Related papers: Calabi problem for manifolds with edge-cone singul…
This paper first generalises the Bogomolov-Tian-Todorov unobstructedness theorem to the case of Calabi-Yau threefolds with canonical singularities. The deformation space of such a Calabi-Yau threefold is no longer smooth, but the general…
Building on work of Du, Gao, and Yau, we give a characterization of smooth solutions, up to normalization, of the complex Plateau problem for strongly pseudoconvex Calabi--Yau CR manifolds of dimension $2n-1 \ge 5$ and in the hypersurface…
We use arithmetic and Hodge-theoretic techniques to study pencils of Calabi-Yau varieties realized as highly symmetric hypersurfaces in Grassmannians and their quotients, demonstrating that their geometric properties are distinct from the…
In this paper, we study the form type Calabi-Yau equation. We define the astheno-Ricci curvature and prove that there exists a solution for the form type Calabi-Yau equation if the astheno-Ricci curvature is non-positive.
A quaternionic version of the Calabi problem on Monge-Ampere equation is introduced. It is a quaternionic Monge-Ampere equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For…
Motivated from mathematical aspects of the superstring theory, we introduce a new equation on a balanced, hermitian manifold, with zero first Chern class. Solving the equation, one will obtain, in each Bott--Chern cohomology class, a…
In the first part of this thesis, we study the Yamabe problem with singularities, that we can announce as follow: Given a compact Riemannian manifold $(M,g)$, find a constant scalar curvature metric, conformal to $g$, when $g$ has not…
M-theory compactification leads one to consider 7-manifolds obtained by rolling Calabi-Yau threefolds in the web of Calabi-Yau moduli spaces. The resulting 7-space in general has singularities governed by the extremal transition undergone.…
We define defect for hypersurfaces with A-D-E singularities in complex projective normal Cohen-Macaulay fourfolds having some vanishing properties of Bott-type and prove formulae for Hodge numbers of big resolutions of such hypersurfaces.…
We give a classification of smooth complex manifolds with a finite abelian group action, such that the quotient is isomorphic to a projective space. The case where the manifold is a Calabi-Yau is studied in detail.
Recent progress in the deformation theory of Calabi-Yau varieties $Y$ with canonical singularities has highlighted the key role played by the higher Du Bois and higher rational singularities, and especially by the so-called $k$-liminal…
We present new examples of affine Calabi--Yau manifolds of Euclidean volume growth and quadratic curvature decay, whose tangent cones at infinity are irregular and have smooth links. In the process, we demonstrate (and provide the relevant…
We produce local Calabi-Yau metrics on $\mathbf C^2$ with conical singularities along three or more complex lines through the origin whose cone angles strictly violate the Troyanov condition. The tangent cone at the origin is a flat…
We revisit the second order estimate for solutions to the quaternionic Calabi-Yau problem on hyperk\"ahler manifolds, originally established by Dinew and Sroka. In this note, we present a simplified argument to derive the estimate.
In this paper, we study the convergence of Calabi-Yau manifolds under K\"{a}hler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a…
We develop the deformation theory of Calabi-Yau threefolds, by which we mean 3-dimensional complex manifolds with a nowhere-vanishing holomorphic 3-form, on manifolds with boundary. The boundary data is a closed, real 3-form on the…
We review briefly the characteristic topological data of Calabi--Yau threefolds and focus on the question of when two threefolds are equivalent through related topological data. This provides an interesting test case for machine learning…
We investigate a method of construction of Calabi--Yau manifolds, that is, by smoothing normal crossing varieties. We develop some theories for calculating the Picard groups of the Calabi--Yau manifolds obtained in this method. Some…
In this paper we establish a necessary and sufficient condition for solving a general class of fully nonlinear elliptic equations on compact almost Hermitian manifolds, extending a recent work of Guo-Song. As applications, we determine the…
In this paper we classify a kind of special Calabi hypersurfaces with negative constant sectional curvature in Calabi affine geometry. Meanwhile, we find a class of new Euclidean complete and Calabi complete affine hypersurfaces, which…