arXiv research

New proof for K-stability of certain Fano varieties.

The paper extends Dunfield's theorem about 3-manifold holonomy maps and volume functions.

After recalling the notion of caustics of plane curves and basic equations, we first show the birationality of the caustic map for a general source point S in the plane. Then we prove more generally a theorem for curves D in the projective space of 3x3 symmetric matrices B. For a general 3x1 vector S the projection to …

New proof for Fano manifolds, showing rigidity and stability.

Let M be an orientable, cusped hyperbolic 3-manifold of finite volume. We show that the restriction map from a Dehn surgery component in the PSL(2,C)-character variety of M to the character variety of the boundary of M is a birational isomorphism onto its image. This generalises a result by Nathan Dunfield. A key step …

New invariant distinguishes real algebraic surfaces.

We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in $\Bbb C^n$, then any local diffeomorphism $F:X \to \Bbb C^n$ of simply connected manifolds which is a $d$-sheeted cover away from $D$ has degree $d=1$ or $d=\infty$ (however all degrees $d > 1$ are poss…

Proves K-stability and superrigidity of certain singular Fano hypersurfaces.

We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kaehler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and tho…

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

Groups of birational transformations with specific properties are conjugate to groups of pseudo-automorphisms.

The study proves Fano complete intersections' rigidity and stability under certain conditions.

A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces…

3D hyperbolic manifolds map one-to-one to their boundary character varieties.

Study properties of algebraic threefolds with specific singularities.

We study holomorphic $(n+1)$-chains $E_n\to E_{n-1} \to >... \to E_0$ consisting of holomorphic vector bundles over a compact Riemann surface and homomorphisms between them. A notion of stability depending on $n$ real parameters was introduced in the work of the first two authors and moduli spaces were constructed by t…

The paper proves residual finiteness of certain lattices and constructs surfaces with specific fundamental groups.

Study on mapping classes of real rational surface automorphisms, focusing on reducible maps and pseudo-Anosov maps.

In the present paper and the companion paper [8] a probabilistic (statistical mechanical) approach to the study of canonical metrics and measures on a complex algebraic variety X is introduced. On any such variety with positive Kodaira dimension a canonical (birationally invariant) random point processes is defined and…