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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]

The deep geometrical relationships holding among the NLS equation, the vortex filament equation, the Heisenberg spin model, and the Schrödinger map equation are extended to the general setting of Hermitian symmetric spaces. New results are obtained by utilizing a generalized Hasimoto variable which arises from applying the general theory of parallel moving frames. The example of complex projective space ℂ P N = S U (N + 1) ∕ U (N) is used to illustrate the method and results. [ABSTRACT FROM AUTHOR]

We show that ruled real hypersurfaces with constant mean curvature in the complex projective and hyperbolic spaces must be minimal. This provides their classification, by virtue of a result of Lohnherr and Reckziegel. [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]

Abstract We construct a family of Poisson structures of hydrodynamic type on the loop space of ℂ P n − 1. This family is parametrized by the moduli space of elliptic curves or, in other words, by the modular parameter τ. This family can be lifted to a homogeneous Poisson structure on the loop space of ℂ n but in order to do that we need to upgrade the modular parameter τ to an additional field τ (x) with Poisson brackets { τ (x) , τ (y) } = 0 , { τ (x) , z a (y) } = 2 π i z a (y) δ ′ (x − y) where z 1 , ... , z n are coordinates on ℂ n. These homogeneous Poisson structures can be written in terms of an elliptic r -matrix of hydrodynamic type. [ABSTRACT FROM AUTHOR]

Abstract We study the automorphism group of curves and surfaces in CP 3 with respect to the conformal group, i.e. the group of all A ∈ P G L (4 , ℂ) commuting with the anti-holomorphic involution j defined by j ((z 0 : z 1 : z 2 : z 3)) = (− z ¯ 1 : z ¯ 0 : z ¯ 3 : − z ¯ 2). For some singular surfaces we check when this group is finite. Among the singular surfaces we handle there are: (1) certain cones; (2) surfaces X containing no line and with j (X) ≠ X ; (3) surfaces containing only finitely many, k , twistor lines with k ≥ 3. In many cases the proofs need results on conformal automorphisms of singular curves. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of supergeometry of complex projective superspaces P n | m . First, we provide formulas for the cohomology of invertible sheaves of the form O P n | m ( ℓ ) , that are pullbacks of ordinary invertible sheaves on the reduced variety P n . Next, by studying the even Picard group Pic 0 ( P n | m ) , classifying invertible sheaves of rank 1 | 0 , we show that the sheaves O P n | m ( ℓ ) are not the only invertible sheaves on P n | m , but there are also new genuinely supersymmetric invertible sheaves that are unipotent elements in the even Picard group. We study the Π -Picard group Pic Π ( P n | m ) , classifying Π -invertible sheaves of rank 1 | 1 , proving that there are also non-split Π -invertible sheaves on supercurves P 1 | m . Further, we investigate infinitesimal automorphisms and first order deformations of P n | m , by studying the cohomology of the tangent sheaf using a supersymmetric generalisation of the Euler exact sequence. A special attention is paid to the meaningful case of supercurves P 1 | m and of Calabi–Yau’s P n | n + 1 . Last, with an eye to applications to physics, we show in full detail how to endow P 1 | 2 with the structure of N = 2 super Riemann surface and we obtain its SUSY-preserving infinitesimal automorphisms from first principles, that prove to be the Lie superalgebra o s p ( 2 | 2 ) . A particular effort has been devoted to keep the exposition as concrete and explicit as possible. [ABSTRACT FROM AUTHOR]

In this paper we prove that Π -projective spaces P Π n arise naturally in supergeometry upon considering a non-projected thickening of P n related to the cotangent sheaf Ω P n 1 . In particular, we prove that for n ⩾ 2 the Π -projective space P Π n can be constructed as the non-projected supermanifold determined by three elements ( P n , Ω P n 1 , λ ) , where P n is the ordinary complex projective space, Ω P n 1 is its cotangent sheaf and λ is a non-zero complex number, representative of the fundamental obstruction class ω ∈ H 1 ( T P n ⊗ ⋀ 2 Ω P n 1 ) ≅ C . Likewise, in the case n = 1 the Π -projective line P Π 1 is the split supermanifold determined by the pair ( P 1 , Ω P 1 1 ≅ O P 1 ( − 2 ) ) . Moreover we show that in any dimension Π -projective spaces are Calabi–Yau supermanifolds. To conclude, we offer pieces of evidence that, more in general, also Π -Grassmannians can be constructed the same way using the cotangent sheaf of their underlying reduced Grassmannians, provided that also higher, possibly fermionic, obstruction classes are taken into account. This suggests that this unexpected connection with the cotangent sheaf is characteristic of Π -geometry. [ABSTRACT FROM AUTHOR]

