1,154 research outputs found
Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces
We construct a compactification of the Uhlenbeck-Donaldson type
for the moduli space of slope stable framed bundles. This is a kind of a moduli
space of slope semistable framed sheaves. We show that there exists a
projective morphism , where is
the moduli space of S-equivalence classes of Gieseker-semistable framed
sheaves. The space has a natural set-theoretic stratification
which allows one, via a Hitchin-Kobayashi correspondence, to compare it with
the moduli spaces of framed ideal instantons.Comment: 18 pages. v2: a few very minor changes. v3: 27 pages. Several proofs
have been considerably expanded, and more explanations have been added. v4:
28 pages. A few minor changes. Final version accepted for publication in
Math.
Derived categories of cubic fourfolds
We discuss the structure of the derived category of coherent sheaves on cubic
fourfolds of three types: Pfaffian cubics, cubics containing a plane and
singular cubics, and discuss its relation to the rationality of these cubics.Comment: 18 page
Exceptional Sequences of Line Bundles and Spherical Twists - a Toric Example
Exceptional sequences of line bundles on a smooth projective toric surface
are automatically full when they can be constructed via augmentation. By using
spherical twists, we give examples that there are also exceptional sequences
which can not be constructed this way but are nevertheless full.Comment: 12 pages, 3 figure
The Moduli of Reducible Vector Bundles
A procedure for computing the dimensions of the moduli spaces of reducible,
holomorphic vector bundles on elliptically fibered Calabi-Yau threefolds X is
presented. This procedure is applied to poly-stable rank n+m bundles of the
form V + pi* M, where V is a stable vector bundle with structure group SU(n) on
X and M is a stable vector bundle with structure group SU(m) on the base
surface B of X. Such bundles arise from small instanton transitions involving
five-branes wrapped on fibers of the elliptic fibration. The structure and
physical meaning of these transitions are discussed.Comment: 33+1 page
Webs of Lagrangian Tori in Projective Symplectic Manifolds
For a Lagrangian torus A in a simply-connected projective symplectic manifold
M, we prove that M has a hypersurface disjoint from a deformation of A. This
implies that a Lagrangian torus in a compact hyperk\"ahler manifold is a fiber
of an almost holomorphic Lagrangian fibration, giving an affirmative answer to
a question of Beauville's. Our proof employs two different tools: the theory of
action-angle variables for algebraically completely integrable Hamiltonian
systems and Wielandt's theory of subnormal subgroups.Comment: 18 pages, minor latex problem fixe
Introduction to derived categories of coherent sheaves
In these notes, an introduction to derived categories and derived functors is
given. The main focus is the bounded derived category of coherent sheaves on a
smooth projective variety.Comment: 24 pages, minor changes, same content as published versio
Abel-Jacobi maps for hypersurfaces and non commutative Calabi-Yau's
It is well known that the Fano scheme of lines on a cubic 4-fold is a
symplectic variety. We generalize this fact by constructing a closed p-form
with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y
of degree n. We provide several definitions of this form - via the Abel-Jacobi
map, via Hochschild homology, and via the linkage class, and compute it
explicitly for n = 4. In the special case of a Pfaffian hypersurface Y we show
that the Fano scheme is birational to a certain moduli space of sheaves on a
p-dimensional Calabi--Yau variety X arising naturally in the context of
homological projective duality, and that the constructed form is induced by the
holomorphic volume form on X. This remains true for a general non Pfaffian
hypersurface but the dual Calabi-Yau becomes non commutative.Comment: 34 pages; exposition of Hochschild homology expanded; references
added; introduction re-written; some imrecisions, typos and the orbit diagram
in the last section correcte
On the Heterotic World-sheet Instanton Superpotential and its individual Contributions
For supersymmetric heterotic string compactifications on a Calabi-Yau
threefold endowed with a vector bundle the world-sheet superpotential
is a sum of contributions from isolated rational curves \C in ; the
individual contribution is given by an exponential in the K\"ahler class of the
curve times a prefactor given essentially by the Pfaffian which depends on the
moduli of and the complex structure moduli of . Solutions of (or
even of ) can arise either by nontrivial cancellations between the
individual terms in the summation over all contributing curves or because each
of these terms is zero already individually. Concerning the latter case
conditions on the moduli making a single Pfaffian vanish (for special moduli
values) have been investigated. However, even if corresponding moduli -
fulfilling these constraints - for the individual contribution of one curve are
known it is not at all clear whether {\em one} choice of moduli exists which
fulfills the corresponding constraints {\em for all contributing curves
simultaneously}. Clearly this will in general happen only if the conditions on
the 'individual zeroes' had already a conceptual origin which allows them to
fit together consistently. We show that this happens for a class of cases. In
the special case of spectral cover bundles we show that a relevant solution set
has an interesting location in moduli space and is related to transitions which
change the generation number.Comment: 47 page
- …