arXiv subjects
Publications and source records attributed to Soham Chanda.
Adapting the construction of global Kuranishi charts to the contact setting, we associate to any non-degenerate contact manifold a flow category based on Reeb orbits and moduli spaces of pseudo-holomorphic buildings. The construction lifts contact homology and is natural in the sense that to any exact symplectic cobordism we can associate a flow bimodule between the flow categories of its ends.
We construct a new surgery type operation by switching between two exact fillings of Legendrians which we call a BSP surgery. In certain cases, this surgery can preserve monotonicity of Lagrangians. We prove a wall-crossing type formula for the change of the disk-potential under surgery with Bohr-Sommerfeld profiles. As an application, we show that Biran's circle-bundle lifts admit a Bohr-Sommerfeld type surgery. We use the wall-crossing theorem about disk-potentials to construct exotic monotone Lagrangian tori in $\bP^n$.
This is the second in a sequence of papers in which we construct Chekanov-Eliashberg algebras for Legendrians in circle-fibered contact manifolds and study the associated augmentation varieties. In this part, we first define the Chekanov-Eliashberg algebra and its Legendrian contact homology. For a tame Lagrangian cobordism between Legendrians, we define a chain map between their Chekanov-Eliashberg algebras.
This is the third in a series of papers in which we construct Chekanov-Eliashberg algebras for Legendrians in circle-fibered contact manifolds and study the associated augmentation varieties. In this part, we prove that for connected Legendrian covers of monotone Lagrangian tori, the augmentation variety is equal to the image of the zero level set of the disk potential, as suggested by Dimitroglou-Rizell-Golovko. In particular, we show that Legendrian lifts of Vianna's exotic tori are not Legendrian isotopic. Using related ideas, we show that the Legendrian lift of the Clifford torus admits no exact fillings, extending results of Dimitroglou-Rizell and Treumann-Zaslow in dimension two. We consider certain disconnected Legendrians, and show, similar to another suggestion of Aganagic-Ekholm-Ng-Vafa that the components of the augmentation variety correspond to certain partitions and each component is defined by a (not necessarily exact) Lagrangian filling.
This is the first in a sequence of papers where we show that Lagrangian fillings such as the Harvey-Lawson filling in any dimension define augmentations of Chekanov-Eliashberg differential graded algebras by counting configurations of holomorphic disks connected by gradient trajectories, as in Aganagic-Ekholm-Ng-Vafa; we also prove that for Legendrian lifts of monotone tori, the augmentation variety is the zero level set of the Landau-Ginzburg potential of the Lagrangian projection, as suggested by Dimitroglou-Rizell-Golovko. In this part, we set up the foundations of moduli spaces of pseudoholomorphic buildings.
Vianna constructed infinitely many exotic Lagrangian tori in the complex projective plane. We lift these tori to higher-dimensional projective spaces and show that they remain non-symplectomorphic. Our proof is elementary except for an application of the wall-crossing formula by Pascaleff-Tonkonog.
We extend the construction of higher mutation as introduced in Pascaleff-Tonkonog to local higher mutation, which is applicable to a larger class of monotone Lagrangians. In two-dimensional Lagrangians, local higher mutation is the same as performing a Lagrangian anti-surgery in the sense of Haug followed by a Lagrangian surgery. We prove that up to a change of local systems, the Lagrangian intersection Floer cohomology of a pair of Lagrangians is invariant under local mutation. This result generalizes the wall-crossing formula in Pascaleff-Tonkonog. For two-dimensional Lagrangians, this result agrees with the invariance result in Palmer-Woodward.