1,346 research outputs found
D-orbifolds and d-bordism
No full textThe purpose of this thesis is to study d-manifolds and d-orbifolds and their bordism groups. D-manifolds and d-orbifolds were recently introduced by Joyce as a new class of geometric objects to study moduli problems in algebraic and symplectic geometry. In the spirit of Joyce we will introduce the notion of (stable) nearly and homotopy complex structures on these 2-categories and study their unitary bordism groups. Fukaya and Ono proved that the moduli space of n-pointed, genus g, J-holomorphic curves Mg,n(M,J,β) carries a so called stably almost complex structure, and as Kuranishi spaces are closely related to d-orbifolds, the introduction of complex structures will be essential in studying symplectic Gromov-Witten invariants using d-orbifolds. We furthermore introduce the notion of representable d-orbifolds and prove that these representable d-orbifolds can be embedded into an orbifold. We will then explain how a result of Kresch can be used to show that many important moduli spaces in algebraic geometry, are representable and thus embeddable d-orbifolds. Moreover we will sketch how one could prove an analogous result in the symplectic case. We then prove as one of our main results, that for a compact manifold the unitary d-bordism group is isomorphic to its ‘classical’ unitary bordism group. This result extends a result by Joyce who proved a similar statement for oriented manifolds and d-manifolds. Furthermore we will introduce the notion of blowups in the 2-category of d-manifolds and prove that these d-blowups satisfy a universal property. Finally, we sketch how our results may be used to make a step towards a proof of the Gopakumar–Vafa integrality conjecture and a “resolution of singularities” theorem for d-orbifolds
Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds
No full textLet be a separated, -shifted symplectic derived -scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension , and the underlying complex analytic topological space. We prove that can be given the structure of a derived mooth manifold , of real virtual dimension . This is not canonical, but is independent of choices up to bordisms fixing the underlying topological pace . There is a 1-1 correspondence between orientations on and orientations on .
Because compact, oriented derived manifolds have virtual classes, this means that proper, oriented -shifted symplectic derived -schemes have virtual classes, in either homology or bordism. This is surprising, as conventional algebro-geometric virtual cycle methods fail in this case. Our virtual classes have half the expected dimension, and from purely complex algebraic input, can yield a virtual class of odd real dimension.
Now derived moduli schemes of coherent sheaves on a Calabi-Yau 4-fold are expected to be -shifted symplectic (this holds for stacks). We propose to use our virtual classes to define new Donaldson-Thomas style invariants andapos;countingandapos; (semi)stable coherent sheaves on Calabi-Yau 4-folds over , which should be unchanged under deformations of
Authorship, ethics and the reader Blake, Dickens, Joyce
No full textRelations between literature and ethics are currently the subject of much discussion amongst critics and philosophers alike. Dominic Rainsford furthers this debate by examining ways in which texts may appear to comment on their authors' own ethical status - problematical disclosures which are significant for any reader who wishes to relate literature to moral issues in extra-literary life. He pursues these matters through readings of Blake, Dickens and Joyce, three authors who find vivid ways of casting doubt on their own moral authority, with the result that the reader's perception of the author becomes closely linked to the social ills exposed within his texts. Combining the desire to find ethical significance in literature with a sceptical mode of reading, informed by post-structuralist theory, the book thus develops a type of radical humanism with applications far beyond the three authors with whom it is immediately concerned
Moduli spaces of complexes of coherent sheaves
No full textIn this thesis we consider problems related to Joyce’s vertex algebra construction and the topology of stabilized moduli spaces. We first compute the homology of the moduli stack of objects in the derived category of a smooth complex projective variety X in class D, showing that the rational cohomology ring is freely generated by tautological classes. This is used to identify Joyce’s construction with a generalized super-lattice vertex algebra on the rational K-theory of X^an. Then, we prove orientability of moduli spaces of (complexes of) coherent sheaves on projective Calabi–Yau 4-folds–this has applications to defining a C-linear enumerative invariant theory for Calabi–Yau 4-folds. This result is based on joint work with Yalong Cao and Dominic Joyce. Lastly, we consider a connective even complex-oriented homology theory E with associated formal group law F and show that, given a Künneth isomorphism, replacing ordinary homology with E in Joyce’s construction yields a vertex F -algebra–this result is based on joint work with Markus Upmeier
On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds
No full textWall-crossing and orientations for invariants counting coherent sheaves on CY fourfolds
No full textBorisov–Joyce and Oh–Thomas defined virtual invariants counting sheaves on Calabi–Yau fourfolds. Similarly to Donaldson invariants, these depend on existence and choice of orientations on moduli spaces
