arXiv research

In this paper we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space o…

We consider projective varieties with degenerate Gauss image whose focal hypersurfaces are non-reduced schemes. Examples of this situation are provided by the secant varieties of Severi and Scorza varieties. The Severi varieties are moreover characterized by a uniqueness property.

New proof of divisibility property for certain algebraic varieties.

Each of the four critical Severi varieties arises from a minimal holomorphic nilpotent orbit in a simple regular rank 3 hermitian Lie algebra and each such variety lies as singular locus in a cubic--the chordal variety--in the corresponding complex projective space; the cubic and projective space are identified in term…

Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projec…

The Hermitian symmetric space $M=\mathrm{EIII}$ appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure. This means the existence of a real oriented Euclidean vector bundle $E$ over it together with an algebra bundle morphism $\varphi:\mathrm{Cl}^0(E) …

A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes.

The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones,…

Continuity of Kähler-Einstein potentials at singularities proven.

We study the local differential geometry of varieties $X^n\subset \Bbb C\Bbb P^{n+a}$ with degenerate secant and tangential varieties. We show that the second fundamental form of a smooth variety with degenerate tangential variety is subject to certain rank restrictions. The rank restrictions imply a slightly refined v…

We give an exposition of a theorem of Hirzebruch, Kodaira and Yau which proves the uniqueness of the Kahler structure of complex projective space, and of Yau's resolution of the Severi Conjecture.

Let $X$ be an irreducible smooth geometrically integral projective surface over a field. In this paper we give an effective bound in terms of the Neron--Severi rank $ρ(X)$ of $X$ for the number of irreducible curves $C$ on $X$ with negative self-intersection and geometric genus less than $b_1(X)/4$, where $b_1(X)$ is t…

A remarkable and elementary fact that a locally compact set F of Euclidean space is a smooth manifold if and only if the lower and upper paratangent cones to F coincide at every point, is proved. The celebrated von Neumann's result (1929) that a locally compact subgroup of the general linear group is a smooth manifold,…

We list up to Möbius equivalence all possible degrees and embedding dimensions of real surfaces that are covered by at least two pencils of circles, together with the number of such pencils. In addition, we classify incidences between the contained circles, complex lines and isolated singularities. Such geometric chara…

Study deformation invariants of projective surfaces using twisted sheaves and Azumaya algebras.

This research extends quasiplurisubharmonic functions on compact Kähler manifolds.

This paper proves all Pfaffian varieties are area-minimizing except hypersurfaces.

The paper shows how certain complex projective varieties can be broken down into simpler types.

Study of braid varieties and their Legendrian isotopy.

We introduce the fibred toric varieties as equivariant $\mathbb{C}P^r$ bundles over lower dimensional toric varieties. An equivalent characterization is that the natural morphisms on them degenerate to bundle projections in the context of variation of toric varieties as GIT quotients. Our main observation is that these…

The study shows boundedness of certain fibered varieties in algebraic geometry.

Study identifies subvarieties of projective varieties mapping to models.

Paper describes holomorphic polyvector fields on toric varieties.

The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.

Cominuscule subvarieties found in flag varieties.

Kähler-Einstein metrics found on special types of symmetric varieties.

Study empty polar varieties' impact on singular function-germs.

We investigate the secant dimensions and the identifiablity of flag varieties parametrizing flag of sub vector spaces of a fixed vector space. We give numerical conditions ensuring that secant varieties of flag varieties have the expected dimension, and that a general point on these secant varieties is identifiable.

Character varieties get a natural Poisson structure.

Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomog…

The dual variety X* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X* is a hypersurface in the dual space (P^N)*. If dim X* < N - 1, then the variety X is called dually degenerate. The authors refine these definitions for a varie…

The paper classifies certain singular projective varieties with specific properties.

Proves minimality of tensor varieties, generalizing previous results.

Study projective KLT varieties with projectively flat cotangent sheaves.

The paper examines when real matrix Schubert varieties are minimal submanifolds.

New connections on symmetric spaces with invariant properties.

Solves open problems on curved projective varieties.

Stratifies representation varieties of twisted Hopf links.

Classifies holomorphic parabolic geometries on complex manifolds.

Character variety of Borromean link solved, Alexander polynomial formula found.

The article examines twisted cohomologies on algebraic and analytic varieties.

Proves properness of K-moduli spaces for Fano varieties.

Extends Kollár's result to fibered Calabi-Yau varieties with cohomological assumption.

BNSR invariants are contained in the complement of tropical varieties.

Integral points are potentially dense in character varieties of quasi-projective varieties.

A family of new algebraic Poisson varieties will be constructed, generalising the complex character varieties of Riemann surfaces. Then the well-known (Poisson) mapping class group actions on the character varieties will be generalised.

Study characteristic classes of a specific type of determinantal varieties.

In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the {\it rigidity} of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieti…