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Mirko Mauri

Publications and source records attributed to Mirko Mauri.

At least 19 recordsLinked to original sources
arXiv2025

Baily--Borel compactifications of period images and the b-semiampleness conjecture

We address two questions related to the semiampleness of line bundles arising from Hodge theory. First, we prove there is a functorial compactification of the image of a period map of a polarizable integral pure variation of Hodge structures for which the Griffiths bundle extends amply. In particular the Griffiths bundle is semiample. We prove more generally that the Hodge bundle of a Calabi--Yau variation of Hodge structures is semiample subject to some extra conditions, and as our second result deduce the b-semiampleness conjecture and the existence of a functorial Hodge-theoretic compactification of moduli spaces of polarized Calabi--Yau varieties. The semiampleness results (and the construction of the Baily--Borel compactifications) crucially use o-minimal GAGA, and the deduction of the b-semiampleness conjecture uses work of Ambro and results of Koll\'ar on the geometry of minimal lc centers to verify the extra conditions.

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arXiv2025

Boundedness of some fibered K-trivial varieties

We prove that irreducible Calabi-Yau varieties of a fixed dimension, admitting a fibration by abelian varieties or primitive symplectic varieties of a fixed analytic deformation class, are birationally bounded. We prove that there are only finitely many deformation classes of primitive symplectic varieties of a fixed dimension, admitting a Lagrangian fibration. We also show that fibered Calabi-Yau 3-folds are bounded. Conditional on the generalized abundance or hyperk\"ahler SYZ conjecture, our results prove that there are only finitely many deformation classes of hyperk\"ahler varieties, of a fixed dimension, with $b_2 \geq 5$.

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arXiv2025

Hitchin fibrations are Ng\^{o} fibrations

We study the geometry of the Hitchin fibration for $\mathcal{L}$-valued $G$-Higgs bundles over a smooth projective curve of genus $g$, where $G$ is a reductive group and $\mathcal{L}$ is a suitably positive line bundle. We show that the Hitchin fibration admits the structure of a weak Abelian fibration. In the case when the line bundle $\mathcal{L}$ is a twist of the canonical bundle of the curve by a (possibly empty) reduced effective divisor, we prove a cohomological bound and $\delta$-regularity of the weak Abelian fibration.

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arXiv2024

Perverse-Hodge octahedron

The perverse-Hodge octahedron is a 3D enhancement of the Hodge diamond of a compact hyperk\"{a}hler manifold. Its existence is equivalent to Nagai's conjecture, which holds for all known deformation types. The octahedron appears implicitly in Huybrechts-Mauri and Shen-Yin.

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arXiv2024

Birational complexity and dual complexes

We introduce the notion of birational complexity of a log Calabi-Yau pair. This invariant measures how far the log Calabi-Yau pair is to being birational to a toric pair. We study fundamental properties of the new invariant, with a particular focus on the geometry of dual complexes.

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arXiv2024

Terminalizations of quotients of compact hyperk\"ahler manifolds by induced symplectic automorphisms

Terminalizations of symplectic quotients are sources of new deformation types of irreducible symplectic varieties. We classify all terminalizations of quotients of Hilbert schemes of K3 surfaces or of generalized Kummer varieties, by finite groups of symplectic automorphisms induced from the underlying K3 or abelian surface. We determine their second Betti number and the fundamental group of their regular locus. In the Kummer case, we prove that the terminalizations have quotient singularities, and determine the singularities of their universal quasi-\'etale cover. In particular, we obtain at least nine new deformation types of irreducible symplectic varieties of dimension four. Finally, we compare our deformation types with those in [FM21; Men22]. The smooth terminalizations are only three and of K$3^{[n]}$-type, and surprisingly they all appeared in different places in the literature [Fuj83; Kaw09; Flo22].

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arXiv2022

Homological Bondal-Orlov localization conjecture for rational singularities

Given a resolution of rational singularities $\pi\colon \tilde{X} \to X$ over a field of characteristic zero we use a Hodge-theoretic argument to prove that the image of the functor $\mathbf{R}\pi_*\colon \mathbf{D}(\tilde{X}) \to \mathbf{D}(X)$ between bounded derived categories of coherent sheaves generates $\mathbf{D}(X)$ as a triangulated category. This gives a weak version of the Bondal-Orlov localization conjecture, answering a question of Pavic and Shinder. The same result is established more generally for proper (non-necessarily birational) morphisms $\pi\colon \tilde{X} \to X$, with $\tilde{X}$ smooth, satisfying $\mathbf{R}\pi_*(\mathcal{O}_{\tilde{X}}) = \mathcal{O}_X$.

