100,389 research outputs found
Virtual classes and virtual motives of Quot schemes on threefolds
No full textFor a simple, rigid vector bundle F on a Calabi–Yau 3-fold Y, we construct a symmetric obstruction theory on the Quot scheme QuotY(F,n), and we solve the associated enumerative theory. We discuss the case of other 3-folds. Exploiting the critical structure on the local model QuotAjavax.xml.bind.JAXBElement@e117367(O⊕r,n), we construct a virtual motive (in the sense of Behrend–Bryan–Szendrői) for QuotY(F,n) for an arbitrary vector bundle F on a smooth 3-fold Y. We compute the associated motivic partition function. We obtain new examples of higher rank (motivic) Donaldson–Thomas invariants
On the motive of the Quot scheme of finite quotients of a locally free sheaf
No full textLet X be a smooth variety, E a locally free sheaf on X. We express the generating function of the motives [QuotX(E,n)] in terms of the power structure on the Grothendieck ring of varieties. This extends a recent result of Bagnarol, Fantechi and Perroni for curves, and a result of Gusein-Zade, Luengo and Melle-Hernández for Hilbert schemes. We compute this generating function for curves and we express the relative motive [QuotAjavax.xml.bind.JAXBElement@13d85294(O⊕r)→SymAd] as a plethystic exponential
Local Contributions to Donaldson–Thomas Invariants
No full textLet C be a smooth curve embedded in a smooth quasi-projective three-fold Y, and let QnC = Quotn(IC) be the Quot scheme of length n quotients of its ideal sheaf. We show the identity ?(QnC ) = (-1)n?(QnC ), where ? is the Behrend weighted Euler characteristic. When Y is a projective CalabiYau three-fold, this shows that the Donaldson Thomas (DT) contribution of a smooth rigid curve is the signed Euler characteristic of the moduli space. This can be rephrased as a DT/PT wall-crossing type formula, which can be formulated for arbitrary smooth curves. In general, such wall-crossing formula is shown to be equivalent to a certain Behrend function identity
The equivariant Atiyah class
No full textLet X be a complex scheme acted on by an affine algebraic group G. We prove that the Atiyah class of a G-equivariant perfect complex on X, as constructed by Huybrechts and Thomas, is G-equivariant in a precise sense. As an application, we show that, if G is reductive, the obstruction theory on the fine relative moduli space M → B of simple perfect complexes on a G-invariant smooth projective family Y → B is Gequivariant. The results contained here are meant to suggest how to check the equivariance of the natural obstruction theories on a wide variety of moduli spaces equipped with a torus action, arising for instance in Donaldson-Thomas theory and Vafa-Witten theory
The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves
No full textLet be a hyperelliptic curve embedded in its Jacobian via an Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme component of containing the Abel-Jacobi curve as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism along the hyperelliptic locus. As an application, we determine the scheme structure of the moduli space of Picard sheaves (introduced by Mukai) on a hyperelliptic Jacobian
The DT/PT correspondence for smooth curves
No full textWe show a version of the DT/PT correspondence relating local curve counting invariants, encoding the contribution of a fixed smooth curve in a Calabi–Yau threefold. We exploit a local study of the Hilbert–Chow morphism about the cycle of a smooth curve. We compute, via Quot schemes, the global Donaldson–Thomas theory of a general Abel–Jacobi curve of genus 3
An Invitation to Modern Enumerative Geometry
No full textThis book is based on a series of lectures given by the author at SISSA, Trieste, within the PhD courses Techniques in enumerative geometry (2019) and Localisation in enumerative geometry (2021). The goal of this book is to provide a gentle introduction, aimed mainly at graduate students, to the fast-growing subject of enumerative geometry and, more specifically, counting invariants in algebraic geometry. In addition to the more advanced techniques explained and applied in full detail to concrete calculations, the book contains the proofs of several background results, important for the foundations of the theory. In this respect, this text is conceived for PhD students or research “beginners” in the field of enumerative geometry or related areas. This book can be read as an introduction to Hilbert schemes and Quot schemes on 3-folds but also as an introduction to localisation formulae in enumerative geometry. It is meant to be accessible without a strong background in algebraic geometry; however, three appendices (one on deformation theory, one on intersection theory, one on virtual fundamental classes) are meant to help the reader dive deeper into the main material of the book and to make the text itself as self-contained as possible
Jet bundles on Gorenstein curves and applications
No full textIn the last twenty years a number of papers appeared aiming to construct locally free replacements of the sheaf of principal parts for families of Gorenstein curves. The main goal of this survey is to present to the widest possible audience of mathematical readers a catalogue of such constructions, discussing the related literature and reporting on a few applications to classical problems in Enumerative Algebraic Geometry
Framed motivic Donaldson–Thomas invariants of small crepant resolutions
No full textFor an arbitrary integer r >= 1, we compute r-framed motivic DT and PT invariants of small crepant resolutions of toric Calabi-Yau 3-folds, establishing a "higher rank" version of the motivic DT/PT wall-crossing formula. This generalises the work of Morrison and Nagao. Our formulae, in particular their relationship with the r=1 theory, fit nicely in the current development of higher rank refined DT invariants
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