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Roland Abuaf

Publications and source records attributed to Roland Abuaf.

16 recordsLinked to original sources
arXiv2026

Hirzebruch signature theorem on Hochschild homology, with applications to derived invariance of Hodge numbers

We prove a refinement of Hirzebruch's signature formula on the individual Hochschild diagonals of a smooth projective complex variety. Natural Hermitian forms, obtained by symmetrizing a Todd-corrected pairing, have explicitly computable positive and negative indices and are preserved by derived equivalences. On diagonals of the same parity as the dimension, their signatures separate the even and odd contributions to the Hochschild-Kostant-Rosenberg decomposition; on the other diagonals, their radicals and signatures are governed by the first Chern class. We deduce the derived invariance of $h^{n-2,0}$, $h^{n-1,1}$ and $h^{n-3,1}$ in any dimension, and of $h^{2,0}$ when the canonical bundle is trivial. Combining the invariance of $h^{2,0}$ with the multiplicative structure of Hochschild cohomology and Verbitsky's orthogonal action, we give a new short proof of the theorem of Huybrechts and Nieper-Wißkirchen that a derived partner of an irreducible holomorphic symplectic variety is again irreducible holomorphic symplectic. In dimension five with trivial canonical bundle, the signature invariants and the Libgober-Wood identity together establish the derived invariance of all Hodge numbers

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arXiv2024

Holomorphic symplectic manifolds from semistable Higgs bundles

Let $\mathcal{M}_{C}(2, 0)$ be the moduli space of semistable rank two and degree zero Higgs bundles on a smooth complex hyperelliptic curve $C$ of genus three. We prove that the quotient of $\mathcal{M}_{C}(2, 0)$ by a twisted version of the hyperelliptic involution is an 18-dimensional holomorphic symplectic variety admitting a crepant resolution, whose local model was studied by Kaledin and Lehn to describe O'Grady's singularities. Similarly, by considering the moduli space of Higgs bundles with trivial determinant $\mathcal{M}_C(2, \mathcal{O}_{C})\subseteq \mathcal{M}_C(2, 0)$, we show that the quotient of $\mathcal{M}_C(2, \mathcal{O}_{C})$ by the hyperelliptic involution is a 12-dimensional holomorphic symplectic variety admitting a crepant resolution.

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arXiv2021

On the twisted octonionic eigenvalue problem and some sextics hypersurfaces related to the Cartan cubic

We revisit the octonionic eigenvalue problem from a geometric perspective. In particular, we study a tautological sheaf defined on a sextic related to this problem, the Ogievetskiî-Dray-Manogue sextic. We then define and study a twisted version of the octonionic eigenvalue problem. A new sextic arises in this setting and we study the corresponding tautological sheaf supported on it. This twisted version of the octonionic eigenvalue problem is eminently more symmetric than the original one, as reflected by the last result we prove in this paper : the automorphism group of the twisted octonionic eigenvalue problem, though not isomorphic to $\mathrm{E}_6$, acts prehomogeneously on the exceptional Jordan algebra $\mathrm{J}_{3}(\mathbb{O})$. This is in sharp contrast with the fact that the generic orbit for the action of the automorphism group of the classical octonionic eigenvalue problem has (at least) codimension $6$ in $\mathrm{J}_{3}(\mathbb{O})$.

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arXiv2019

Derived invariance of the numbers $h^{0,p}(X)$

Let $X_1$ and $X_2$ be derived equivalent smooth projective varieties over the field of complex numbers. We prove that the numbers $h^{0,p}(X_1)$ and $h^{0,p}(X_2)$ are equal for any $p$.

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arXiv2019

Gradings of Lie algebras, magical spin geometries and matrix factorizations

We describe a remarkable rank fourtenn matrix factorization of the octic Spin(14)-invariant polynomial on either of its half-spin representations. We observe that this representation can be, in a suitable sense, identified with a tensor product of two octonion algebras. Moreover the matrix factorisation can be deduced from a particular Z-grading of the exceptional Lie algebra $\mathfrak{e}_8$. Intriguingly, the whole story can be extended to the whole Freudenthal-Tits magic square and yields matrix factorizations on other spin representations, as well as for the degree seven invariant on the space of three-forms in several variables. As an application of our results on Spin(14), we construct a special rank seven vector bundle on a double-octic threefold, that we conjecture to be spherical.

