Lagrangian tori in A n Milnor fibres and their Floer cohomology 2 . 1 A

We study the symplectic topology of some finite algebraic quotients of the An Milnor fibre which are diffeomorphic to the rational homology balls that appear in Fintushel and Stern’s rational blowdown construction. We prove that these affine surfaces have no closed exact Lagrangian submanifolds by using the already available and deep understanding of the Fukaya category of the An Milnor fibre coming from homological mirror symmetry. On the other hand, we find Floer theoretically essential monotone Lagrangian tori, finitely covered by the monotone tori which we study in the An Milnor fibre. We conclude that these affine surfaces have non-vanishing symplectic cohomology.

[1]  Maksim Maydanskiy,et al.  The symplectic topology of some rational homology balls , 2012, 1202.5625.

[2]  P. Seidel.  Lagrangian homology spheres in (A_m) Milnor fibres , 2012, 1202.1955.

[3]  Nick Sheridan.  On the homological mirror symmetry conjecture for pairs of pants , 2010, 1012.3238.

[4]  P. Seidel,et al.  Altering symplectic manifolds by homologous recombination , 2010, 1007.3281.

[5]  B. Ozbagci,et al.  Milnor fillable contact structures are universally tight , 2010, 1005.2385.

[6]  Alexander F. Ritter.  Deformations of Symplectic Cohomology and Exact Lagrangians in ALE Spaces , 2009, 0903.0240.

[7]  H. Uehara,et al.  Stability Conditions on An-Singularities , 2010 .

[8]  Maksim Maydanskiy,et al.  Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres , 2009, 0906.2230.

[9]  O. Cornea,et al.  Rigidity and uniruling for Lagrangian submanifolds , 2008, 0808.2440.

[10]  C. Wendl.  Strongly fillable contact manifolds and $J$-holomorphic foliations , 2008, 0806.3193.

[11]  Paul Seidel,et al.  Fukaya Categories and Picard-Lefschetz Theory , 2008 .

[12]  D. Auroux.  Mirror symmetry and T-duality in the complement of an anticanonical divisor , 2007, 0706.3207.

[13]  P. Seidel.  Symplectic homology as Hochschild homology , 2006, math/0609037.

[14]  Y. Oh.  Relative Floer and quantum cohomology and the symplectic topology of Lagrangian submanifolds , 2005 .

[15]  H. Uehara,et al.  Autoequivalences of derived categories on the minimal resolutions of An-singularities on surfaces , 2004, math/0409151.

[16]  P. Seidel,et al.  A link invariant from the symplectic geometry of nilpotent slices , 2004, math/0405089.

[17]  P. Lisca.  On symplectic fillings of lens spaces , 2003, math/0312354.

[18]  Cheol-hyun Cho,et al.  Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus , 2003, math/0308224.

[19]  P. Seidel.  A long exact sequence for symplectic Floer cohomology , 2001, math/0105186.

[20]  L. Polterovich.  The Geometry of the Group of Symplectic Diffeomorphism , 2001 .

[21]  L. Lazzarini.  Existence of a somewhere injective pseudo-holomorphic disc , 2000 .

[22]  D. Salamon,et al.  Loops of Lagrangian submanifolds and pseudoholomorphic discs , 2000, math/0003079.

[23]  Richard P. Thomas,et al.  Braid group actions on derived categories of coherent sheaves , 2000, math/0001043.

[24]  Emmanuel Giroux.  Structures de contact en dimension trois et bifurcations des feuilletages de surfaces , 1999, math/9908178.

[25]  K. Honda.  On the classification of tight contact structures I , 1999, math/9910127.

[26]  K. Fukaya.  MORSE HOMOTOPY AND ITS QUANTIZATION , 1999 .

[27]  Jongil Park.  Seiberg-Witten invariants of generalised rational blow-downs , 1997, Bulletin of the Australian Mathematical Society.

[28]  Y. Oh.  Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings , 1996 .

[29]  R. Stern,et al.  Rational blowdowns of smooth 4-manifolds , 1995, alg-geom/9505018.

[30]  J. Kollár,et al.  Threefolds and deformations of surface singularities , 1988 .

[31]  M. Gromov.  Pseudo holomorphic curves in symplectic manifolds , 1985 .

[32]  J. Harer,et al.  Some homology lens spaces which bound rational homology balls. , 1981 .

[33]  J. Wahl.  Smoothings of normal surface singularities , 1981 .