In this paper we study harmonic maps from super Riemann surfaces in complex projective spaces and projective spaces associated with the super skew-field D . In both cases, we develop the theory of Gauß transforms and study the notion of isotropy, in particular its relation to holomorphic differentials on the super Riemann surface. Moreover, we give a definition of finite type harmonic maps for a special class of maps into C P n | n + 1 and thus obtain a classification for certain harmonic super tori. Furthermore, we investigate the equations satisfied by the underlying objects and give an example of a harmonic super torus in D P 2 whose underlying map is not harmonic. [ABSTRACT FROM AUTHOR]

We present generating functions for extensions of multiplicative invariants of wreath symmetric products of orbifolds presented as the quotient by the locally free action of a compact, connected Lie group in terms of orbifold sector decompositions. Particularly interesting instances of these product formulas occur for the Euler and Euler–Satake characteristics, which we compute for a class of weighted projective spaces. This generalizes results known for global quotients by finite groups to all closed, effective orbifolds. We also describe a combinatorial approach to extensions of multiplicative invariants using decomposable functors that recovers the formula for the Euler–Satake characteristic of a wreath product of a global quotient orbifold. [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]

We study the expectation of the number of components b 0 ( X ) of a random algebraic hypersurface X defined by the zero set in projective space R P n of a random homogeneous polynomial f of degree d . Specifically, we consider invariant ensembles , that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables. Fixing n , under some rescaling assumptions on the family of ensembles (as d → ∞ ), we prove that E b 0 ( X ) has the same order of growth as [ E b 0 ( X ∩ R P 1 ) ] n . This relates the average number of components of X to the classical problem of M. Kac (1943) on the number of zeros of the random univariate polynomial f | R P 1 . The proof requires an upper bound for E b 0 ( X ) , which we obtain by counting extrema using Random Matrix Theory methods from Fyodorov (2013), and it also requires a lower bound, which we obtain by a modification of the barrier method from Lerario and Lundberg (2015) and Nazarov and Sodin (2009). We also provide quantitative upper bounds on implied constants; for the real Fubini–Study model these estimates provide super-exponential decay (as n → ∞ ) of the leading coefficient (in d ) of E b 0 ( X ) . [ABSTRACT FROM AUTHOR]

It is shown that the coordinate algebra of the quantum 2 n + 1 -dimensional lens space O ( L q 2 n + 1 ( ∏ i = 0 n m i ; m 0 , … , m n ) ) is a principal C Z -comodule algebra or the coordinate algebra of a circle principal bundle over the weighted quantum projective space W P q n ( m 0 , … , m n ) . Furthermore, the weighted U ( 1 ) -action or the C Z -coaction on the quantum odd dimensional sphere algebra O ( S q 2 n + 1 ) that defines W P q n ( 1 , m 1 , … , m n ) is free or principal. Analogous results are proven for quantum real weighted projective spaces R P q 2 n ( m 0 , … , m n ) . The K -groups of W P q n ( 1 , … , 1 , m ) and R P q 2 n ( 1 , … , 1 , m ) and the K 1 -group of L q 2 n + 1 ( N ; m 0 , … , m n ) are computed. [ABSTRACT FROM AUTHOR]

We give a complete classification of Lagrangian immersions of homogeneous 3-manifolds (the Berger spheres, the Heisenberg group Nil 3 , the universal covering of the Lie group PSL ( 2 , R ) and the Lie group Sol 3 ) in 3-dimensional complex space forms. As a corollary, we get a new characterization of the Berger sphere in complex projective space. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we estimate the eigenvalues of the twisted Dirac operator on Kähler submanifolds of the complex projective space and we discuss the sharpness of this estimate for the embedding . [Copyright &y& Elsevier]

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]

Abstract: Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor of the Fubini-Study metric of the complex projective space and the curvature form of the Hopf fibration. We also estimate the number of conjugate points of a sub-Riemannian extremal in terms of the bounds of the sectional curvature and the curvature form. It presents a typical example for the study of curvature maps and comparison theorems for a general corank 1 sub-Riemannian structure with symmetries done by C. Li and I. Zelenko (2011) in [2]. [Copyright &y& Elsevier]

