The Schrödinger equation can be derived using the minimum Fisher information principle. I discuss why such an approach should work, and also show that the Kähler and Hilbert space structures of quantum mechanics result from combining the symplectic structure of the hydrodynamical model with the Fisher information metric. PACS: 03.65.Bz; 89.70.+c

It is well-known (see eg [22]) that the topology of a compact Kähler manifold X is strongly restricted by Hodge theory. In fact, Hodge theory provides two sets of data on the cohomology of a compact Kähler manifold. The first data are the Hodge decompositions on the cohomology spaces H(X,C) (see (1.1) where V = H(X,Q)); they depend only on the complex structure. The second data, known as the Le...

Abstract We formulate a Calabi–Yau-type conjecture in generalized Kähler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for structures generalizing notion class, we unique solvability Gualtieri’s Calabi–Yau equation within this class. establish uniqueness, and moreover show that all such solutions are actually hyper-Kähle...

We show that the almost complex structure underlying a nonKähler, nearly Kähler 6-manifold (in particular, the standard almost complex structure of S) cannot be compatible with any symplectic form, even locally.

Examples of Kähler metrics of constant scalar curvature are relatively scarce. Over the past two decades, several workers in geometry and physics have used symmetry reduction to construct complete Kähler metrics of constant scalar curvature by ODE methods. One fruitful idea—the “Calabi ansatz”—is to begin with an Hermitian line bundle p : (L, h)→ (M, gM ) over a Kähler manifold, and to search f...

The moduli of N = 1 compactifications of IIB string theory can be stabilized by a combination of fluxes (which freeze complex structure moduli and the dilaton) and nonperturbative superpotentials (which freeze Kähler moduli), typically leading to supersymmetric AdS vacua. We show that stringy corrections to the Kähler potential qualitatively alter the structure of the effective scalar potential...

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach discussed in this ...

A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kähler metric of constant scalar curvature on the blow-up according to [18]. We present a generalization of this construction to the case of parabolically polystable ruled surfaces. Thus we can produce numerous examples of Kähle...

We prove a structure theorem for the Gromov-Witten invariants of compact Kähler surfaces with geometric genus pg > 0. Under the technical assumption that there is a canonical divisor that is a disjoint union of smooth components, the theorem shows that the GW invariants are universal functions determined by the genus of this canonical divisor components and the holomorphic Euler characteristic ...

Divergence functions play a central role in information geometry. Given a manifold M, a divergence function D is a smooth, nonnegative function on the product manifold M ×M that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold ∆M ⊂ M ×M. In this Chapter, we review how such divergence functions induce i) a statistical st...