Infinitely many monotone Lagrangian Tori in CP^2

28 October 2014
Renato Vianna
In previous work, we constructed an exotic monotone Lagrangian torus in $\mathbb{CP}^2$ (not Hamiltonian isotopic to the known Clifford and Chekanov tori) using techniques motivated by mirror symmetry. We named it $T(1,4,25)$ because, when following a degeneration of $\mathbb{CP}^2$ to the weighted projective space $\mathbb{CP}(1,4,25)$, it degenerates to the central fibre of the moment map for the standard torus action on $\mathbb{CP}(1,4,25)$. Related to each degeneration from $\mathbb{CP}^2$ to $\mathbb{CP}(a^2,b^2,c^2)$, for $(a,b,c)$ a Markov triple -- $a^2 + b^2 + c^2 = 3abc$ -- there is a monotone Lagrangian torus, which we call $T(a^2,b^2,c^2)$.  We employ techniques from symplectic field theory to prove that no two of them are Hamiltonian isotopic to each other.
  • Algebraic and Symplectic Geometry Seminar