Abstract: A holomorphy potential is a complex valued function whose complex gradient, with respect to some Kähler metric, is a holomorphic vector field. Given holomorphic vector fields on a compact complex manifold, form, for a given Kähler metric, a product of the following type: a function of the scalar curvature multiplied by functions of the holomorphy potentials of each of the vector fields. It is shown that the stipulation that such a product be itself a holomorphy potential for yet another vector field singles out critical metrics for a particular functional. This may be regarded as a generalization of the extremal metric variation of Calabi, where and the functional is the square of the -norm of the scalar curvature. The existence question for such metrics is examined in a number of special cases. Examples are constructed in the case of certain multifactored product manifolds. For the SKR metrics investigated by Derdzinski and Maschler and residing in the complex projective space, it is shown that only one type of nontrivial criticality holds in dimension three and above. [Copyright &y& Elsevier]

Abstract: Two hierarchies of quantum principal bundles over quantum real projective spaces are constructed. One hierarchy contains bundles with as a structure group, the other has the quantum group as a fibre. Both hierarchies are obtained by the process of prolongation from bundles with the cyclic group of order 2 as a fibre. The triviality or otherwise of these bundles is determined by using a general criterion for a prolongation of a comodule algebra to be a cleft Hopf–Galois extension. [Copyright &y& Elsevier]

Abstract: We define holomorphic structures on canonical line bundles of the quantum projective space and identify their space of holomorphic sections. This determines the quantum homogeneous coordinate ring of the quantum projective space. We show that the fundamental class of is naturally presented by a twisted positive Hochschild cocycle. Finally, we verify the main statements of the Riemann–Roch formula and the Serre duality for and . [Copyright &y& Elsevier]

Abstract: In this paper we study orbits of - and -actions on the spaces of binary and ternary polynomials as well as rational forms and find criteria for their equivalence. Similar results are also valid for real forms. [Copyright &y& Elsevier]

Abstract: We study Kustin–Miller unprojections between Calabi–Yau threefolds, or more precisely the geometric transitions they induce. We use them to connect many families of Calabi–Yau threefolds with Picard number one to the web of Calabi–Yau complete intersections. This result enables us to find explicit description of a few known families of Calabi–Yau threefolds in terms of equations. Moreover, we find two new examples of Calabi–Yau threefolds with Picard group of rank one, which are described by Pfaffian equations in weighted projective spaces. [Copyright &y& Elsevier]

Abstract: We present explicit constructions of complete Ricci-flat Kähler metrics that are asymptotic to cones over non-regular Sasaki–Einstein manifolds. The metrics are constructed from a complete Kähler–Einstein manifold of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kähler metrics on the total spaces of (i) holomorphic orbifold fibrations over , (ii) holomorphic orbifold fibrations over weighted projective spaces , with generic fibres being the canonical complex cone over , and (iii) the canonical orbifold line bundle over a family of Fano orbifolds. As special cases, we also obtain smooth complete Ricci-flat Kähler metrics on the total spaces of (a) rank two holomorphic vector bundles over , and (b) the canonical line bundle over a family of geometrically ruled Fano manifolds with base . When our results give Ricci-flat Kähler orbifold metrics on various toric partial resolutions of the cone over the Sasaki–Einstein manifolds . [Copyright &y& Elsevier]

We consider complex manifolds with a class of holomorphic coordinate functions satisfying the condition that each transition function is given by the standard action on CP2n−1 of some element in Sp(2n,C)/Z2. We show that such a manifold has a natural contact structure. Given any contact manifold, one can associate with it a symplectic manifold. It is shown that the symplectic manifolds arising from complex manifolds with special coordinate functions of the above type admit a canonical quantization. [Copyright &y& Elsevier]

We consider a d -dimensional well-formed weighted projective space P (w ¯) as a toric variety associated with a fan Σ (w ¯) in N w ¯ ⊗ R whose 1-dimensional cones are spanned by primitive vectors v 0 , v 1 , ... , v d ∈ N w ¯ generating a lattice N w ¯ and satisfying the linear relation ∑ i w i v i = 0. For any fixed dimension d , there exist only finitely many weight vectors w ¯ = (w 0 , ... , w d) such that P (w ¯) contains a quasi-smooth Calabi–Yau hypersurface X w defined by a transverse weighted homogeneous polynomial W of degree w = ∑ i = 0 d w i . Using a formula of Vafa for the orbifold Euler number χ orb (X w) , we show that for any quasi-smooth Calabi–Yau hypersurface X w the number (− 1) d − 1 χ orb (X w) equals the stringy Euler number χ str (X w ¯ ∗) of Calabi–Yau compactifications X w ¯ ∗ of affine toric hypersurfaces Z w ¯ defined by non-degenerate Laurent polynomials f w ¯ ∈ ℂ [ N w ¯ ] with Newton polytope conv ({ v 0 , ... , v d }). In the moduli space of Laurent polynomials f w ¯ there always exists a special point f w ¯ 0 defining a mirror X w ¯ ∗ with a Z ∕ w Z -symmetry group such that X w ¯ ∗ is birational to a quotient of a Fermat hypersurface via a Shioda map. [ABSTRACT FROM AUTHOR]

