The Crossing as a New Approach for the Urban Transformation of Traditional Cities Towards the Sustainability

The topic of urban transformations has attracted the attention of researchers as it is one of the basic issues through which cities can be transformed towards sustainability. A specific level of transformation levels according to a philosophical concept known as a crossing. This article has relied on a specific methodology that aims to find a new approach for urban transformation based on the crossing concept. This concept derives from philosophical entrances based on the concepts of (being, process, becoming, and integration). Four levels have been for the crossing are (normal, ascending, leap, and descending). Each of these levels includes specific characteristics that distinguish it. The results showed that there is no descending crossing in the study area and that four indicators that have achieved leap crossing are: (coverage ratio, compactness, Effectiveness of the pedestrian and bicycle movement, and proximity. As for ascending crossing achieved in three indicators are: mix land use, blocks' length, and connectivity, as for normal crossing achieved in three indicators are: diversity of building heights, population density, and housing density. These results help in making the planning decision to transform traditional Karbala city into a sustainable city, by giving priority in development to the indicators that have achieved ascending crossing because it takes less time and effort to reach the leap crossing in comparison with indicators that have achieved normal crossing.

A Torelli type theorem for nodal curves

The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal-crossing singularities and it provides flat degeneration of the moduli of vector bundles over a smooth projective curve. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than [Formula: see text]) and fixed degree such that rank and degree are co-prime.

Smoothing toroidal crossing spaces

We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is to study log structures that are incoherent on a subspace of codimension 2 and prove a Hodge–de Rham degeneration theorem for such log spaces that also settles a conjecture by Danilov. We show that the homotopy equivalence between Maurer–Cartan solutions and deformations combined with Batalin–Vilkovisky theory can be used to obtain smoothings. The construction of new Calabi–Yau and Fano manifolds as well as Frobenius manifold structures on moduli spaces provides potential applications.

Maximal time existence of unnormalized conical Kähler–Ricci flow

We generalize the maximal time existence of Kähler–Ricci flow in [G. Tian and Z. Zhang, On the Kähler–Ricci flow on projective manifolds of general type, Chin. Ann. Math. Ser. B 27 (2006), no. 2, 179–192] and [J. Song and G. Tian, The Kähler–Ricci flow through singularities, Invent. Math. 207 (2017), no. 2, 519–595] to the conical case. Furthermore, if the log canonical bundle {K_{M}+(1-\beta)[D]} is big or big and nef, we can examine the limit behaviors of such conical Kähler–Ricci flow. Moreover, these results still hold when D is a simple normal crossing divisor.

SNC Log Symplectic Structures on Fano Products

This paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.

Asymptotically conical Calabi-Yau metrics with cone singularities along a compact divisor

We construct Asymptotically Locally Euclidean (ALE) and, more generally, asymptotically conical Calabi–Yau metrics with cone singularities along a compact simple normal crossing divisor. In particular, this includes the case of the minimal resolution of 2D quotient singularities for any finite subgroup $\Gamma \subset U(2)$ acting freely on the three-sphere, hence generalizing Kronheimer’s construction of smooth ALE gravitational instantons.

LOGARITHMIC DE RHAM COMPARISON FOR OPEN RIGID SPACES

In this note, we prove the logarithmic $p$ -adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally $K(\unicode[STIX]{x1D70B},1)$ (in a certain sense) with respect to $\mathbb{F}_{p}$ -local systems and ramified coverings along the divisor. We follow Scholze’s method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.

A BOGOMOLOV UNOBSTRUCTEDNESS THEOREM FOR LOG-SYMPLECTIC MANIFOLDS IN GENERAL POSITION

We consider compact Kählerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure $\unicode[STIX]{x1D6F1}$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor $D(\unicode[STIX]{x1D6F1})$. We prove that $(X,\unicode[STIX]{x1D6F1})$ has unobstructed deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on $H^{2}$ of the open symplectic manifold $X\setminus D(\unicode[STIX]{x1D6F1})$, and in fact coincides with this $H^{2}$ provided the Hodge number $h_{X}^{2,0}=0$, and finally that the degeneracy locus $D(\unicode[STIX]{x1D6F1})$ deforms locally trivially under deformations of $(X,\unicode[STIX]{x1D6F1})$.

Symmetric differentials and variations of Hodge structures

Let D be a simple normal crossing divisor in a smooth complex projective variety X. We show that the existence on X-D of a non-trivial polarized complex variation of Hodge structures with integral monodromy implies that the pair (X,D) has a non-zero logarithmic symmetric differential (a section of a symmetric power of the logarithmic cotangent bundle). When the corresponding period map is generically immersive, we show more precisely that the logarithmic cotangent bundle is big.