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Adrian Langer

Publications and source records attributed to Adrian Langer.

At least 19 recordsLinked to original sources
arXiv2026

Ample vector bundles on non-proper schemes

We solve two open problems on ample vector bundles posed by Hartshorne in 1966. We prove that tensor products of ample vector bundles on schemes of finite type over an algebraically closed field are ample in arbitrary characteristic, extending Hartshorne's characteristic-zero result and Barton's projective positive-characteristic theorem. More generally, let f: X -> S be a morphism of schemes. Then we prove that tensor products of f-ample vector bundles are f-ample. Moreover, if E is an f-ample vector bundle of rank r>0 and W is a finite locally free polynomial GL(r,S)-module of positive rank with W_0 = 0, then E(W) is f-ample. In particular, Gamma^n E is f-ample for every n>0. Finally, adapting a construction of Ejiri-Fujino-Iwai, we show that in every characteristic a smooth quasi-projective surface carries a non-ample extension of an ample line bundle by itself.

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arXiv2026

Isocrystals on unibranch varieties and open immersions

We prove that for a connected, geometrically unibranch variety $X$ over a perfect field of positive characteristic and a dense open subset $U \subset X$, the canonical homomorphism between the Tannaka duals of the categories of overconvergent isocrystals is a quotient map. We also show that the analogous statement for convergent isocrystals, and for all isocrystals, fails. The counterexample is based on Crew's construction of a unit-root $F$-isocrystal that is convergent but not overconvergent.

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arXiv2025

$F$-divided bundles on normal $F$-finite schemes

In this paper we study $F$-divided bundles on irreducible Noetherian normal $F$-finite $\mathbb{F}_p$-schemes and we show that their Tannakian category is governed by the behaviour at the generic point. In particular, if $U\subset X$ is an open subset of a normal variety defined over an algebraically closed field then the corresponding homomorphism of $F$-divided fundamental groups is faithfully flat. This is analogous to a known fact about the topological fundamental group of an open subset of a normal complex analytic variety. We use this result to show that simply connected, proper, normal varieties in positive characteristic admit no nontrivial $F$-divided bundles. This generalizes an earlier result of H. Esnault and V. Mehta concerning smooth projective varieties, and settles Gieseker's conjecture in a more general setting.

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arXiv2025

Relative Gieseker's problem on $F$-divided bundles

Let $f: X\to Y$ be a proper surjective morphism of varieties defined over an algebraically closed field of positive characteristic. We prove that if $f$ has geometrically connected fibers then the induced homomorphism of $F$-divided fundamental groups is faithfully flat. An important new ingredient in our proof is an analogue of B. Bhatt's and P. Scholze's descent theorem \cite[Theorem 1.3]{Bhatt-Scholze2017} for $F$-divided bundles. As a corollary, we prove that in general if $X$ is normal, $Y$ is smooth, both $X$ and $Y$ are projective, and the induced map on \'etale fundamental groups is surjective, then the corresponding homomorphism on $F$-divided fundamental groups is faithfully flat. We also establish an analogous result for isomorphisms. This generalizes and strengthens a recent result of X. Sun and L. Zhang \cite{Sun-Zhang2025}, which in turn generalized earlier results of H. Esnault and V. Mehta \cite{Esnault-Mehta2010} and I. Biswas, M. Kumar, and A. J. Parameswaran \cite{Biswas-Parameswaran-Kumar2025}.

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arXiv2024

Projective contact log varieties

We study contact structures on smooth complex projective varieties with a simple normal crossing divisor, generalizing some well-known results concerning the non-logarithmic case. In particular, we describe the structure of elementary log contractions of such log varieties and we construct the corresponding contact structures.

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arXiv2023

Bridgeland stability conditions on normal surfaces

We prove a new version of Bogomolov's inequality on normal proper surfaces. This allows to construct Bridgeland's stability condition on such surfaces. In particular, this gives the first known examples of stability conditions on non-projective, proper schemes.

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arXiv2023

Simpson's correspondence on singular varieties in positive characteristic

The main aim of the paper is to provide analogues of Simpson's correspondence on singular projective varieties defined over an algebraically closed field of characteristic $p>0$. There are two main cases. In the first case, we consider analogues of numerically flat vector bundles on a big open subset of a normal projective variety (with arbitrary singularities). Here we introduce the S-fundamental group scheme for quasi-projective varieties that admit compactifications with complement of codimension $\ge 2$. We prove that this group scheme coincides with the S-funda\-men\-tal group scheme of the regular locus of any of its (small) projective compactification. In particular, it provides a new invariant for normal projective varieties isomorphic in codimension $1$. In the second case, we consider vector bundles with an integrable $\lambda$-connection on a normal projective variety $X$ that is (almost) liftable modulo $p^2$. In this case we restrict to varieties with $F$-liftable singularities that are analogous to log canonical singularities. We prove that a semistable Higgs vector bundle on the regular locus of $X$, with appropriately defined vanishing Chern classes, admits a canonical Higgs--de Rham flow. This provides an analogue of some results due to D. Greb, S. Kebekus, B. Taji and T. Peternell. Finally, we give some applications of the obtained results, also to varieties defined in characteristic zero.

