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The moduli space of projective structures on a compact oriented surface Σ has a holomorphic symplectic structure, which is constructed by pulling back, using the monodromy map, the Atiyah–Bott–Goldman symplectic form on the character variety Hom (π 1 (Σ) , PSL (2 , C)) / / PSL (2 , C). We produce another construction of this symplectic form. This construction shows that the symplectic form on the moduli space is actually algebraic. Note that the monodromy map is only holomorphic and not algebraic, so the first construction does not give algebraicity of the pulled back form. [ABSTRACT FROM AUTHOR]

We investigate the moduli space P g of smooth complex projective curves of genus g equipped with a projective structure. When g ≥ 3 , it is shown that this moduli space P g does not admit any nonconstant algebraic function. This is in contrast with the case of P 2 which is known to be an affine variety. [ABSTRACT FROM AUTHOR]

We define the analogue of instanton sheaves on the blow-up P n ˜ of the n -dimensional projective space at a point. We choose an appropriate polarisation on P n ˜ and construct rank 2 examples of locally free and non locally free (but torsion free) type. In general, the defined instantons also turn out to be the cohomology of monads, although non-linear ones. Moreover, in the five dimensional case, we show that there are continuous families of them that fill, at least, a smooth component in the moduli of semi-stable sheaves. [ABSTRACT FROM AUTHOR]

We construct an explicit map from a generic minimal δ (2) -ideal Lagrangian submanifold of C n to the quaternionic projective space H P n − 1 , whose image is either a point or a minimal totally complex surface. A stronger result is obtained for n = 3 , since the above mentioned map then provides a one-to-one correspondence between minimal δ (2) -ideal Lagrangian submanifolds of C 3 and minimal totally complex surfaces in H P 2 which are moreover anti-symmetric. Finally, we also show that there is a one-to-one correspondence between such surfaces in H P 2 and minimal Lagrangian surfaces in C P 2. [ABSTRACT FROM AUTHOR]

C ∞ -Rings are R -algebras equipped with operations ϕ f for every f ∈ C ∞ (R n) and every n ∈ N. Therefore, a C ∞ -version of algebraic geometry can be developed using C ∞ -rings instead of ordinary rings and many classical constructions can be performed in this context. In particular, C ∞ -schemes are the C ∞ counterpart of classical schemes. Examples of schemes are often obtained by gluing schemes or using fiber products. Another useful way to give examples of schemes is looking for representable functors F : Schemes → Sets. In this work, we show that constructions such as gluing schemes and fiber products can be done in the context of C ∞ -algebraic geometry and they can be used to exhibit some examples of C ∞ -schemes such as projective spaces and Grassmannians as well as necessary and sufficient conditions for a functor F : C ∞ − Schemes → Sets to be representable. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective threefold where the generalized Bogomolov–Gieseker inequality holds. In this paper we examine the conditions that an object E with ch 0 (E) = 0 has to satisfy in order for it to be asymptotically (semi)stable with regard to Weak or Bridgeland stability conditions. This notion turned out to equivalent to sheaf Giesker–Simpson (semi)stability or a dual of it, depending on the curve considered. [ABSTRACT FROM AUTHOR]

We consider quantum mechanics on the noncommutative spaces characterized by the commutation relations [ x a , x b ] = i θ f a b c x c , where f a b c are the structure constants of a Lie algebra. We note that the quantum problems in this noncommutative setting can be reformulated as ordinary quantum problems in the commuting momentum space. The coordinates are then represented as linear differential operators x ˆ a = i E a b (p) ∂ / ∂ p b. Generically, the matrix E a b (p) represents a certain infinite series over the deformation parameter θ : E a b = δ a b + .... For semisimple compact Lie algebras, the naturally chosen Hamiltonian, H ˆ = 1 2 x ˆ a 2 , coincides with the Laplace-Beltrami operator describing the motion along the corresponding group manifold endowed by the metric invariant under left and right group rotations. Then E a b have the meaning of vielbeins. The characteristic size of the manifold is of order θ − 1. For the algebras u (N) , the operators x ˆ a can be represented in a simple finite form with only two terms in the expansion in θ. When N = 2 , this gives rise to the Hermitian Hamiltonian describing the motion along a non-compact cover of U (2). When N ≥ 3 , such representation involves a non-Hermitian Hamiltonian. A byproduct of our study are new nonstandard formulas for the metrics on all the spheres S n , on the corresponding projective spaces R P n and on the cover of U (2). [ABSTRACT FROM AUTHOR]

We study the quantum Riemannian geometry of quantum projective spaces of any dimension. In particular, we compute the Riemann and Ricci tensors using previously introduced quantum metrics and quantum Levi-Civita connections. We show that the Riemann tensor is a bimodule map and derive various consequences of this fact. We prove that the Ricci tensor is proportional to the quantum metric, giving a quantum analogue of the Einstein condition, and compute the corresponding scalar curvature. [ABSTRACT FROM AUTHOR]

Suppose given a holomorphic and Hamiltonian action of a compact torus T on a polarized Hodge manifold M. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of T on the associated Hardy space. If in addition the moment map is nowhere zero, for each weight ν the ν -th isotypical component in the Hardy space of the polarization is finite-dimensional. Assuming that the moment map is transverse to the ray through ν , we give a geometric interpretation of the isotypical components associated to the weights k ν , k → + ∞ , in terms of certain polarized orbifolds associated to the Hamiltonian action and the weight. These orbifolds are generally not reductions of M in the usual sense, but arise rather as quotients of certain loci in the unit circle bundle of the polarization; this construction generalizes the one of weighted projective spaces as quotients of the unit sphere, viewed as the domain of the Hopf map. [ABSTRACT FROM AUTHOR]

We review noncommutative Poisson structures on affine and projective spaces over C. We also construct a class of examples of noncommutative Poisson structures on C P n − 1 for n > 2. These noncommutative Poisson structures depend on a modular parameter τ ∈ C and an additional discrete parameter k ∈ Z , where 1 ≤ k < n and k , n are coprime. The abelianization of these Poisson structures can be lifted to the quadratic elliptic Poisson algebras q n , k (τ). [ABSTRACT FROM AUTHOR]