arXiv subjects
Publications and source records attributed to Sushmita Venugopalan.
Hofer-Wysocki-Zehnder and Bourgeois proved that a finite energy punctured pseudoholomorphic curve in the symplectization of a Morse-Bott contact manifold either has a removable singularity or asymptotes to a Reeb orbit. We give an alternate proof of this result.
In previous work, we introduced a version of the Fukaya algebra associated to a degeneration of a symplectic manifold, whose structure maps count collections of maps in the components of the degeneration satisfying matching conditions. In this paper, we introduce a further degeneration of the matching conditions (similar in spirit to Bourgeois' version of symplectic field theory) which results in a "split Fukaya algebra" whose structure maps are, in good cases, sums of products over vertices of tropical graphs. In the case of toric Lagrangians contained in a toric component of the degeneration, an invariance argument implies the existence of projective Maurer-Cartan solutions, which gives an alternate proof of the unobstructedness result of Fukaya-Oh-Ohta-Ono for toric manifolds. Our result also proves unobstructedness in more general cases, such as for toric Lagrangians in almost toric four-manifolds.
We compute the Fukaya category of the symplectic blowup of a compact rational symplectic manifold at a point in the following sense: Suppose a collection of Lagrangian branes satisfy Abouzaid's criterion for split-generation of a bulk-deformed Fukaya category of cleanly-intersecting Lagrangian branes. We show that for a small blow-up parameter, their inverse images in the blowup together with a collection of branes near the exceptional locus split-generate the Fukaya category of the blowup. This categorifies a result on quantum cohomology by Bayer and is an example of a more general conjectural description of the behavior of the Fukaya category under transitions occuring in the minimal model program, namely that mmp transitions generate additional summands.
We introduce a tropical version of the Fukaya algebra of a Lagrangian submanifold. Tropical graphs arise as large-scale behavior of pseudoholomorphic disks under a multiple cut operation on a symplectic manifold that produces a collection of cut spaces each containing relative normal crossing divisors, following works of Ionel and Brett Parker. Given a Lagrangian submanifold in the complement of the relative divisors in one of the cut spaces, the structure maps of the broken Fukaya algebra count broken disks associated to rigid tropical graphs. We apply the results to give various computations of potentials, such as those of Lagrangians in cubic surfaces and flag varieties.
Novikov's theorem is a rigidity result on the class of taut foliations on three-manifolds. For higher dimensional manifolds, foliations with a strong symplectic form have been suggested as the class of foliations having similar rigidity properties to taut foliations on three-manifolds. This leads to the natural question of whether strong symplectic foliations satisfy an analogue of Novikov's theorem. In this paper, we construct a five-dimensional manifold with a smooth foliation and a strong symplectic form that does not satisfy the expected analogue of Novikov's theorem. Our example is a foliated Lefschetz fibration.
We classify symplectically foliated fillings of certain foliated manifolds with a contact structure on the leaves. We show that for the foliated sphere cotangent bundle of the Reeb foliation on the three-sphere, the corresponding foliated disk cotangent bundle is the unique strong symplectic foliated filling up to blowups and symplectic deformation equivalence. En route to the proof, we study another foliated manifold, namely the product of a circle and an annulus with an almost horizontal foliation. In this case, the foliated filling of the foliated sphere cotangent bundle is not unique. We show that any such filling is a foliated Lefschetz fibration, and is determined up to symplectic deformation equivalence, by combinatorial invariants arising from the singular locus of the Lefschetz fibration.
We show that the moduli space of regular affine vortices, which are solutions of the symplectic vortex equation over the complex plane, has the structure of a smooth manifold. The construction uses Ziltener's Fredholm theory results [31]. We also extend the result to the case of affine vortices over the upper half plane. These results are necessary ingredients in defining the "open quantum Kirwan map" proposed by Woodward [24].
We study the gradient flow lines of a Yang-Mills-type functional on the space of gauged holomorphic maps $\mathcal{H}(P,X)$, where $P$ is a principal bundle on a Riemann surface $Σ$ and $X$ is a Kähler Hamiltonian $G$-manifold. For compact $Σ$, possibly with boundary, we prove long time existence of the gradient flow. The flow lines converge to critical points of the functional. So, there is a stratification on $\mathcal{H}(P,X)$ that is invariant under the action of the complexified gauge group. Symplectic vortices are the zeros of the functional we study. When $Σ$ has boundary, similar to Donaldson's result for the Hermitian Yang-Mills equations, we show that there is only a single stratum - any element of $\mathcal{H}(P,X)$ can be complex gauge transformed to a symplectic vortex. This is a version of Mundet's Hitchin-Kobayashi result on a surface with boundary.
We consider Riemann surfaces obtained from nodal curves with infinite cylinders in the place of nodal and marked points, and study the space of finite energy vortices defined on these surfaces. To compactify the space of vortices, we need to consider stable vortices - these incorporate breaking of cylinders and sphere bubbling in the fibers. In this paper, we prove that the space of gauge equivalence classes of stable vortices representing a fixed equivariant homology class is compact and Hausdorff under the Gromov topology. We also show that this space is homeomorphic to the moduli space of quasimaps defined by Ciocan-Fontanine, Kim and Maulik.
We prove a Hitchin-Kobayashi correspondence for affine vortices generalizing a result of Jaffe-Taubes for the action of the circle on the affine line. Namely, suppose a compact Lie group K has a Hamiltonian action on a Kaehler manifold X which is either compact or convex at infinity with a proper moment map, and so that stable=semistable for the action of the complexified Lie group G. Then, for some sufficiently divisible integer n, there is a bijection between gauge equivalence classes of K-vortices with target X modulo gauge and isomorphism classes of maps from the weighted projective line P(1,n) to X/G that map the stacky point at infinity P(n) to the semistable locus in X. The results allow the construction and partial computation of the quantum Kirwan map in Woodward, and play a role in the conjectures of Dimofte, Gukov, and Hollande relating vortex counts to knot invariants.