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3 results

  • RIGID STABLE VECTOR BUNDLES ON HYPERKÄHLER VARIETIES OF TYPE $K3^{[n]}$

  • Kieran G. O’Grady
  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 23 / Issue 5 / September 2024
    Published online by Cambridge University Press:
    22 December 2023, pp. 2051-2080
    Print publication:
    September 2024
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  • We prove existence and unicity of slope-stable vector bundles on a general polarized hyperkähler (HK) variety of type $K3^{[n]}$ with certain discrete invariants, provided the rank and the first two Chern classes of the vector bundle satisfy certain equalities. The latter hypotheses at first glance appear to be quite restrictive, but, in fact, we might have listed almost all slope-stable rigid projectively hyperholomorphic vector bundles on polarized HK varieties of type $K3^{[n]}$ with $20$ moduli.

  • A DERIVED LAGRANGIAN FIBRATION ON THE DERIVED CRITICAL LOCUS

  • Part of
  • Albin Grataloup
  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 23 / Issue 1 / January 2024
    Published online by Cambridge University Press:
    31 August 2022, pp. 311-345
    Print publication:
    January 2024
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  • We study the symplectic geometry of derived intersections of Lagrangian morphisms. In particular, we show that for a functional $f : X \rightarrow \mathbb {A}_{k}^{1}$, the derived critical locus has a natural Lagrangian fibration $\textbf {Crit}(f) \rightarrow X$. In the case where f is nondegenerate and the strict critical locus is smooth, we show that the Lagrangian fibration on the derived critical locus is determined by the Hessian quadratic form.