We solve the problem of determining the fundamental degrees of freedom underlying a generalized Kähler structure of symplectic type. For a usual Kähler structure, it is well-known that the geometry is determined by a complex structure, a Kähler class, and the choice of a positive $(1, 1)$-form in this class, which depends locally on only a single real-valued function: the Kähler potential. Such a description for generalized Kähler geometry has been sought since it was discovered in 1984. We show that a generalized Kähler structure of symplectic type is determined by a pair of holomorphic Poisson manifolds, a holomorphic symplectic Morita equivalence between them, and the choice of a positive Lagrangian brane bisection, which depends locally on only a single real-valued function, which we call the generalized Kähler potential. Our solution draws upon, and specializes to, the many results in the physics literature which solve the problem under the assumption (which we do not make) that the Poisson structures involved have constant rank. To solve the problem we make use of, and generalize, two main tools: the first is the notion of symplectic Morita equivalence, developed by Weinstein and Xu to study Poisson manifolds; the second is Donaldson’s interpretation of a Kähler metric as a real Lagrangian submanifold in a deformation of the holomorphic cotangent bundle.

We prove the LeBrun–Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold $X$ of dimension $2n+1$ that has reductive automorphism group of rank at least $n-2$ is necessarily homogeneous. This implies that any positive quaternion-Kähler manifold of real dimension at most $16$ is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most $9$ with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contribute to such problem in general, by studying the action of algebraic torus on varieties and exploiting Białynicki-Birula decomposition and equivariant Riemann–Roch theorems. From the point of view of $T$ varieties (that is, varieties with a torus action), our result is about high complexity T-manifolds. The complexity here is at most $\frac{1}{2} (\operatorname{dim}X+5)$ with $\operatorname{dim}$ arbitrarily high, but we require this special (contact) structure of $X$. Previous methods for studying $T$-varieties in general usually only apply for complexity at most $2$ or $3$.

Let $X$ be a smooth, irreducible, complex projective surface, $H$ a polarization on $X$. Let $\gamma = (r, c, \Delta)$ be a Chern character. In this paper, we study the cohomology of moduli spaces of Gieseker semistable sheaves $M_{X,H} (\gamma)$. When the rank $r = 1$, the Betti numbers were computed by Göttsche. We conjecture that if we fix the rank $r \geq 1$ and the first Chern class $c$, then the Betti numbers (and more generally the Hodge numbers) of $M_{X,H} (r, c, \Delta)$ stabilize as the discriminant $\Delta$ tends to infinity and that the stable Betti numbers are independent of $r$ and $c$. In particular, the conjectural stable Betti numbers are determined by Göttsche’s calculation. We present evidence for the conjecture. We analyze the validity of the conjecture under blowup and wall-crossing. We prove that when $X$ is a rational surface and $K_X \cdot H \lt 0$, then the classes $[M_{X,H} (\gamma)]$ stabilize in an appropriate completion of the Grothendieck ring of varieties as $\Delta$ tends to $\infty$. Consequently, the virtual Poincaré and Hodge polynomials stabilize to the conjectural value. In particular, the conjecture holds when $X$ is a rational surface, $H \cdot K_X \lt 0$ and there are no strictly semistable objects in $M_{X,H} (\gamma)$.

As the first step of proving the Hodge-FVH correspondence recently proposed in [19], we derive the Virasoro constraints and the Dubrovin–Zhang loop equation for special cubic Hodge integrals. We show that this loop equation has a unique solution, and provide a new algorithm for the computation of these Hodge integrals. We also observe the gap phenomenon for certain special cubic Hodge free energies.

We prove that no eigenvalue of the clamped disk can have multiplicity greater than six. Our method of proof is based on a new recursion formula, linear algebra arguments and a transcendence theorem due to Siegel and Shidlovskii.