of coherent sheaves. The first part of the thesis addresses this question
for quasi-projective Calabi–Yau fourfolds, generalizing the work of Cao–
Gross–Joyce. The orientations on compactly supported perfect complexes are expressed in terms of a pull-back of gauge-theoretic ones which
live on the classifying space CcsX of compactly supported K-theory. The
proof relies on a choice of a compactification, which allows us to directly
obtain orientability of moduli spaces of stable pairs. In the second part
of the thesis, we study the conjectural wall-crossing formulae of Gross–
Joyce–Tanaka. We begin, by addressing the conjecture of Cao–Kool
, which expresses the virtual integrals of a tautological line bundle L[n]
on the Hilbert scheme of points Hilbn
(X) in terms of the MacMahon function. We also obtain a prediction for the K-theoretic refinement of this
invariant proposed by Nekrasov, which coincides with the expectations from the result for C
4
. Studying the invariants further, we find a
universal transformation relating them to integrals on Hilbert schemes of
points for elliptic surfaces. To understand this, we recover the previously
known results for Quot-schemes on elliptic surfaces using similar wallcrossing arguments. We will further study this in to recover and generalize the full result of Arbesfeld–Johnson–Lim–Oprea–Pandharipande for surfaces including divisor contributions
A reading of selected writings of James Joyce in relation to the works of Gilles Deleuze (and Félix Guattari)
No full textChapter One consists in a more complete survey of the writings of Gilles Deleuze and Felix Guattari on the works of James Joyce than has previously been available,
together with an overview of Deleuzian philosophy. The focus in the first chapter is on Deleuze and Deleuze and Guattari's reading of philosophers and writers alike as
4 symptornatologists' of their times and the ethico-political beliefs which they implicitly share with Joyce. I relate this to Hardt and Negri's political speculations.
The conceptual 'tools' which make up 'schizoanalysis' are set out. The second chapter uses these tools in a 'symptomatological' diagnosis by first setting out and
then going beyond Joyce's depiction of the 'paralysis' of the populace in Dubliners and A Portrait to his fuller understanding of our problematic situation in modernity
depicted and diagnosed in the masochism of Bloom in Ulysses. In Chapter Three, I look at the epiphany, Deleuze's concept of Joyce's 'epiphanic machine', 'duration' as understood by Bergson and Deleuze, and the Deleuzian concept of 'affect' as potentially liberatory insights, after the preceding focus on negative
4 symptornatological' diagnoses. Together with a critique of the prevailing views of Joyce's epiphany I analyse three stories in Dubliners as illustrative of Deleuze's
understanding of the concept of the epiphany. In the fourth and fifth chapters I focus on Issy in the Wake read in terms of the 'bird-girl' of A Portrait and couple this with
the Deleuzian concept of the 'girl' as a crucial, but misunderstood, node in what can be seen as the 'rhizomatic assemblage' or 'network' constituting the 'epiphanic
machine' of the Wake. In Chapter Four, after first setting out the range of readings of Issy available in current Joycean criticism, I look at 'The Mime of Mick Nick and the
Maggies' (FW219.18-252.21) in terms of a further Joycean challenge to modernity's 'oedipalising' tendencies through Izod/ Issy. Here, I place a final emphasis on the
significance of incest and the incest taboo in 'the Mime' as the culmination of Joyce's 'symptomatological' diagnosis of modernity, and in counterbalancing this, his use of
the 'affect' of colour to offer us a productive 'line of flight'. In Chapter Five I recapitulate on Deleuze's highlighting of the letter in his positive comments on the
Wake and then, by using some established discussions in Joycean criticism as an introduction, engage in a reading of Issy's letter (FW 279F 1) as the Wakean 'line of
flight' by reading 'her' as liberatory 'desiring machine' with all of its ethico-political potentialities
Lectures on Calabi-Yau and special Lagrangian geometry
No full textThis paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the subject, and is designed to be fairly self-contained. It is based on lecture courses given at Nordfjordeid, Norway and MSRI, Berkeley in June and July 2001. We introduce Calabi-Yau m-folds via holonomy groups, Kahler geometry and the Calabi Conjecture, and special Lagrangian m-folds via calibrated geometry. `Almost Calabi-Yau m-folds' (a generalization of Calabi-Yau m-folds useful in special Lagrangian geometry) are explained and the deformation theory and moduli spaces of compact special Lagrangian submanifolds in (almost) Calabi-Yau m-folds is described. In the final part we consider isolated singularities of special Lagrangian m-folds, focussing mainly on singularities locally modelled on cones, and the expected behaviour of singularities of compact special Lagrangian m-folds in generic (almost) Calabi-Yau m-folds. String Theory, Mirror Symmetry and the SYZ Conjecture are briefly discussed, and some results of the author on singularities of special Lagrangian fibrations of Calabi-Yau 3-folds are described
Bordism invariance of orientations and real APS index theory
No full textThe author thanks Dominic Joyce for many discussion
- …