The moduli spaces of symplectic vector bundles of arbitrary rank on projective space P 3 are far from being well-understood. By now the only type of such bundles having satisfactory description are the so-called tame symplectic instantons. It is shown by U. Bruzzo, D. Markushevich and the first author in two papers from 2012 and 2016 that the moduli spaces of tame symplectic instantons are irreducible generically reduced algebraic spaces of dimension prescribed by the deformation theory. In the present paper we construct an infinite series of smooth irreducible moduli components of symplectic vector bundles of an arbitrary even rank 2 r , r ≥ 1 , obtained by an iterative use of the monad construction applied to tame symplectic instantons. As a particular case we obtain an infinite series of irreducible moduli components of stable rank 2 vector bundles on P 3. We show that this series contains as a subseries a large part of an infinite series of moduli components constructed by the authors and S. Tikhomirov in 2019. We also prove that, for any integers n , r , where r ≥ 1 and n ≥ r + 147 , there exists a moduli component, not necessarily unique, of our series such that symplectic bundles from this component have rank 2 r and second Chern class n. [ABSTRACT FROM AUTHOR]

DeTurck and Yang have shown that in the neighborhood of every point of a 3-dimensional Riemannian manifold, there exists a system of orthogonal coordinates (that is, with respect to which the metric has diagonal form). We show that this property does not generalize to higher dimensions. In particular, the complex projective spaces CP m and the quaternionic projective spaces HP q , endowed with their canonical metrics, do not have local systems of orthogonal coordinates for m , q ≥ 2. [ABSTRACT FROM AUTHOR]

Attempt to represent the self-interacting quantum electron as the cyclic motion has been discussed. This motion obeys the quantum inertia principle expressed by the parallel transported energy–momentum generator along a closed geodesic in the space of the unlocated quantum states C P (3) . The affine gauge potential in the complex projective state space (similar to the Higgs potential) seriously deforms the Jacobi fields in the vicinity of the "north pole". It was assumed that the divergency of the Jacobi field may be compensated by the fields of the Poincaré generators representing EM-like "field shell" of the electron in the dynamical spacetime. Thereby, the spacetime looks as ultimately deprecated in the role of the "container of matter" and it appears as the accompanied to the quantum electron functional space (dynamical spacetime). Meanwhile, the dynamics of the self-interacting electron is essentially non-linear and deterministic. [ABSTRACT FROM AUTHOR]

We describe generating functions for arbitrary-genus Gromov–Witten invariants of the projective space with any number of marked points explicitly. The structural portion of this description gives rise to uniform energy bounds and vanishing results for these invariants. They suggest deep conjectures relating Gromov–Witten invariants of symplectic manifolds to the energy of pseudo-holomorphic maps and the expected dimension of their moduli space. [ABSTRACT FROM AUTHOR]

As a continuation of a general program started in two previous publications, in the present paper we study the Kähler quotient resolution of the orbifold ℂ 3 ∕ Z 4 , comparing with the results of a toric description of the same. In this way we determine the algebraic structure of the exceptional divisor, whose compact component is the second Hirzebruch surface F 2. We determine the explicit Kähler geometry of the smooth resolved manifold Y , which is the total space of the canonical bundle of F 2. We study in detail the chamber structure of the space of stability parameters (corresponding in gauge theory to the Fayet–Iliopoulos parameters) that are involved in the construction of the desingularizations either by generalized Kronheimer quotient, or as algebro-geometric quotients. The walls of the chambers correspond to two degenerations; one is a partial desingularization of the quotient, which is the total space of the canonical bundle of the weighted projective space P [ 1 , 1 , 2 ] , while the other is the product of the ALE space A 1 by a line, and is related to the full resolution in a subtler way. These geometrical results will be used to look for exact supergravity brane solutions and dual superconformal gauge theories. [ABSTRACT FROM AUTHOR]