arXiv subjects
Publications and source records attributed to Mohan Swaminathan.
Let $X$ be a smooth projective variety over $\mathbb{C}$ with a simple normal crossings divisor $D\subset X$. We compare the notions of stable log maps to $(X,D)$ in algebraic geometry and symplectic topology. In particular, we prove an equivalence between fine (basic) algebraic log maps and symplectic log maps, and we define the symplectic analogue of fine saturated algebraic log maps by refining the notion of log Gromov convergence.
We construct global Kuranishi charts for the moduli spaces of stable pseudoholomorphic maps to a closed symplectic manifold in all genera. This is used to prove a product formula for symplectic Gromov-Witten invariants. As a consequence we obtain a Künneth formula for quantum cohomology.
We describe an obstruction to smoothing stable maps in smooth projective varieties, which generalizes some previously known obstructions. Our obstruction comes from the non-existence of certain rational functions on the ghost components, with prescribed simple poles and residues.
We define an integer-valued virtual count of embedded pseudo-holomorphic curves of two times a primitive homology class and arbitrary genus in symplectic Calabi--Yau $3$-folds, which can be viewed as an extension of Taubes' Gromov invariant. The construction depends on a detailed study of bifurcations of moduli spaces of embedded pseudo-holomorphic curves which is partially motivated by Wendl's recent solution of Bryan--Pandharipande's super-rigidity conjecture.
Let $X$ be a smooth projective variety. Define a stable map $f:C\to X$ to be "eventually smoothable" if there is an embedding $X\hookrightarrow\mathbb{P}^N$ such that $(C,f)$ occurs as the limit of a $1$-parameter family of stable maps to $\mathbb{P}^N$ with smooth domain curves. Via an explicit deformation-theoretic construction, we produce a large class of stable maps (called "stable maps with model ghosts"), and show that they are eventually smoothable.
We establish a quantitative version of the Gromov compactness theorem for closed genus 0 pseudoholomorphic curves in the setting of a tamed almost complex manifold with bounded geometry.
We show that moduli spaces of transversely cut-out (perturbed) pseudo-holomorphic curves in an almost complex manifold carry canonical relative smooth structures ("relative to the moduli space of domain curves"). The main point is that these structures can be characterized by a universal property. The tools required are ordinary gluing analysis combined with some fundamental results from the polyfold theory of Hofer--Wysocki--Zehnder.