Related papers: A generic global Torelli theorem for certain Horik…

The weighted triangulation algebras associated to triangulation quivers and their socle deformations were recently introduced and studied in [15]-[20] and [2]. These algebras, based on surface triangulations and originated from the theory…

Given a compact K\"ahler manifold, the Infinitesimal Torelli problem asks whether the differential of the period map of a Kuranishi family is injective. Unlike the classical Torelli theorem for curves, there is a negative answer for example…

In the 1970s O. Zariski introduced a general theory of equisingularity for algebroid and algebraic hypersurfaces over an algebraically closed field of characteristic zero. His theory builds up on understanding the dimensionality type of…

We construct a surface of general type with invariants \( \chi = K^2 = 1 \) and torsion group \( \Bbb{Z}/{2} \). We use a double plane construction by finding a plane curve with certain singularities, resolving these, and taking the double…

We introduce the notions of strong local Torelli and T-class for polarized manifolds, and prove that strong local Torelli implies global Torelli theorem on the Torelli spaces for polarized manifolds in the T-class. We discuss many new…

We construct a new family of minimal surfaces of general type with $p_g=q=2$ and $K^2=6$, whose Albanese map is a quadruple cover of an abelian surface with polarization of type $(1,3)$. We also show that this family provides an irreducible…

We establish a global Torelli theorem for the complete family of Calabi-Yau threefolds arising from cyclic triple covers of $\mathbb P^3$ branched along stable hyperplane arrangements.

We prove an equivalence between the infinitesimal Torelli theorem for top forms on a hypersurface contained inside a Grassmannian $\mathbb G$ and the theory of adjoint volume forms presented in L. Rizzi, F. Zucconi, "Generalized adjoint…

We classify all projective surfaces with only T-singularities, ample canonical class, and $K^2=2p_g-4$. In this way, we identify all surfaces, smoothable or not, with only T-singularities in the Koll\'ar--Shepherd-Barron--Alexeev (KSBA)…

This paper is devoted to the classification of irregular surfaces of general type with $p_g=q=2$ and non birational bicanonical map. The main result is that, if $S$ is such a surface and if $S$ is minimal with no pencil of curves of genus…

For each closed orientable surface we introduce a simplical complex with some additional structure which is a version of the complex of curves of this surface adjusted to investigation of its Torelli group. We call this complex the Torelli…

Minimal algebraic surfaces of general type $X$ such that $K^2_X=2\chi(\mathcal{O}_X)-6$ or $K^2_X=2\chi(\mathcal{O}_X)-5$ are called Horikawa surfaces. In this note we study $\mathbb{Z}^2_2$-actions on Horikawa surfaces. The main result is…

We prove a big monodromy result for a smooth family of complex algebraic surfaces of general type, with invariants p_g=q=1 and K^2=3, that has been introduced by Catanese and Ciliberto. This is accomplished via a careful study of…

We construct a new minimal complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$ (in fact $\pi_1^{\text{alg}}=\mathbb{Z}/4\mathbb{Z}$), which settles the existence question for numerical Campedelli surfaces…

In this paper, by using the Kuranishi coordinates on the Teichm\"uller space and the explicit deformation formula of holomorphic one-forms on Riemann surface, we give an explicit expression of the period map and derive new differential…

We classify minimal complex surfaces of general type with $p_g=q=3$. More precisely, we show that such a surface is either the symmetric product of a curve of genus 3 or a free $\Z_2-$quotient of the product of a curve of genus 2 and a…

We give a constructive proof of the Hodge conjecture for complex $K3$ surfaces that does not rely on Torelli-type results. Starting with an arbitrary rational $(1,1)$-class $\alpha\in H^{1,1}(X,\mathbb{Q})$, we algorithmically build a…

Surfaces of amplitude 1 in ordinary projective space are of general type, but this need not be the case in weighted projective spaces. Indeed, there are 4 classes of quasi-smooth weighted hypersurfaces in $\mathbf{P}(1,2,a,b)$ of amplitude…

This is an improved version of the eprint previously entitled "Unexpected isomorphisms between hyperk\"ahler fourfolds." We study smooth projective hyperk\"ahler fourfolds that are deformations of Hilbert squares of K3 surfaces and are…

In this paper we construct a new family of simply connected minimal complex surfaces of general type with $p_g=1$, $q=0$, and $K^2=3, 4, 5, 6, 8$ using a $\mathbb{Q}$-Gorenstein smoothing theory. We also reconstruct minimal complex surfaces…