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arXiv2022

Index of coregularity zero log Calabi-Yau pairs

In this article, we study the index of log Calabi--Yau pairs $(X,B)$ of coregularity 0. We show that $2\lambda(K_X+B)\sim 0$, where $\lambda$ is the Weil index of $(X,B)$. This is in contrast to the case of klt Calabi--Yau varieties, where the index can grow doubly exponentially with the dimension. Our sharp bound on the index extends to the context of generalized log Calabi--Yau pairs, semi-log canonical pairs, and isolated log canonical singularities of coregularity 0. As a consequence, we show that the index of a variety appearing in the Gross--Siebert program or in the Kontsevich--Soibelman program is at most $2$. Finally, we discuss applications to Calabi--Yau varieties endowed with a finite group action, including holomorphic symplectic varieties endowed with a purely non-symplectic automorphism.

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arXiv2022

Hodge-to-singular correspondence for reduced curves

We study the summands of the decomposition theorem for the Hitchin system for $\mathrm{GL}_n$, in arbitrary degree, over the locus of reduced spectral curves. A key ingredient is a new correspondence between these summands and the topology of hypertoric quiver varieties. In contrast to the case of meromorphic Higgs fields, the intersection cohomology groups of moduli spaces of regular Higgs bundles depend on the degree. We describe this dependence.

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arXiv2021

Lagrangian fibrations

We review the theory of Lagrangian fibrations of hyperk\"ahler manifolds as initiated by Matsushita. We also discuss more recent work of Shen-Yin and Harder-Li-Shen-Yin. Occasionally, we give alternative arguments and complement the discussion by additional observations.

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arXiv2021

On type II degenerations of hyperk\"ahler manifolds

We give a simple argument to prove Nagai's conjecture for type II degenerations of compact hyperk\"ahler manifolds and cohomology classes of middle degree. Under an additional assumption, the techniques yield the conjecture in arbitrary degree. This would complete the proof of Nagai's conjecture in general, as it was proved already for type I degenerations by Koll\'ar, Laza, Sacc\`a, and Voisin and independently by Soldatenkov, while it is immediate for type III degenerations. Our arguments are close in spirit to a recent paper by Harder proving similar results for the restrictive class of good degenerations.

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arXiv2021

Intersection cohomology of rank two character varieties of surface groups

For $G = \mathrm{GL}_2, \mathrm{SL}_2, \mathrm{PGL}_2$ we compute the intersection E-polynomials and the intersection Poincar\'e polynomials of the $G$-character variety of a compact Riemann surface $C$ and of the moduli space of $G$-Higgs bundles on $C$ of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.

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arXiv2021

Topological mirror symmetry for rank two character varieties of surface groups

The moduli spaces of flat $\mathrm{SL}_2$- and $\mathrm{PGL}_2$-connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tam\'as Hausel in Remark 3.30 of "Global topology of the Hitchin system".

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arXiv2020

P=W conjectures for character varieties with symplectic resolution

We establish P=W and PI=WI conjectures for character varieties with structural group $\mathrm{GL}_n$ and $\mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-\'{e}tale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and the projectivity of the compactification of the de Rham moduli space. In particular, we study in detail a Dolbeault moduli space which is specialization of the singular irreducible holomorphic symplectic variety of type O'Grady 6.

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arXiv2019

Constructing local models for Lagrangian torus fibrations

We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways. We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi-Yau 3-folds with codimension 2 discriminant locus. Then, we find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base of the fibration is an open subset of the cone over the dual complex of the divisor. This can be used to construct an analogue of the non-archimedean SYZ fibration constructed by Nicaise, Xu and Yu.

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arXiv2018

On the geometric P=W conjecture

We formulate the geometric P=W conjecture for singular character varieties. We establish it for compact Riemann surfaces of genus one, and obtain partial results in arbitrary genus. To this end, we employ non-Archimedean, birational and degeneration techniques to study the topology of the dual boundary complex of certain character varieties. We also clarify the relation between the geometric and the cohomological P=W conjectures.

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arXiv2018

G-birational superrigidity of del Pezzo surfaces of degree 2 and 3

Any minimal Del Pezzo G-surface S of degree smaller than 3 is G-birationally rigid. We classify those which are G-birationally superrigid and for those which fail to be so, we describe the equations of a set of generators for the infinite group of G-birational automorphisms.

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