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Deformation openness of big fundamental groups and applications. New (joint work with Chikako Mese & Botong Wang) arXiv:2412.08636
Abstract
In 2001, de Oliveira, Katzarkov, and Ramachandran conjectured that the property of smooth projective varieties having big fundamental groups is stable under small deformations. This conjecture was proven by Benoît Claudon in 2010 for surfaces and for threefolds under suitable assumptions. In this paper, we prove this conjecture for smooth projective varieties admitting a big complex local system. Moreover, we address a more general conjecture by Campana and Claudon concerning the deformation invariance of the -dimension of projective varieties. As an application, we establish the deformation openness of pseudo-Brody hyperbolicity for projective varieties endowed with a big and semisimple complex local system. To achieve these results, we develop the deformation regularity of equivariant pluriharmonic maps into Euclidean buildings and Riemannian symmetric spaces in families, along with techniques from the reductive and linear Shafarevich conjectures.
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Existence and unicity of pluriharmonic maps to Euclidean buildings and applications. New (joint work with Chikako Mese) arXiv:2410.07871
Abstract
Given a complex smooth quasi-projective variety , a reductive algebraic group defined over some non-archimedean local field and a Zariski dense representation , we construct a -equivariant pluriharmonic map from the universal cover of into the Bruhat-Tits building of , with appropriate asymptotic behavior. We also establish the uniqueness of such a pluriharmonic map in a suitable sense, and provide a geometric characterization of these equivariant maps.
This paper builds upon and extends previous work by the authors jointly with G. Daskalopoulos and D. Brotbek.
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-vanishing theorem and a conjecture of Kollár. New (joint work with Botong Wang) arXiv:2409.11399
Abstract
In 1995, Kollár conjectured that a complex projective -fold with generically large fundamental group has Euler characteristic . In this paper, we confirm the conjecture assuming has linear fundamental group, i.e., there exists an almost faithful representation . We deduce the conjecture by proving a stronger vanishing theorem: for the universal cover of such , its -Dolbeaut cohomology for . The main ingredients of the proof are techniques from the linear Shafarevich conjecture along with some analytic methods.
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Linear Chern-Hopf-Thurston Conjecture. (joint work with Botong Wang) arXiv:2405.12012
Abstract
If is a closed -dimensional aspherical manifold, i.e., the universal cover of is contractible, then the Chern-Hopf-Thurston conjecture predicts that . We prove this conjecture when is a complex projective manifold whose fundamental group admits an almost faithful linear representation over any field. In fact, we prove a much stronger statement that if is a complex projective manifold with large fundamental group and admits an almost faithful linear representation, then for any perverse sheaf on .
To prove the main result, we introduce a vanishing cycle functor of multivalued one-forms. Then using techniques from non-abelian Hodge theories in both archimedean and non-archimedean settings, we deduce the desired positivity from the geometry of pure and mixed period maps.
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Linear Shafarevich Conjecture in Positive Characteristic, Hyperbolicity, and Applications. (joint work with Katsutoshi Yamanoi) arXiv:2403.16199
Abstract
Given a complex quasi-projective normal variety and a linear representation with any field of positive characteristic, we mainly establish the following results:
1. the construction of the Shafarevich morphism associated with .
2. In cases where is projective, is faithful and the -dimension of X is at most two (e.g. ), we prove that the Shafarevich conjecture holds for X.
3. In cases where is big, we prove that the Green-Griffiths-Lang conjecture holds for X.
4. When is big and the Zariski closure of is a semisimple algebraic group, we prove that is pseudo Picard hyperbolic, and strongly of log general type.
5. If is special or -special, then is virtually abelian.
We also prove Claudon-Höring-Kollár's conjecture for complex projective manifolds with linear fundamental groups of any characteristic.
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Quasi-finiteness of morphisms of character varieties. (joint work with Yuan Liu) arXiv:2311.13299
Abstract
Let be a morphism between smooth complex quasi-projective varieties and be the closure of with the inclusion map. We prove that:
1. for any field , there exist finitely many semisimple representations with the minimal field contained in such that if is any representation satisfying , then for some .
