Torus action on the moduli spaces of torsion plane sheaves of multiplicity four

We describe the torus fixed locus of the moduli space of stable sheaves with Hilbert polynomial $4m+1$ on ${\mathbb{P}}^2$. We determine the torus representation of the tangent spaces at the fixed points, which leads to the computation of the Betti and Hodge numbers of the moduli space.     

On a characteristic of the first eigenvalue of the Dirac operator on compact spin symmetric spaces with a Kähler or Quaternion-Kähler structure

It is shown that on a compact spin symmetric space with a Kähler or Quaternion-Kähler structure, the first eigenvalue of the Dirac operator is linked to a “lowest” action of the holonomy, given by the fiberwise action on spinors of the canonical forms characterized by this holonomy. The result is also verified for the symmetric space $F_4/{\mathrm{Spin}}_9$, proving that it is valid for all the “possible” holonomies in Berger’s list occurring in that context. The proof is based on a characterization of the first eigenvalue of the Dirac operator given in Milhorat (2005) and Milhorat (2006). By the way, we review an incorrect statement in the proof of the first lemma in Milhorat (2005).     

Reduction of Vaisman structures in complex and quaternionic geometry

We consider locally conformal Kähler geometry as an equivariant (homothetic) Kähler geometry: a locally conformal Kähler manifold is, up to equivalence, a pair $(K,\Gamma )$, where $K$ is a Kähler manifold and $\Gamma$ is a discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kähler manifold $(K,\Gamma )$ as the rank of a natural quotient of $\Gamma$, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kähler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover, we define locally conformal hyperKähler reduction as an equivariant version of hyperKähler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally, we show that locally conformal hyperKähler reduction induces hyperKähler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperKähler reduction.     

Complex structures adapted to magnetic flows

Let $M$ be a compact real-analytic manifold, equipped with a real-analytic Riemannian metric $g$, and let $\beta$ be a closed real-analytic 2-form on $M$, interpreted as a magnetic field. Consider the Hamiltonian flow on $T^{1}M$ that describes a charged particle moving in the magnetic field $\beta$. Following an idea of T. Thiemann, we construct a complex structure on a tube inside $T^{1}M$ by pushing forward the vertical polarization by the Hamiltonian flow “evaluated at time $i$”. This complex structure fits together with $\omega -{\pi }^1\beta$ to give a Kähler structure on a tube inside $T^{1}M$. When $\beta =0$, our magnetic complex structure is the adapted complex structure of Lempert–Szőke and Guillemin–Stenzel.We describe the magnetic complex structure in terms of its $(1,0)$-tangent bundle, at the level of holomorphic functions, and via a construction using the embeddings of Whitney–Bruhat and Grauert. We describe an antiholomorphic intertwiner between this complex structure and the complex structure induced by $-\beta$, and we give two formulas for local Kähler potentials, which depend on a local choice of vector potential $1$-form for $\beta$. Finally, we compute the magnetic complex structure explicitly for constant magnetic fields on ${\mathbb{R}}^2$ and $S^2$.     

Gaugino masses in modular invariant supergravity

We calculate gaugino masses in string-derived models with hidden-sector gaugino condensation. The linear multiplet formulation for the dilaton superfield is used to implement perturbative modular invariance. The contribution arising from quantum effects in the observable sector includes the term recently found in generic supergravity models. A much larger contribution is present if matter fields with Standard Model gauge couplings also couple to the Green–Schwarz counter term. We comment on the relation of our Kähler U(1) superspace formalism to other calculations.