The Dialectics of Creation and the Projective Structure of Space

The article considers creationism as a historically relevant principle in the scientific and philosophic aspects denoting the ontological structure of the world. Outside of the religious interpretation, the author speaks of the dialectics of creation, which is revealed as an implicative connection of the one and nothing. Logical inversion (logical turn), acting from within this implicative connection, is postulated as the principle of a fundamental negation, which, according to the author, forms the true and dramatic essence of the world as a creation. The author distances himself from the widespread discussion between evolutionism and scientific creationism, stating that it does not correspond to the very subject of creationism, understood as the implication of a real from nothing. The author focuses on considering ‘nothing’ as a purely dialectical / metaphysical principle and relies partly on the Hegel’s dialectic of ‘being’ and ‘nothing’, and partly on the neoplatonic concept of the one. Rejecting the medieval interpretation of the temporal beginning and the Hegel’s identity, he deduces a scheme of the logical connection between the one and the difference, which postulates the inversion (turnover) forming the creation - the one and the difference disjunctively change places, the one becomes the real, and the difference out of the one becomes nothing. It is argued that this postulate, in particular, refutes the thesis about the ‘fall into sin’. In the second part of the article, a spatial-phenomenological hypothesis is presented: the author provides a description of the space as a geometrical-semantic plane (projective structure). This hypothesis follows from the phenomenological problem of the duality of a geometric object, which results in the problem of ontological transition between a point and a line (in the aporia of the Eleats) and the related problem of spatial congruence / parallelism. According to the author, the potential for solving these not essentially mathematical, but metaphysical questions is the projective geometry, in which parallel lines intersect at ‘point at infinity’, and space is complemented by the ‘plane at infinity’. The essence of the solution consists, firstly, in the assumption of the single plane, which underlies the transition, and secondly, in the description of the perceived world as a result of a specific turn over and closure of this plane, forming the projective structure. The key in this part is the demonstration of the surface of a three-dimensional object as a phenomenon of perceptual-semantic unfolding, which can be imagined as an action of consciousness, consistently reducing the usual scheme. An important aspect of considering the projective structure is the correlation with ‘the Plane’ by G. Deleuze. The general idea of the article is that the dialectical scheme of creation and the projective structure of the space coincide: the logical inversion (logical turn), acting in connection of the one and nothing, and projective structural turnover – are the same things.

Zariski closures of images of algebraic subsets under the uniformization map on finite-volume quotients of the complex unit ball

We prove the analogue of the Ax–Lindemann–Weierstrass theorem for not necessarily arithmetic lattices of the automorphism group of the complex unit ball $\mathbb{B}^{n}$ using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ and the associated uniformization map $\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=:X_{\unicode[STIX]{x1D6E4}}$. Given an algebraic subset $S\,\subset \,\mathbb{B}^{n}$ and writing $Z$ for the Zariski closure of $\unicode[STIX]{x1D70B}(S)$ in $X_{\unicode[STIX]{x1D6E4}}$ (which is equipped with a canonical quasi-projective structure), in some precise sense we realize $Z$ as a variety uniruled by images of algebraic subsets under the uniformization map, and study the asymptotic geometry of an irreducible component $\widetilde{Z}$ of $\unicode[STIX]{x1D70B}^{-1}(Z)$ as $\widetilde{Z}$ exits the boundary $\unicode[STIX]{x2202}\mathbb{B}^{n}$ by exploiting the strict pseudoconvexity of $\mathbb{B}^{n}$, culminating in the proof that $\widetilde{Z}\,\subset \,\mathbb{B}^{n}$ is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of $\text{ Aut}(\unicode[STIX]{x1D6FA})$ for (possibly reducible) bounded symmetric domains $\unicode[STIX]{x1D6FA}$.

Bubbling complex projective structures with quasi-Fuchsian holonomy

For a given quasi-Fuchsian representation [Formula: see text] of the fundamental group of a closed surface [Formula: see text] of genus [Formula: see text], we prove that a generic branched complex projective structure on [Formula: see text] with holonomy [Formula: see text] and two branch points can be obtained from some unbranched structure on [Formula: see text] with the same holonomy by bubbling, i.e. a suitable connected sum with a copy of [Formula: see text].

Curvature and Generalized PHGs

Even though it is quite clear at this stage what curvature means, its technical definition turns out to be problematic, since it needs a technical assumption. However, this assumption forces a projective structure to be flat, and it becomes necessary to modify the approach adopted so far.

PROJECTIVE STRUCTURES AND -CONNECTIONS

We extend T. Y. Thomas’s approach to projective structures, over the complex analytic category, by involving the $\unicode[STIX]{x1D70C}$-connections. This way, a better control of projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold $P$ is endowed with a complex projective structure then $P$ can be locally identified, through quaternionic diffeomorphisms, with the quaternionic projective space.

Branched Holomorphic Cartan Geometries and Calabi–Yau Manifolds

We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum [25]. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit a branched flat holomorphic projective structure. We also give an example of a non-flat branched holomorphic normal projective structure on a compact complex surface. It is known that no compact complex surface admits such a structure with empty branching locus. We prove that non-projective compact simply connected Kähler Calabi–Yau manifolds do not admit any branched holomorphic projective structure. The key ingredient of its proof is the following result of independent interest: if E is a holomorphic vector bundle over a compact simply connected Kähler Calabi–Yau manifold and E admits a holomorphic connection, then E is a trivial holomorphic vector bundle and any holomorphic connection on E is trivial.