Search results for "Algebraic Geometry"
showing 10 items of 356 documents
An Introduction to Hodge Structures
2015
We begin by introducing the concept of a Hodge structure and give some of its basic properties, including the Hodge and Lefschetz decompositions. We then define the period map, which relates families of Kahler manifolds to the families of Hodge structures defined on their cohomology, and discuss its properties. This will lead us to the more general definition of a variation of Hodge structure and the Gauss-Manin connection. We then review the basics about mixed Hodge structures with a view towards degenerations of Hodge structures; including the canonical extension of a vector bundle with connection, Schmid’s limiting mixed Hodge structure and Steenbrink’s work in the geometric setting. Fin…
On the topology of surfaces with the generalised simple lift property.
2020
In this paper, we study the geometry of surfaces with the generalised simple lift property. This work generalises previous results by Bernstein and Tinaglia, and it is motivated by the fact that leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the generalised simple lift property.
Projective resolutions associated to projections
2000
In this paper we will describe projective resolutions of d dimensional Cohen–Macaulay spaces X by means of a projection of X to a hypersurface in d+1-dimensional space. We will show that for a certain class of projections, the resulting resolution is minimal.
New degeneration of Fay's identity and its application to integrable systems
2011
In this paper, we find a new degenerated version of Fay's trisecant identity; this degeneration corresponds to the limit when the four points entering the trisecant identity coincide pairwise. This degenerated version of Fay's identity is used to construct algebro-geometric solutions to the multi-component nonlinear Schrodinger equation. This identity also leads to an independent derivation of algebro-geometric solutions to the Davey–Stewartson equations previously obtained in [17] in the framework of the Krichever scheme. We also give the condition of smoothness of the obtained solutions.
Hodge Theory and Algebraic Cycles
2006
Algebraic cycles and Hodge theory, in particular Chow groups, Deligne cohomology and the study of cycle class maps were some of the themes of the Schwerpunkt ”Globale Methoden in der Komplexen Geometrie”. In this survey we report about several projects around the structure of (higher) Chow groups CH(X,n) [3] which the author has studied with his coauthors during this time by using different methods. In my opinion there are two interesting view points: first the internal structure of higher Chow groups, i.e., the existence of interesting elements and nontriviality of parts of their Bloch-Beilinson filtrations. This case has arithmetic and geometric features, and the groups in question show d…
On stability of logarithmic tangent sheaves. Symmetric and generic determinants
2021
We prove stability of logarithmic tangent sheaves of singular hypersurfaces D of the projective space with constraints on the dimension and degree of the singularities of D. As main application, we prove that determinants and symmetric determinants have stable logarithmic tangent sheaves and we describe an open dense piece of the associated moduli space.
Logarithmic bundles of deformed Weyl arrangements of type $A_2$
2016
We consider deformations of the Weyl arrangement of type $A_2$, which include the extended Shi and Catalan arrangements. These last ones are well-known to be free. We study their sheaves of logarithmic vector fields in all other cases, and show that they are Steiner bundles. Also, we determine explicitly their unstable lines. As a corollary, some counter-examples to the shift isomorphism problem are given.
On Higgs bundles over Shimura varieties of ball quotient type
2016
We prove the generic exclusion of certain Shimura varieties of unitary and orthogonal types from the Torelli locus. The proof relies on a slope inequality on surface fibration due to G. Xiao, and the main result implies that certain Shimura varieties only meet the Torelli locus in dimension zero.
Cluster tilting for one-dimensional hypersurface singularities
2008
In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfy $\tau^2=\id$. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities.
Holomorphic de Rham Cohomology
2017
We are going to define a natural comparison isomorphism between algebraic de Rham cohomology and singular cohomology of varieties over the complex numbers with coefficients in \(\mathbb {C}\). The link is provided by holomorphic de Rham cohomology, which we study in this chapter.