Unexpected loss of maximality: the case of Hilbert squares of real surfaces

Kharlamov, Viatcheslav; Răsdeaconu, Rareş

doi:10.48550/arxiv.2303.02796

Cited by 2 publications

(14 citation statements)

References 24 publications

(68 reference statements)

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“…The statement does not hold for maps of even degree. As a counter-example, take a maximal real surface S with non-maximal Hilbert square S [2] (such surface exists by [53], see Remark 7.4), then the degree-2 quotient map Bl ∆ (S × S) → S [2] is from a maximal variety to a non-maximal one.…”

Section: Remark 36 (Flips and Flops)mentioning

confidence: 99%

“…Remark 7.4 (Loss of maximality in Hilbert squares). It is quite surprising that Kharlamov and Rȃsdeaconu [53] recently discovered the existence of maximal real surfaces with non-maximal Hilbert squares. In fact, they showed in [53,Theorem 1.2] that for a real surface X with H 1 (X, F 2 ) = 0, its Hilbert square X [2] is maximal if and only if X(R) is connected.…”

Section: Hilbert Squares and Hilbert Cubesmentioning

confidence: 99%

“…It is quite surprising that Kharlamov and Rȃsdeaconu [53] recently discovered the existence of maximal real surfaces with non-maximal Hilbert squares. In fact, they showed in [53,Theorem 1.2] that for a real surface X with H 1 (X, F 2 ) = 0, its Hilbert square X [2] is maximal if and only if X(R) is connected. This finding led to the discovery of many examples of maximal surfaces with non-maximal Hilbert square, such as K3 surfaces.…”

Section: Hilbert Squares and Hilbert Cubesmentioning

confidence: 99%

“…• Section 2 provides a brief overview of equivariant cohomology, Atiyah's KR-theory, and Chern classes. [53]. Furthermore, I am grateful for the conference "Real Aspects of Geometry" held at CIRM in 2022, which provided me with many sources of inspiration.…”

mentioning

confidence: 99%

“…On the one hand, in (complex) dimension 2, maximal real K3 surfaces and abelian surfaces exist and have been thoroughly studied; see [52], [77,Chapters IV,VIII]. On the other hand, Kharlamov and Rȃsdeaconu [53] recently discovered that the Hilbert squares of maximal real K3 surfaces and 1 The distinction between (ABA) and (AAB) depends on the action of the involution on the holomorphic symplectic form: (ABA) if it is preserved and (AAB) if it is reversed.…”

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confidence: 99%

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Maximal real varieties from moduli constructions

Fu¹

2023

Preprint

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For a complex manifold equipped with an anti-holomorphic involution, which is referred to as a real variety, the Smith-Thom inequality states that the total F 2 -Betti number of the real locus is not greater than the total F 2 -Betti number of the ambient complex manifold. A real variety is called maximal if the equality holds. In this paper, we present a series of new constructions of maximal real varieties by exploring moduli spaces of certain objects on a maximal real variety. Our results establish the maximality of the following real varieties:• Moduli spaces of stable vector bundles of coprime rank and degree over a maximal smooth projective real curve (known as Brugallé-Schaffhauser's theorem [16], with a short new proof presented in this work); the same result holds for moduli spaces of stable parabolic vector bundles. • Moduli spaces of stable Higgs bundles of coprime rank and degree over a maximal smooth projective real curve, providing maximal hyper-Kähler examples. • If a real variety has non-empty real locus and maximal Hilbert square, then the variety itself and its Hilbert cube are maximal. This is always the case for maximal real smooth cubic threefolds, but never the case for maximal real smooth cubic fourfolds. • Punctual Hilbert schemes on a maximal real projective surface with vanishing first F 2 -Betti number and connected real locus, such as R-rational maximal real surfaces and some generalized Dolgachev surfaces. • Moduli spaces of stable sheaves on the real projective plane, or more generally, on an R-rational maximal Poisson surface. We also observe that maximality is a motivic property when interpreted as equivariant formality. Furthermore, any smooth projective real variety motivated by maximal ones is also maximal.

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Section: Remark 36 (Flips and Flops)mentioning

confidence: 99%

Section: Hilbert Squares and Hilbert Cubesmentioning

confidence: 99%

Section: Hilbert Squares and Hilbert Cubesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Maximal real varieties from moduli constructions

Fu¹

2023

Preprint

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On the Smith–Thom deficiency of Hilbert squares

Kharlamov,

Răsdeaconu

2024

Journal of Topology

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We give an expression for the Smith–Thom deficiency of the Hilbert square of a smooth real algebraic variety in terms of the rank of a suitable Mayer– Vietoris mapping in several situations. As a consequence, we establish a necessary and sufficient condition for the maximality of in the case of projective complete intersections, and show that with a few exceptions, no real nonsingular projective complete intersection of even dimension has maximal Hilbert square. We also provide new examples of smooth real algebraic varieties with maximal Hilbert square.

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Unexpected loss of maximality: the case of Hilbert squares of real surfaces

Cited by 2 publications

References 24 publications

Maximal real varieties from moduli constructions

Maximal real varieties from moduli constructions

On the Smith–Thom deficiency of Hilbert squares

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