2. The induced morphism between -character varieties (of any characteristic) of and is quasi-finite if is a finite index subgroup of .
These results extend the main results by Lasell in 1995 and Lasell-Ramachandran in 1996 from smooth complex projective varieties to quasi-projective cases with richer structures.
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Reductive Shafarevich Conjecture. (joint work with Katsutoshi Yamanoi, Ludmil Katzarkov) arXiv:2306.03070
Abstract
In this paper, we present a more accessible proof of Eyssidieux's proof of the reductive Shafarevich conjecture in 2004, along with several generalizations. In a nutshell, we prove the holomorphic convexity of the covering of a projective normal variety , which corresponds to the intersection of kernels of reductive representations . Our approach avoids the necessity of using the reduction mod p method employed in Eyssidieux's original proof. Moreover, we extend the theorems to singular normal varieties under a weaker condition of absolutely constructible subsets, thereby answering a question by Eyssidieux, Katzarkov, Pantev, and Ramachandran. Additionally, we construct the Shafarevich morphism for reductive representations over quasi-projective varieties unconditionally, and proving its algebraic nature at the function field level.
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Hyperbolicity and fundamental groups of complex quasi-projective varieties. (joint work with Benoit Cadorel and Katsutoshi Yamanoi) arXiv:2212.12225 Oberwolfach report
Abstract
It is natural to ask how fundamental groups of complex quasi-projective varieties determine their hyperbolicity. In this paper we address this question when there is a linear representation . First, if $\varrho$ is big and the Zariski closure of in is a semisimple algebraic group, then we prove that there is a proper Zariski closed subset such that any closed irreducible subvariety of not contained in $Z$ is of log general type; any holomorphic map from the punctured disk to with image not contained in does not have essential singularity at the origin. In particular, is of log general type and all entire curves in $X$ lie on . We provide examples to illustrate that the condition is sharp.
Second, if is a special or hyperbolically special quasi-projective manifold in the sense of Campana, we prove that is virtually nilpotent. We also construct examples to show that this result is optimal, thus disproving Campana's abelianity conjecture for quasi-projective manifolds.
To prove the above theorems we develop new feature in non-abelian Hodge, geometric group and Nevanlinna theories. Along the way we prove the strong Green-Griffiths-Lang conjecture for quasi-projective varieties admitting a morphism with
where is a semi-abelian variety; a reduction theorem for Zariski dense representations where is a reductive group over a non-Archimedean local field .
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Pluriharmonic maps into buildings and symmetric differentials. (joint work with D. Brotbek, G. Daskalopoulos and C. Mese) arXiv:2206.11835
Abstract
In this paper we develop some non-abelian Hodge techniques over complex quasi-projective manifolds in both archimedean and non-archimedean contexts. In the non-archimedean case, we first generalize a theorem by Gromov-Schoen: for any Zariski dense representation , where is a semisimple algebraic group defined over some non-archimedean local field $K$, we construct a -equivariant harmonic map from into the Bruhat-Tits building of with some suitable asymptotic behavior. We then construct logarithmic symmetric differential forms over when the image of such is unbounded. Our main result in the archimedean case is that any semisimple representation is rigid provided that does not admit logarithmic symmetric differential forms. Furthermore, such representation is conjugate to where is the ring of integer of some number field , so that is a complex direct factor of a -variation of Hodge structures.
As an application we prove that a complex quasi-projective manifold has nonzero global logarithmic symmetric differential forms if there is linear representation with infinite images.
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Picard hyperbolicity for manifolds admitting nilpotent harmonic bundles. (joint work with Benoît Cadorel). arXiv:2107.07550
Abstract
For a quasi-compact Kähler manifold endowed with a nilpotent harmonic bundle whose Higgs field is injective at one point, we prove that is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic, and is of log general type. Moreover, we prove that there is a finite unramified cover of from a quasi-projective manifold so that any projective compactification of is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic and is of general type. As a byproduct, we establish some criterion of pseudo-Picard hyperbolicity and pseudo-algebraic hyperbolicity for quasi-compact Kähler manifolds.