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Christian Schnell

Publications and source records attributed to Christian Schnell.

At least 19 recordsLinked to original sources
arXiv2026

Meromorphic Group Actions and the Support Theorem for Lagrangian Fibrations

We prove several new results about Lagrangian fibrations on holomorphic symplectic complex spaces, under the assumption that the total space is K\"ahler (but possibly non-compact or singular) and that the base is a complex manifold. First, we construct a meromorphic action by a family of meromorphic groups. Second, we use this structure, together with Hodge-theoretic methods, to prove a version of Ng\^o's support theorem for Lagrangian fibrations. Along the way, we prove a freeness theorem for the cohomology of compact K\"ahler spaces equipped with a meromorphic group action.

math.AG↗
arXiv2024

A log resolution for the theta divisor of a hyperelliptic curve

In this paper, we prove that the theta divisor of a smooth hyperelliptic curve has a natural and explicit embedded resolution of singularities using iterated blowups of Brill-Noether subvarieties. We also show that the Brill-Noether stratification of the hyperelliptic Jacobian is a Whitney stratification.

math.AG↗
arXiv2023

Higher multiplier ideals

We associate a family of ideal sheaves to any Q-effective divisor on a complex manifold, called higher multiplier ideals, using the theory of mixed Hodge modules and V-filtrations. This family is indexed by two parameters, an integer indicating the Hodge level and a rational number, and these ideals admit a weight filtration. When the Hodge level is zero, they recover the usual multiplier ideals. We study the local and global properties of higher multiplier ideals systematically. In particular, we prove vanishing theorems and restriction theorems, provide criteria for the nontriviality, and introduce the center of minimal exponent (generalizing the notion of minimal log canonical center). The main idea is to exploit the global structure of the V-filtration along an effective divisor using the notion of twisted Hodge modules. As applications, we prove new cases of conjectures by Debarre, Casalaina-Martin and Grushevsky on singularities of theta divisors on principally polarized abelian varieties.

math.AG↗
arXiv2023

Singular hermitian metrics and the decomposition theorem of Catanese, Fujita, and Kawamata

We prove that a torsion-free sheaf $\mathcal F$ endowed with a singular hermitian metric with semi-positive curvature and satisfying the minimal extension property admits a direct-sum decomposition $\mathcal F \simeq \mathcal U \oplus \mathcal A$ where $\mathcal U$ is a hermitian flat bundle and $\mathcal A$ is a generically ample sheaf. The result applies to the case of direct images of relative pluricanonical bundles $f_* ω_{X/Y}^{\otimes m}$ under a surjective morphism $f\colon X \to Y$ of smooth projective varieties with $m\geq 2$. This extends previous results of Fujita, Catanese--Kawamata, and Iwai.

math.AG↗
arXiv2023

Hodge theory and Lagrangian fibrations on holomorphic symplectic manifolds

The purpose of this paper is to establish several new results about the Hodge theory of Lagrangian fibrations on (not necessarily compact) holomorphic symplectic manifolds. Let $M$ be a holomorphic symplectic manifold of dimension $2n$ that is K\"ahler but not necessarily compact, and let $\pi \colon M \to B$ be a Lagrangian fibration. We establish a relationship between the bundle of holomorphic $(n+i)$-forms on $M$ and the $i$-th perverse sheaf $P_i$ in the decomposition theorem for $\pi$. This is formulated using Saito's theory of Hodge modules and the BGG correspondence (between graded modules over the symmetric and exterior algebra). Along the way, we prove a relative Hard Lefschetz theorem for the action by the symplectic form; we prove two recent conjectures by Maulik, Shen, and Yin; we give a short proof for Matsushita's theorem (about higher direct images of the structure sheaf); and we show, without using hyperk\"ahler metrics, that every Lagrangian fibration gives rise to an action by the Lie algebra $\mathfrak{sl}_3(\mathbb{C})$ (in the noncompact case) or $\mathfrak{sl}_4(\mathbb{C})$ (in the compact case). [See the comment below.]

math.AG↗
arXiv2022

Hodge modules and Singular Hermitian Metrics

The purpose of this paper is to study certain notions of metric positivity for the lowest nonzero piece in the Hodge filtration of a Hodge module. We show that the Hodge metric satisfies the minimal extension property. In particular, this singular Hermitian metric has semi-positive curvature.

math.AG↗
arXiv2022

Degenerating complex variations of Hodge structure in dimension one

We analyze the behavior of polarized complex variations of Hodge structure on the punctured unit disk. For integral variations of Hodge structure, this analysis was first carried out by Wilfried Schmid. We get rid of the assumption that the eigenvalues of the monodromy transformation are roots of unity. In this generality, we give new (and, we think, more conceptual) proofs for all the major results in Schmid's paper, such as the estimates for the rate of growth of the Hodge norm; the existence of a limiting mixed Hodge structure; the nilpotent orbit theorem; and a simplified (but still sufficiently powerful) version of the SL(2)-orbit theorem.

