IP Studies in Advanced Mathematics 46, Amer. Math. Soc., Providence, RI (2009)
• K Fukaya, Y-G Oh, H Ohta, K Ono, Lagrangian Floer theory on compact toric manifolds, I, Duke Math. J. 151 (2010) 23–174
• W Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton Univ. Press (1993)
• S Galkin, The conifold point
• S Ganatra, Symplectic cohomology and duality for the wrapped Fukaya category, PhD thesis, Massachusetts Institute of Technology (2012) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/1238001248 {\unhbox0
• A B Givental, Equivariant Gromov–Witten invariants, Int. Math. Res. Notices (1996) 613–663
• A Givental, A mirror theorem for toric complete intersections, from: “Topological field theory, primitive forms and related topics”, (M Kashiwara, A Matsuo, K Saito, I Satake, editors), Progr. Math. 160, Birkhäuser, Boston, MA (1998) 141–175
• P Griffiths, J Harris, Principles of algebraic geometry, Wiley-Interscience, New York (1978)
• V Guillemin, Moment maps and combinatorial invariants of Hamiltonian $T\sp n$–spaces, Progress in Mathematics 122, Birkhäuser, Boston, MA (1994)
• H Hofer, D A Salamon, Floer homology and Novikov rings, from: “The Floer memorial volume”, (H Hofer, C H Taubes, A Weinstein, E Zehnder, editors), Progr. Math. 133, Birkhäuser, Basel (1995) 483–524
• H Iritani, Convergence of quantum cohomology by quantum Lefschetz, J. Reine Angew. Math. 610 (2007) 29–69
• T Kato, Perturbation theory for linear operators, corrected 2nd edition, Grundl. Math. Wissen. 132, Springer, Berlin (1980)
• J Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. 32, Springer, Berlin (1996)
• Y-P Lee, Quantum Lefschetz hyperplane theorem, Invent. Math. 145 (2001) 121–149
• D McDuff, D Salamon, $J$–holomorphic curves and symplectic topology, AMS Colloquium Publications 52, Amer. Math. Soc., Providence, RI (2004)
• D McDuff, S Tolman, Topological properties of Hamiltonian circle actions, Int. Math. Res. Pap. (2006) Art. ID 72826, 1–77
• D McDuff, S Tolman, Polytopes with mass linear functions, II: The four-dimensional case, Int. Math. Res. Not. 2013 (2013) 3509–3599
• Y Ostrover, I Tyomkin, On the quantum homology algebra of toric Fano manifolds, Selecta Math. 15 (2009) 121–149
• A F Ritter, Novikov-symplectic cohomology and exact Lagrangian embeddings, Geom. Topol. 13 (2009) 943–978
• A F Ritter, Topological quantum field theory structure on symplectic cohomology, J. Topol. 6 (2013) 391–489
• A F Ritter, Floer theory for negative line bundles via Gromov–Witten invariants, Adv. Math. 262 (2014) 1035–1106
• A F Ritter, I Smith, The open-closed string map revisited
• A F Ritter, I Smith, The monotone wrapped Fukaya category and the open-closed string map (2015)
• P Seidel, $\pi\sb 1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997) 1046–1095
• P Seidel, Fukaya categories and deformations, from: “Proceedings of the International Congress of Mathematicians, II”, (T Li, editor), Higher Ed. Press, Beijing (2002) 351–360
• P Seidel, A biased view of symplectic cohomology, from: “Current developments in mathematics, 2006”, (B Mazur, T Mrowka, W Schmid, R Stanley, S-T Yau, editors), Int. Press, Somerville, MA (2008) 211–253
• C Viterbo, Functors and computations in Floer homology with applications, I, Geom. Funct. Anal. 9 (1999) 985–1033