International Journal of Mathematics

The symplectic vortex equations admit a variational description as global minimum of the Yang–Mills–Higgs functional. We study its negative gradient flow on holomorphic pairs $(A,u)$ where $A$ is a connection on a principal $G$-bundle $P$ over a closed Riemann surface $\mathrm{\Sigma }$ and $u:P\to X$ is an equivariant map into a Kähler Hamiltonian $G$-manifold. The connection $A$ induces a holomorphic structure on the Kähler fibration $P{\times }_{G}X$ and we require that $u$ descends to a holomorphic section of this fibration.

We prove a Łojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the $W^{1,2}\times W^{2,2}$-topology when $X$ is equivariantly convex at infinity with proper moment map, $X$ is holomorphically aspherical and its Kähler metric is analytic.

As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang–Mills–Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet’s Kobayashi–Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment–weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem.

Let $\mathcal{\mathscr{H}}\simeq G/K$ be the symmetric tube domain associated with the Jordan algebra ${Herm}_r(𝔽)$, $𝔽=ℝ$, $ℂ$, or $ℍ$, and $X=G/\overline{P}$ be its Shilov boundary. Also, let $I_P^G(\chi )$ be a degenerate principal series representation of $G$. Then we investigate the Bessel integrals assigned to functions in general $K$-types of $I_P^G(\chi )$. We give individual upper bounds of their supports, when $I_P^G(\chi )$ is reducible. We also use the upper bounds to give a partition for the set of all $K$-types in $I_P^G(\chi )$, that turns out to explain the $G$-module structure of $I_P^G(\chi )$. Thus, our results concretely realize a relationship observed by Kashiwara and Vergne [$K$-types and singular spectrum, in Noncommutative Harmonic analysis, Lecture Notes in Mathematics, Vol. 728 (Springer, 1979), pp. 177–200] between the Fourier supports and the asymptotic $K$-supports assigned to $G$-submodules in $I_P^G(\chi )$.

Given two open books with equal pages, we show the existence of an exact symplectic cobordism whose negative end equals the disjoint union of the contact manifolds associated to the given open books, and whose positive end induces the contact manifold associated to the open book with the same page and concatenated monodromy. Using similar methods, we show the existence of strong fillings for contact manifolds associated with doubled open books, a certain class of fiber bundles over the circle obtained by performing the binding sum of two open books with equal pages and inverse monodromies. From this we conclude, following an outline by Wendl, that the complement of the binding of an open book cannot contain any local filling obstruction. Given a contact $3$-manifold, according to Eliashberg there is a symplectic cobordism to a fibration over the circle with symplectic fibers. We extend this result to higher dimensions recovering a recent result by Dörner–Geiges–Zehmisch. Our cobordisms can also be thought of as the result of the attachment of a generalized symplectic $1$-handle.

In a recent paper, we obtained a WDVV-type relation for real genus 0 Gromov–Witten invariants with conjugate pairs of insertions; it specializes to a complete recursion in the case of odd-dimensional projective spaces. This paper provides another, more complex-geometric, proof of the latter. The main part of this approach readily extends to real symplectic manifolds with empty real locus, but not to the general case.

Landstad–Vaes theory deals with the structure of the crossed product of a ${\mathrm{\text{C}}}^{\ast }$-algebra by an action of locally compact (quantum) group. In particular, it describes the position of original algebra inside crossed product. The problem was solved in 1979 by Landstad for locally compact groups and in 2005 by Vaes for regular locally compact quantum groups. To extend the result to non-regular groups we modify the notion of $G$-dynamical system introducing the concept of weak action of quantum groups on ${\mathrm{\text{C}}}^{\ast }$-algebras. It is still possible to define crossed product (by weak action) and characterize the position of original algebra inside the crossed product. The crossed product is unique up to an isomorphism. At the end we discuss a few applications.

In this series of papers, we develop the theory of a class of locally compact quantum groupoids, which is motivated by the purely algebraic notion of weak multiplier Hopf algebras. In this Part I, we provide motivation and formulate the definition in the $C^{\ast }$-algebra framework. Existence of a certain canonical idempotent element is required and it plays a fundamental role, including the establishment of the coassociativity of the comultiplication. This class contains locally compact quantum groups as a subclass.

Let $G$ be a complex affine algebraic group, and let ${\sigma }_1$ and ${\sigma }_2$ be commuting anti-holomorphic involutions of $G$. We construct an algebraic family of algebraic groups over the complex projective line and a real structure on the family that interpolates between the real forms $G^{{\sigma }_1}$ and $G^{{\sigma }_2}$.

Given a 2-sheeted torus over the circle with winding number 1, we prove that its polynomial hull is a union of 2-sheeted holomorphic discs. Moreover, when the hull is non-degenerate its boundary is a Levi-flat solid torus foliated by such discs.

Recently, Chang and Li generalized the theory of virtual fundamental class to the setting of semi-perfect obstruction theory. A semi-perfect obstruction theory requires only the local existence of a perfect obstruction theory with compatibility conditions. In this paper, we generalize the torus localization of Graber–Pandharipande [T. Graber and R. Pandharipande, Localization of virtual cycles, Invent. Math. 135(2) (1999) 487–518], the cosection localization [Y.-H. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26(4) (2013) 1025–1050] and their combination [H.-L. Chang, Y.-H. Kiem and J. Li, Torus localization and wall crossing for cosection localized virtual cycles, Adv. Math. 308 (2017) 964–986], to the setting of semi-perfect obstruction theory. As an application, we show that the Jiang-Thomas theory [Y. Jiang and R. Thomas, Virtual signed Euler characteristics, preprint (2014), arXiv:1408.2541] of virtual signed Euler characteristic works without the technical quasi-smoothness assumption from derived algebraic geometry.

We prove W${}^{\ast }$-superrigidity for a large class of coinduced actions. We prove that if $\mathrm{\Sigma }$ is an amenable almost-malnormal subgroup of an infinite conjugagy class (icc) property (T) countable group $\mathrm{\Gamma }$, the coinduced action $\mathrm{\Gamma }\curvearrowright X$ from an arbitrary probability measure preserving action $\mathrm{\Sigma }\curvearrowright X_0$ is W${}^{\ast }$-superrigid. We also prove a similar statement if $\mathrm{\Gamma }$ is an icc non-amenable group which is measure equivalent to a product of two infinite groups. In particular, we obtain that any Bernoulli action of such a group $\mathrm{\Gamma }$ is W${}^{\ast }$-superrigid.