math.AG↗
arXiv2022

Singular metrics and a conjecture by Campana and Peternell

A conjecture by Campana and Peternell says that if a positive multiple of $K_X$ is linearly equivalent to an effective divisor $D$ plus a pseudo-effective divisor, then the Kodaira dimension of $X$ should be at least as big as the Iitaka dimension of $D$. This is a very useful generalization of the non-vanishing conjecture (which is the case $D = 0$). We use recent work about singular metrics on pluri-adjoint bundles to show that the Campana-Peternell conjecture is almost equivalent to the non-vanishing conjecture.

math.AG↗
arXiv2021

Finiteness for self-dual classes in integral variations of Hodge structure

We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in the o-minimal structure $\mathbb{R}_{\mathrm{an},\exp}$.

math.AG↗
arXiv2020

Pushforwards of pluricanonical bundles under morphisms to abelian varieties

Let $f \colon X \to A$ be a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). We show that the sheaves $f_* ω_X^{\otimes m}$ become globally generated after pullback by an isogeny. We use this to deduce a decomposition theorem for these sheaves when $m \ge 2$, analogous to that obtained by Chen-Jiang when $m = 1$. This is in turn applied to effective results for pluricanonical linear series on irregular varieties with canonical singularities.

math.AG↗
arXiv2020

Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities

We investigate under what conditions holomorphic forms defined on the regular locus of a reduced complex space extend to holomorphic (or logarithmic) forms on a resolution of singularities. We give a simple necessary and sufficient condition for this, whose proof relies on the Decomposition Theorem and Saito's theory of mixed Hodge modules. We use it to generalize the theorem of Greb-Kebekus-Kovács-Peternell to complex spaces with rational singularities, and to prove the existence of a functorial pull-back for reflexive differentials on such spaces. We also use our methods to settle the "local vanishing conjecture" proposed by Mustaţă, Olano, and Popa.

math.AG↗
arXiv2019

The Fourier-Mukai transform made easy

We propose a slightly modified definition for the Fourier-Mukai transform (on abelian varieties) that makes it much easier to remember various formulas. As an application, we give relatively short proofs for two important theorems: the characterization of GV-sheaves in terms of vanishing, due to Hacon; and fact that M-regularity implies (continuous) global generation, due to Pareschi and Popa.

math.AG↗
arXiv2017

On a theorem of Campana and Păun

Let $X$ be a smooth projective variety over the complex numbers, and $Δ\subseteq X$ a reduced divisor with normal crossings. We present a slightly simplified proof for the following theorem of Campana and Păun: If some tensor power of the bundle $Ω_X^1(\log Δ)$ contains a subsheaf with big determinant, then $(X, Δ)$ is of log general type. This result is a key step in the recent proof of Viehweg's hyperbolicity conjecture.

math.AG↗
arXiv2017

Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Paun

We present a simplified proof for a recent theorem by Junyan Cao and Mihai Paun, confirming a special case of Iitaka's conjecture: if $f \colon X\to Y$ is an algebraic fiber space, and if the Albanese mapping of $Y$ is generically finite over its image, then we have the inequality of Kodaira dimensions $κ(X)\geq κ(Y)+κ(F)$, where $F$ denotes a general fiber of $f$. We include a detailed survey of the main algebraic and analytic techniques, especially the construction of singular hermitian metrics on pushforwards of adjoint bundles (due to Berndtsson, Paun, and Takayama).

math.AG↗
arXiv2017

Vanishing theorems for perverse sheaves on abelian varieties, revisited

We revisit some of the basic results of generic vanishing theory, as pioneered by Green and Lazarsfeld, in the context of constructible sheaves. Using the language of perverse sheaves, we give new proofs of some of the basic results of this theory. Our approach is topological/arithmetic, and avoids Hodge theory.

math.AG↗
arXiv2016

The Cremmer-Scherk Mechanism in F-theory Compactifications on K3 Manifolds

It is well understood --- through string dualities --- that there are 20 massless vector fields in the spectrum of eight-dimensional F-theory compactifications on smooth elliptically fibered K3 surfaces at a generic point in the K3 moduli space. Such F-theory vacua, which do not have any enhanced gauge symmetries, can be thought of as supersymmetric type IIB compactifications on P1 with 24 (p,q) seven-branes. Naively, one might expect there to be 24 massless vector fields in the eight-dimensional effective theory coming from world-volume gauge fields of the 24 branes. In this paper, we show how the vector field spectrum of the eight-dimensional effective theory can be obtained from the point of view of type IIB supergravity coupled to the world-volume theory of the seven-branes. In particular, we first show that the two-forms of the type IIB theory absorb the seven-brane world-volume gauge fields via the Cremmer-Scherk mechanism. We then proceed to show that the massless vector fields of the eight-dimensional theory come from KK-reducing the SL(2,Z) doublet two-forms of type IIB theory along SL(2,Z) doublet one-forms on the P1. We also discuss the relation between these vector fields and the "eaten" world-volume vector fields of the seven-branes.

hep-th↗