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arXiv2018

Scheme-theoretic Whitney conditions and applications to tangency of projective varieties

We investigate a scheme-theoretic variant of Whitney condition a. If X is a projec-tive variety over the field of complex numbers and Y $\subset$ X a subvariety, then X satisfies generically the scheme-theoretic Whitney condition a along Y provided that the pro-jective dual of X is smooth. We give applications to tangency of projective varieties over C and to convex real algebraic geometry. In particular, we prove a Bertini-type theorem for osculating plane of smooth complex space curves and a generalization of a Theorem of Ranestad and Sturmfels describing the algebraic boundary of an affine compact real variety.

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arXiv2018

Hodge numbers and Hodge structures for Calabi-Yau categories of dimension three

Let $\mathcal{A}$ be a smooth proper C-linear triangulated category Calabi-Yau of dimension 3 endowed with a (non-trivial) rank function. Using the homological unit of $\mathcal{A}$ with respect to the given rank function, we define Hodge numbers for $\mathcal{A}$. If the classes of unitary objects generate the complexified numerical K-theory of $\mathcal{A}$ (hypothesis satisfied for many examples of smooth proper Calabi-Yau categories of dimension 3), it is proved that these numbers are independent of the chosen rank function : they are intrinsic invariants of the triangulated category $\mathcal{A}$. In the special case where $\mathcal{A}$ is a semi-orthogonal component of the derived category of a smooth complex projective variety and the homological unit of $\mathcal{A}$ is $\mathbb{C} \oplus \mathbb{C}[3]$ (that is $\mathcal{A}$ is strict Calabi-Yau with respect to the rank function), we define a Hodge structure on the Hochschild homology of $\mathcal{A}$. The dimensions of the Hodge spaces of this structure are the Hodge numbers aforementioned. Finally, we give some numerical applications toward the Homological Mirror Symmetry conjecture for cubic sevenfolds and double quartic fivefolds.

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arXiv2017

On quartic double fivefolds and the matrix factorizations of exceptional quaternionic representations

We study quartic double fivefolds from the perspective of Fano manifolds of Calabi-Yau type and that of exceptional quaternionic representations. We first prove that the generic quartic double fivefold can be represented, in a finite number of ways, as a double cover of P^5 ramified along a linear section of the Sp 12-invariant quartic in P^31. Then, using the geometry of the Vinberg's type II decomposition of some exceptional quaternionic representations, and backed by some cohomological computations performed by Macaulay2, we prove the existence of a spherical rank 6 vector bundle on such a generic quartic double fivefold. We finally use the existence this vector bundle to prove that the homological unit of the CY-3 category associated by Kuznetsov to the derived category of a generic quartic double fivefold is C $\oplus$ C[3].

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arXiv2017

Compact hyper-Kähler categories

We define and study the notion of hyper-Kähler category. On the theoretical side, we focus on construction techniques and deformation theory of such categories. We also study in details some examples : non-commutative Hilbert schemes of points on a K3 surface and a categorical resolution of a relative compactified Prymian constructed by Markushevich and Tikhomirov.

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arXiv2015

Homological units

We define and study the invariance properties of homological units. Some applications are given to the derived invariance of Hodge numbers. In particular, we prove that if X and Y are derived equivalent smooth projective varieties of dimension 4 having the same Picard number, then they have the same Hodge numbers. We also give a geometric interpretation of the conjectural invariance of homological units in terms of derived Jacobians.

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arXiv2013

Orthogonal bundles and skew-Hamiltonian matrices

Using properties of skew-Hamiltonian matrices and classic connectedness results, we prove that the moduli space $M_{ort}^0(r,n)$ of stable rank $r$ orthogonal vector bundles on $\mathbb{P}^2$, with Chern classes $(c_1,c_2)=(0,n)$, and trivial splitting on the general line, is smooth irreducible of dimension $(r-2)n-{r \choose 2}$ for specific values of $r$ and $n$.

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arXiv2013

Wonderful resolutions and categorical crepant resolutions of singularities

Let $X$ be an algebraic variety with Gorenstein singularities. We define the notion of a wonderful resolution of singularities of $X$ by analogy with the theory of wonderful compactifications of semi-simple linear algebraic groups. We prove that if $X$ has rational singularities and has a wonderful resolution of singularities, then $X$ admits a categorical crepant resolution of singularities. As an immediate corollary, we get that all determinantal varieties defined by the minors of a generic square/symmetric/skew-symmetric matrix admit categorical crepant resolution of singularities. We also discuss notions of minimality for a categorical resolution of singularities and we explore some links between minimality and crepancy for such resolutions.

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