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Existence and unicity of pluriharmonic maps to Euclidean buildings and applications. (joint work with Chikako Mese) arXiv:2410.07871
Abstract
Given a complex smooth quasi-projective variety \(X\), a reductive algebraic group \(G\) defined over some non-archimedean local field \(K\) and a Zariski dense representation \(\varrho:\pi_1(X)\to G(K)\), we construct a \(\varrho\)-equivariant pluriharmonic map from the universal cover of \(X\) into the Bruhat-Tits building \(\Delta(G)\) of \(G\), 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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\(L^2\) -vanishing theorem and a conjecture of Kollár. (joint work with Botong Wang) arXiv:2409.11399
Abstract
In 1995, Kollár conjectured that a complex projective \(n\)-fold \(X\) with generically large fundamental group has Euler characteristic \(\chi(X, K_X)\geq 0\). In this paper, we confirm the conjecture assuming \(X\) has linear fundamental group, i.e., there exists an almost faithful representation \(\pi_1(X)\to {\rm GL}_N(\mathbb{C})\). We deduce the conjecture by proving a stronger \(L^2\) vanishing theorem: for the universal cover \(\widetilde{X}\) of such \(X\), its \(L^2\)-Dolbeaut cohomology \(H_{(2)}^{n,q}(\widetilde{X})=0\) for \(q\neq 0\). The main ingredients of the proof are techniques from the linear Shafarevich conjecture along with some analytic methods.
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Linear Singer-Hopf Conjecture. (joint work with Botong Wang) arXiv:2405.12012
Abstract
If \(X\) is a closed \(2n\)-dimensional aspherical manifold, i.e., the universal cover of \(X\) is contractible, then the Singer-Hopf conjecture predicts that \((-1)^n\chi(X)\geq 0\). We prove this conjecture when \(X\) 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 \(X\)is a complex projective manifold with large fundamental group and \(\pi_1(X)\) admits an almost faithful linear representation, then \(\chi(X, \mathcal{P})\geq 0\) for any perverse sheaf \(\mathcal{P}\) on \(X\).
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 \(X\) and a linear representation \(\rho:\pi_1(X)\to {\rm GL}_N(K)\) with \(K\) any field of positive characteristic, we mainly establish the following results:
1. the construction of the Shafarevich morphism associated with \(\rho\).
2. In cases where \(X\) is projective, \(\rho\) is faithful and the \(\Gamma\)-dimension of X is at most two (e.g. \({\rm dim} X=2\)), we prove that the Shafarevich conjecture holds for X.
3. In cases where \(\rho\) is big, we prove that the Green-Griffiths-Lang conjecture holds for X.
4. When \(\rho\) is big and the Zariski closure of \(\rho(\pi_1(X))\) is a semisimple algebraic group, we prove that \(X\) is pseudo Picard hyperbolic, and strongly of log general type.
5. If \(X\) is special or \(h\)-special, then \(\rho(\pi_1(X))\) 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 \(f: Y\to X\) be a morphism between smooth complex quasi-projective varieties and \(Z\) be the closure of \(f(Y)\) with \(\iota: Z\to X\) the inclusion map. We prove that:
1. for any field \(K\), there exist finitely many semisimple representations \(\{\tau_i:\pi_1(Z)\to {\rm GL}_N(\overline{k})\}_{i=1,\ldots,\ell}\) with \(k\subset K\) the minimal field contained in \(K\) such that if \(\varrho:\pi_1(X)\to {\rm GL}_{N}(K)\) is any representation satisfying \([f^*\varrho]=1\), then \([\iota^*\varrho]=[\tau_i]\) for some \(i\).
2. The induced morphism between \({\rm GL}_N\)-character varieties (of any characteristic) of \(\pi_1(X)\) and \(\pi_1(Y)\) is quasi-finite if \({\rm Im}[\pi_1(Z)\to \pi_1(X)]\) is a finite index subgroup of \(\pi_1(X)\).
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 \(X\), which corresponds to the intersection of kernels of reductive representations \(\varrho:\pi_1(X)\to {\rm GL}_N(\mathbb{C})\). 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 \(\pi_1(X)\) of complex quasi-projective varieties \(X\) determine their hyperbolicity. In this paper we address this question when there is a linear representation \(\varrho:\pi_1(X)\to GL_N(\mathbb{C})\). First, if $\varrho$ is big and the Zariski closure of \(\varrho\big(\pi_1(X)\big)\) in \(GL_N(\mathbb{C})\) is a semisimple algebraic group, then we prove that there is a proper Zariski closed subset \(Z\subsetneqq X\) such that any closed irreducible subvariety \(V\) of \(X\) not contained in $Z$ is of log general type; any holomorphic map from the punctured disk \(\Delta^*\) to \(X\) with image not contained in \(Z\) does not have essential singularity at the origin. In particular, \(X\) is of log general type and all entire curves in $X$ lie on \(Z\). We provide examples to illustrate that the condition is sharp.
Second, if \(X\) is a special or hyperbolically special quasi-projective manifold in the sense of Campana, we prove that \(\varrho\big(\pi_1(X)\big)\) 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 \(X\) admitting a morphism \(a:X\to A\) with \(\dim X=\dim a(X)\)
where \(A\) is a semi-abelian variety; a reduction theorem for Zariski dense representations \(\pi_1(X)\to G(K)\) where \(G\) is a reductive group over a non-Archimedean local field \(K\).
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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 \(X\) 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 \(\rho:\pi_1(X)\to G(K)\), where \(G\) is a semisimple algebraic group defined over some non-archimedean local field $K$, we construct a \(\rho\)-equivariant harmonic map from \(X\) into the Bruhat-Tits building \(\Delta(G)\) of \(G\) with some suitable asymptotic behavior. We then construct logarithmic symmetric differential forms over \(X\) when the image of such \(\rho\) is unbounded. Our main result in the archimedean case is that any semisimple representation \(\sigma: \pi_1(X)\to\operatorname{GL}_N(\mathbb{C})\) is rigid provided that \(X\) does not admit logarithmic symmetric differential forms. Furthermore, such representation \(\sigma\) is conjugate to \(\sigma':\pi_1(X)\to\operatorname{GL}_N(\mathcal{O}_L)\) where \(\mathcal{O}_L\) is the ring of integer of some number field \(L\), so that \(\sigma'\) is a complex direct factor of a \(\mathbb{Z}\)-variation of Hodge structures.
As an application we prove that a complex quasi-projective manifold \(X\) has nonzero global logarithmic symmetric differential forms if there is linear representation \(\pi_1(X)\to\operatorname{GL}_N(\mathbb{C})\) 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 \(U\) endowed with a nilpotent harmonic bundle whose Higgs field is injective at one point, we prove that \(U\) is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic, and is of log general type. Moreover, we prove that there is a finite unramified cover \(\tilde{U}\) of \(U\) from a quasi-projective manifold \(\tilde{U}\) so that any projective compactification of \(\tilde{U}\) 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.
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