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arXiv2022

Intersection theory and Chern classes on normal varieties

We study intersection theory and Chern classes of reflexive sheaves on normal varieties. In particular, we define generalization of Mumford's intersection theory on normal surfaces to higher dimensions. We also define and study the second Chern class for reflexive sheaves on normal varieties. We use these results to prove some Bogomolov type inequalities on normal varieties in positive characteristic. We also prove some new boundedness results on normal varieties in positive characteristic.

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arXiv2022

Bogomolov's inequality and Higgs sheaves on normal varieties in positive characteristic

We prove Bogomolov's inequality on a normal projective variety in positive characteristic and we use it to show some new restriction theorems and a new boundedness result. Then we redefine Higgs sheaves on normal varieties and we prove restriction theorems and Bogomolov type inequalities for semistable logarithmic Higgs sheaves on some normal varieties in an arbitrary characteristic.

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arXiv2022

Approximation of semistable bundles on smooth algebraic varieties

We prove some strong results on approximation of strongly semistable bundles with vanishing numerical Chern classes by filtrations, whose quotients are line bundles of similar slope. This generalizes some earlier results of Parameswaran-Subramanian in the curve case and Koley-Parameswaran in the surface case and it confirms the conjecture posed by Koley and Parameswaran.

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arXiv2021

On algebraic Chern classes of flat vector bundles

We show that under some assumptions on the monodromy group some combinations of higher Chern classes of flat vector bundles are torsion in the Chow group. Similar results hold for flat vector bundles that deform to such flat vector bundles (also in case of quasi-projective varieties). The results are motivated by Bloch's conjecture on Chern classes of flat vector bundles on smooth complex projective varities but in some cases they give a more precise information. We also study Higgs version of Bloch's conjecture and analogous problems in the positive characteristic case.

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arXiv2021

Moduli spaces of semistable modules over Lie algebroids

We show a few basic results about moduli spaces of semistable modules over Lie algebroids. The first result shows that such moduli spaces exist for relative projective morphisms of noetherian schemes, removing some earlier constraints. The second result proves general separatedness Langton type theorem for such moduli spaces. More precisely, we prove S-completness of some moduli stacks of semistable modules. In some special cases this result identifies closed points of the moduli space of Gieseker semistable sheaves on a projective scheme and of the Donaldson--Uhlenbeck compactification of the moduli space of slope stable locally free sheaves on a smooth projective surface. The last result generalizes properness of Hitchin's morphism and it shows properness of so called Hodge-Hitchin morphism defined in positive characteristic on the moduli space of Gieseker semistable integrable t-connections in terms of the p-curvature morphism. This last result was proven in the curve case by de Cataldo and Zhang using completely different methods.

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arXiv2021

On boundedness of semistable sheaves

We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolov's inequality for semistable sheaves on a smooth projective variety of any dimension $\ge 2$ without using any restriction theorems.

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arXiv2019

On birational boundedness of foliated surfaces

In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function $P: \mathbb Z_{\geq 0}\to \mathbb Z $, then there exists an integer $N_1>0$ such that if $(X,\mathcal F)$ is a canonical or nef model of a foliation of general type with Hilbert polynomial $\chi (X, mK_{\mathcal F})=P(m)$ for all $m\in \mathbb Z_{\geq 0}$, then $|mK_{\mathcal F}|$ defines a birational map for all $m\geq N_1$. We also prove a Grauert-Riemannschneider type vanishing theorem for foliated surfaces with canonical singularities.

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arXiv2019

On smooth projective D-affine varieties

We show various properties of smooth projective D-affine varieties. In particular, any smooth projective D-affine variety is algebraically simply connected and its image under a fibration is D-affine. In characteristic zero such D-affine varieties are also uniruled. We also show that (apart from a few small characteristics) a smooth projective surface is D-affine if and only if it is isomorphic to either ${\mathbb P}^2$ or ${\mathbb P}^1\times {\mathbb P}^1$. In positive characteristic, a basic tool in the proof is a new generalization of Miyaoka's generic semipositivity theorem.

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arXiv2019

Birational geometry of compactifications of Drinfeld half-spaces over a finite field

We study compactifications of Drinfeld half-spaces over a finite field. In particular, we construct a purely inseparable endomorphism of Drinfeld's half-space $Ω(V)$ over a finite field $k$ that does not extend to an endomorphism of the projective space $P (V)$. This should be compared with theorem of Rémy, Thuillier and Werner that every $k$-automorphism of $Ω(V)$ extends to a $k$-automorphism of $P (V)$. Our construction uses an inseparable analogue of the Cremona transformation. We also study foliations on Drinfeld's half-spaces. This leads to various examples of interesting varieties in positive characteristic. In particular, we show a new example of a non-liftable projective Calabi-Yau threefold in characteristic $2$ and we show examples of rational surfaces with klt singularities, whose cotangent bundle contains an ample line bundle.

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arXiv2019

Nearby cycles and semipositivity in positive characteristic

We study restriction of logarithmic Higgs bundles to the boundary divisor and we construct the corresponding nearby-cycles functor in positive characteristic. As applications we prove some strong semipositivity theorems for analogs of complex polarized variations of Hodge structures and their generalizations. This implies, e.g., semipositivity for the relative canonical divisor of a semistable reduction in positive characteristic and it gives some new strong results generalizing semipositivity even for complex varieties.

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