## Periodic Reporting for period 1 - ALKAGE (Algebraic and Kähler geometry)

**Reporting period:**2015-09-01

**to**2017-02-28

## Summary of the context and overall objectives of the project

"The goal of the ALKAGE project is to address a number of fundamental questions in algebraic and analytic geometry, in the following three directions

1) Entire curves, hyperbolic algebraic manifolds and the Green-Griffiths-Lang conjecture

2) Further investigations on the curvature of complex varieties and K\""ahler geometry

3) L² vanishing and extension theorems, and the abundance conjecture

All three directions are connected with major aspects of contemporary research in complex geometry. There are active developpements pursued in a large number of mathematical departments worldwide, and improvements of the related techniques are eagerly expected. In the long term, these questions might also have far reaching consequences in other domains of mathematics such as arithmetic geometry."

1) Entire curves, hyperbolic algebraic manifolds and the Green-Griffiths-Lang conjecture

2) Further investigations on the curvature of complex varieties and K\""ahler geometry

3) L² vanishing and extension theorems, and the abundance conjecture

All three directions are connected with major aspects of contemporary research in complex geometry. There are active developpements pursued in a large number of mathematical departments worldwide, and improvements of the related techniques are eagerly expected. In the long term, these questions might also have far reaching consequences in other domains of mathematics such as arithmetic geometry."

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

"Here is an informal summary of the progress made during the first 18 months of the ALKAGE project, in each of the 3 above directions. We refer to the publication list and corresponding abstracts for details, and only sketch here the main results that have been obtained.

1) Ya Deng (PhD student of the PI) has obtained in 2016 an effective proof of the Kobayashi conjecture (1970): for each dimension n>1, there exists an effectively computable degree d_0(n) such that any sufficiently general n-dimensional algebraic hypersurface of complex projective space of degree d >= d_0(n) is Kobayashi hyperbolic.

The proof is based on a similar (but non effective) result of Damian Brotbek that was achieved earlier in 2016. It uses in depth jet differential techniques introduced by J.-P. Demailly in the last 2 decades. Related results had also been announced 15 years ago by Y.T. Siu, but the new proof by Ya Deng is more geometric, more effective and considerably simpler to follow.

One should mention that these results would follow from the much deeper Green-Griffiths-Lang conjecture (which is one of the very long term goals of after ALKAGE). However, the GGL conjecture is still out of reach at this point, even though recent work of J.-P. Demailly seems to open a new strategy of attack.

2) In Kähler geometry, Philippe Eyssidieux has studied some examples of orbifold K\""ahler groups and solved a related special case of the famous Shafarevich conjecture (stating that the universal cover of a projective manifold is holomorphically convex). Stéphane Druel has extended the Beauville-Bogomolov decomposition theorem to singular projective varieties with trivial canonical

class and dimension at most $5$, assuming that the singularities are klt (the general case being still elusive - and a major unsolved problem in the classification theory of projective varieties). Youngjun Choi (PostDoc) has obtained an interesting result on the positivity of direct images of fiberwise Ricci-flat metrics on Calabi-Yau fibrations. Tao Zheng (PostDoc) has proved deep results on the existence time of the almost complex Chern-Ricci flow; especially, he gave a new "parabolic proof" of the Gauduchon conjecture asserting the existence of Gauduchon metrics of prescribed volume form.

3) In the direction of the abundance conjecture (a major unsolved problem of algebraic geometry), J.-P. Demailly has obtained new and much more general versions of the L² extension theorem due to Ohsawa and Takegoshi. First, end of 2015, the extension has been established for non necessarily reduced subvarieties, provided the latter are defined as zero varieties of a multiplier ideal sheaf. The proof is based on refined Bochner-Kodara-Nakano inequalities and L² approximation techniques. More recently (February 2017), in a joint paper with Junyan Cao and Shin-ichi Matsumura, a further very wide generalization has been obtained: holomorphic sections and higher cohomology classes are extendable under suitable almost optimal curvature conditions. This holds even when the hermitian metric involved are singular and the ambient manifold is Kähler and holomorphically convex, but not necessarily compact. The latter result also covers the case of relative morphisms over a base, and leaves some hope of using induction on dimension for the study of the generalized MMP and abundance conjectures."

1) Ya Deng (PhD student of the PI) has obtained in 2016 an effective proof of the Kobayashi conjecture (1970): for each dimension n>1, there exists an effectively computable degree d_0(n) such that any sufficiently general n-dimensional algebraic hypersurface of complex projective space of degree d >= d_0(n) is Kobayashi hyperbolic.

The proof is based on a similar (but non effective) result of Damian Brotbek that was achieved earlier in 2016. It uses in depth jet differential techniques introduced by J.-P. Demailly in the last 2 decades. Related results had also been announced 15 years ago by Y.T. Siu, but the new proof by Ya Deng is more geometric, more effective and considerably simpler to follow.

One should mention that these results would follow from the much deeper Green-Griffiths-Lang conjecture (which is one of the very long term goals of after ALKAGE). However, the GGL conjecture is still out of reach at this point, even though recent work of J.-P. Demailly seems to open a new strategy of attack.

2) In Kähler geometry, Philippe Eyssidieux has studied some examples of orbifold K\""ahler groups and solved a related special case of the famous Shafarevich conjecture (stating that the universal cover of a projective manifold is holomorphically convex). Stéphane Druel has extended the Beauville-Bogomolov decomposition theorem to singular projective varieties with trivial canonical

class and dimension at most $5$, assuming that the singularities are klt (the general case being still elusive - and a major unsolved problem in the classification theory of projective varieties). Youngjun Choi (PostDoc) has obtained an interesting result on the positivity of direct images of fiberwise Ricci-flat metrics on Calabi-Yau fibrations. Tao Zheng (PostDoc) has proved deep results on the existence time of the almost complex Chern-Ricci flow; especially, he gave a new "parabolic proof" of the Gauduchon conjecture asserting the existence of Gauduchon metrics of prescribed volume form.

3) In the direction of the abundance conjecture (a major unsolved problem of algebraic geometry), J.-P. Demailly has obtained new and much more general versions of the L² extension theorem due to Ohsawa and Takegoshi. First, end of 2015, the extension has been established for non necessarily reduced subvarieties, provided the latter are defined as zero varieties of a multiplier ideal sheaf. The proof is based on refined Bochner-Kodara-Nakano inequalities and L² approximation techniques. More recently (February 2017), in a joint paper with Junyan Cao and Shin-ichi Matsumura, a further very wide generalization has been obtained: holomorphic sections and higher cohomology classes are extendable under suitable almost optimal curvature conditions. This holds even when the hermitian metric involved are singular and the ambient manifold is Kähler and holomorphically convex, but not necessarily compact. The latter result also covers the case of relative morphisms over a base, and leaves some hope of using induction on dimension for the study of the generalized MMP and abundance conjectures."

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

"One of the very long term goals of our research would be a general classification theory of projective varieties and compact Kähler spaces. There is still a very long route to reach a complete understanding of the structure of such varieties. Among the many steps required, one can cite basic vanishing, non vanishing, and extension theorems, in connection with an analysis of fundamental fibrations in relation with the "curvature" properties of the tangent sheaf. Our recent results are substantial steps in these directions."