Gergely Berczi | Searcharxiv

arXiv2015

Thom polynomials of Morin singularities and the Green-Griffiths-Lang conjecture

The Green-Griffiths-Lang conjecture says that for every complex projective algebraic variety $X$ of general type there exists a proper algebraic subvariety of $X$ containing all nonconstant entire holomorphic curves $f:\mathbb{C} \to X$. We construct a compactification of the invariant jet differentials bundle over complex manifolds motivated by an algebraic model of Morin singularities and we develop an iterated residue formula using equivariant localisation for tautological integrals over it. We show that the polynomial GGL conjecture for a generic projective hypersurface of degree $\mathrm{deg}(X)>2n^{10}$ follows from a positivity conjecture for Thom polynomials of Morin singularities.

math.AG↗

arXiv2015

Towards the Green-Griffiths-Lang conjecture via equivariant localisation

The Green-Griffiths-Lang conjecture says that for every complex projective algebraic variety $X$ of general type there exists a proper algebraic subvariety of $X$ containing all nonconstant entire holomorphic curves $f:\mathbb{C} \to X$. Using equivariant localisation on the Demailly-Semple jet differentials bundle we give an affirmative answer to this conjecture for generic projective hypersurfaces $X \subset \mathbb{P}^{n+1}$ of degree $\mathrm{deg}(X) \ge n^{9n}$.

math.AG↗

arXiv2012

A geometric construction for invariant jet differentials

Motivated by Demailly's strategy towards the Kobayashi hyperbolicity conjecture, we study the action on the k-jets of germs of holomorphic discs in a complex manifold X of the reparametrization group of k-jets of germs of biholomorphisms of the source. This reparametrization group is a subgroup of the general linear group GL(k) which is not reductive, but nonetheless we show that its invariants for any linear action which extends to GL(k) form a finitely generated algebra, and give a new geometric description of the Demailly-Semple algebra of invariant jet differentials.

math.AG↗

arXiv2008

Thom polynomials of Morin singularities

We prove a formula for Thom polynomials of Morin (or A_d) singularities in any codimension. We use a combination of the test-curve method of Porteous, and the localization methods in equivariant cohomology. Our formulas are independent of the codimension, and they are computationally efficient for d less than 7.

math.AT↗