Mapping class group and a global Torelli theorem for hyperkähler manifolds

Verbitsky, Misha

doi:10.1215/00127094-2382680

Cited by 161 publications

(177 citation statements)

References 43 publications

Supporting

Mentioning

176

Contrasting

Order By: Relevance

“…(X ′ , L ′ )) to (X 0 , L 0 ). Then the global Torelli theorem of Verbitsky [Ver13] shows that if P(X, L, φ) = P(X ′ , L ′ , ψ), then X and X ′ are birational. In case L and L ′ are ample, this is the statement of [Mar11, Theorem 1.10], and the general case can be deduced either by using a small deformation to the ample case or by using [Mar11, Proposition 1.9] to reduce to the general global Torelli theorem.…”

Section: Remark 34 Our Proof Relies Crucially On the Existence Of Tmentioning

confidence: 99%

See 1 more Smart Citation

Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for K3 surfaces, and the Tate conjecture

Charles

2016

Ann. Math.

View full text Add to dashboard Cite

Abstract. We investigate boundedness results for families of holomorphic symplectic varieties up to birational equivalence. We prove the analogue of Zarhin's trick for K3 surfaces by constructing big line bundles of low degree on certain moduli spaces of stable sheaves, and proving birational versions of Matsusaka's big theorem for holomorphic symplectic varieties.As a consequence of these results, we give a new geometric proof of the Tate conjecture for K3 surfaces over finite fields of characteristic at least 5, and a simple proof of the Tate conjecture for K3 surfaces with Picard number at least 2 over arbitrary finite fields -including fields of characteristic 2.

show abstract

Section: Remark 34 Our Proof Relies Crucially On the Existence Of Tmentioning

confidence: 99%

“…Over the field of complex numbers, we can use the period map and the global Torelli theorem of [Ver13] to answer the question in Theorem 3.3, again possibly changing the bundle L. This has the following consequence. Theorem 1.2.…”

mentioning

confidence: 99%

Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for K3 surfaces, and the Tate conjecture

Charles

2016

Ann. Math.

View full text Add to dashboard Cite

show abstract

“…The present work deals with the GIT side of the story for double EPW-sextics, with a view towards proving that M is identified with Looijenga's compactification associated to a particular arrangement of hypersurfaces in the relevant quotient of a bounded symmetric domain of Type IV (the period map is birational by [33,9,20,21]), namely the hyperfaces S order to show that the standard non-stable strata parametrize non-stable lagrangians it will suffice to express the numerical function µ(A, λ) of a lagrangian A with respect to a 1-PS λ : C × → SL(V ) in terms of the dimension of the intersections of A with the isotypical summands of 3 λ. The proof that any non-stable lagrangian belongs to one of the standard non-stable strata requires more work.…”

Section: Introductionmentioning

confidence: 99%

Moduli of double EPW-sextics

O’Grady¹

2016

Memoirs of the AMS

View full text Add to dashboard Cite

We will study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of 3 C 6 modulo the natural action of SL6, call it M. This is a compactification of the moduli space of smooth double EPW-sextics and hence birational to the moduli space of HK 4-folds of Type K3 [2] polarized by a divisor of square 2 for the Beauville-Bogomolov quadratic form. We will determine the stable points. Our work bears a strong analogy with the work of Voisin, Laza and Looijenga on moduli and periods of cubic 4-folds. We will prove a result which is analogous to a theorem of Laza asserting that cubic 4-folds with simple singularities are stable. We will also describe the irreducible components of the GIT boundary of M. Our final goal (not achieved in this work) is to understand completely the period map from M to the Baily-Borel compactification of the relevant period domain modulo an arithmetic group. We will analyze the locus in the GIT-boundary of M where the period map is not regular. Our results suggest that M is isomorphic to Looijenga's compactification associated to 3 specific hyperplanes in the period domain.

show abstract

“…In [7] (see also [8,9] and [10]), this theorem was used to study subvarieties of generic deformations of a compact holomorphic symplectic manifold M. Recall that the Teichmüller space of M is the quotient Teich = Comp/Diff 0 of the (infinitedimensional) space of all complex structures of hyperkähler type by the group Diff 0 of isotopies [11]. Teich is a complex, non-Hausdorff manifold.…”

Section: Absolutely Trianalytic Subvarieties In Hyperkähler Manifoldsmentioning

confidence: 99%

k-symplectic structures and absolutely trianalytic subvarieties in hyperkähler manifolds

Soldatenkov

Verbitsky

2015

Journal of Geometry and Physics

Self Cite

View full text Add to dashboard Cite

a b s t r a c t Let (M, I, J, K ) be a hyperkähler manifold, and Z ⊂ (M, I) a complex subvariety in (M, I).We say that Z is trianalytic if it is complex analytic with respect to J and K , and absolutely trianalytic if it is trianalytic with respect to any hyperkähler triple of complex structures (M, I, J ′ , K ′ ) containing I. For a generic complex structure I on M, all complex subvarieties of (M, I) are absolutely trianalytic. It is known that the normalization Z ′ of a trianalytic subvariety is smooth; we prove thatTo study absolutely trianalytic subvarieties further, we define a new geometric structure, called k-symplectic structure; this structure is a generalization of hypersymplectic structure. A k-symplectic structure on a 2d-dimensional manifold X is a k-dimensional space R of closed 2-forms on X which all have rank 2d or d. It is called non-degenerate if the set of all degenerate forms in R is a smooth, non-degenerate quadric hypersurface in R.We consider absolutely trianalytic tori in a hyperkähler manifold M of maximal holonomy. We prove that any such torus is equipped with a non-degenerate k-symplectic structure, where k = b 2 (M). We show that the tangent bundle TX of a k-symplectic manifold is a Clifford module over a Clifford algebra Cl(k − 1). Then an absolutely trianalytic torus in a hyperkähler manifold M with b 2 (M) 2r + 1 is at least 2 r−1 -dimensional.

show abstract

Mapping class group and a global Torelli theorem for hyperkähler manifolds

Cited by 161 publications

References 43 publications

Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for K3 surfaces, and the Tate conjecture

Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for K3 surfaces, and the Tate conjecture

Moduli of double EPW-sextics

k-symplectic structures and absolutely trianalytic subvarieties in hyperkähler manifolds

Contact Info

Product

Resources

About