53 research outputs found

### Multiple Derived Lagrangian Intersections

We give a new way to produce examples of Lagrangians in shifted symplectic
derived stacks, based on multiple intersections. Specifically, we show that an
m-fold fiber product of Lagrangians in a shifted symplectic derived stack its
itself Lagrangian in a certain cyclic product of pairwise homotopy fiber
products of the Lagrangians

### Gerbes and the Holomorphic Brauer Group of Complex Tori

The purpose of this paper is to develop the theory of holomorphic gerbes on
complex tori in a manner analogous to the classical theory for line bundles. In
contrast to past studies on this subject, we do not restrict to the case where
these gerbes are torsion or topologically trivial. We give an Appell-Humbert
type description of all holomorphic gerbes on complex tori. This gives an
explicit, simple, cocycle representative (and hence gerbe) for each equivalence
class of holomorphic gerbes. We also prove that a gerbe on the fiber product of
four spaces over a common base is trivial as long as it is trivial upon
restriction to any three out of the four spaces. A fine moduli stack for gerbes
on complex tori is constructed. This involves the construction of a
'Poincar\'e' gerbe which plays a role analogous to the role of the Poincar\'e
bundle in the case of line bundles.Comment: 33 pages, comments appreciate

### Mirror Symmetry and Generalized Complex Manifolds

In this paper we develop a relative version of T-duality in generalized
complex geometry which we propose as a manifestation of mirror symmetry. Let M
be an n-dimensional smooth real manifold, V a rank n real vector bundle on M,
and nabla a flat connection on V. We define the notion of a nabla-semi-flat
generalized complex structure on the total space of V. We show that there is an
explicit bijective correspondence between nabla-semi-flat generalized complex
structures on the total space of V and nabla(dual)-semi-flat generalized
complex structures on the total space of the dual of V. Similarly we define
semi-flat generalized complex structures on real n-torus bundles with section
over an n-dimensional base and establish a similar bijective correspondence
between semi-flat generalized complex structures on pair of dual torus bundles.
Along the way, we give methods of constructing generalized complex structures
on the total spaces of vector bundles and torus bundles with sections. We also
show that semi-flat generalized complex structures give rise to a pair of
transverse Dirac structures on the base manifold. We give interpretations of
these results in terms of relationships between the cohomology of torus bundles
and their duals. We also study the ways in which our results generalize some
well established aspects of mirror symmetry as well as some recent proposals
relating generalized complex geometry to string theory.Comment: Small additions, references adde

### Perversely categorified Lagrangian correspondences

In this article, we construct a $2$-category of Lagrangians in a fixed
shifted symplectic derived stack S. The objects and morphisms are all given by
Lagrangians living on various fiber products. A special case of this gives a
$2$-category of $n$-shifted symplectic derived stacks $Symp^n$. This is a
$2$-category version of Weinstein's symplectic category in the setting of
derived symplectic geometry. We introduce another $2$-category $Symp^{or}$ of
$0$-shifted symplectic derived stacks where the objects and morphisms in
$Symp^0$ are enhanced with orientation data. Using this, we define a partially
linearized $2$-category $LSymp$. Joyce and his collaborators defined a certain
perverse sheaf on any oriented $(-1)$-shifted symplectic derived stack. In
$LSymp$, the $2$-morphisms in $Symp^{or}$ are replaced by the hypercohomology
of the perverse sheaf assigned to the $(-1)$-shifted symplectic derived
Lagrangian intersections. To define the compositions in $LSymp$ we use a
conjecture by Joyce, that Lagrangians in $(-1)$-shifted symplectic stacks
define canonical elements in the hypercohomology of the perverse sheaf over the
Lagrangian. We refine and expand his conjecture and use it to construct $LSymp$
and a $2$-functor from $Symp^{or}$ to $LSymp$. We prove Joyce's conjecture in
the most general local model. Finally, we define a $2$-category of $d$-oriented
derived stacks and fillings. Taking mapping stacks into a $n$-shifted
symplectic stack defines a $2$-functor from this category to $Symp^{n-d}$.Comment: v3: improved exposition; couple of results adde

### Bounded linear endomorphisms of rigid analytic functions

Let $K$ be a field of characteristic zero complete with respect to a
non-trivial, non-Archimedean valuation. We relate the sheaf
$\widehat{\mathcal{D}}$ of infinite order differential operators on smooth
rigid $K$-analytic spaces to the algebra $\mathcal{E}$ of bounded $K$-linear
endomorphisms of the structure sheaf. In the case of complex manifolds,
Ishimura proved that the analogous sheaves are isomorphic. In the rigid
analytic situation, we prove that the natural map $\widehat{\mathcal{D}} \to
\mathcal{E}$ is an isomorphism if and only if the ground field $K$ is
algebraically closed and its residue field is uncountable.Comment: 23 pages. Comments welcom

### Dagger Geometry As Banach Algebraic Geometry

In this article, we apply the approach of relative algebraic geometry towards
analytic geometry to the category of bornological and Ind-Banach spaces
(non-Archimedean or not). We are able to recast the theory of Grosse-Kl\"onne
dagger affinoid domains with their weak G-topology in this new language. We
prove an abstract recognition principle for the generators of their standard
topology (the morphisms appearing in the covers). We end with a sketch of an
emerging theory of dagger affinoid spaces over the integers, or any Banach
ring, where we can see the Archimedean and non-Archimedean worlds coming
together

### Fr\'echet Modules and Descent

We study several aspects of the study of Ind-Banach modules over Banach rings
thereby synthesizing some aspects of homological algebra and functional
analysis. This includes a study of nuclear modules and of modules which are
flat with respect to the projective tensor product. We also study metrizable
and Fr\'{e}chet Ind-Banach modules. We give explicit descriptions of projective
limits of Banach rings as ind-objects. We study exactness properties of
projective tensor product with respect to kernels and countable products. As
applications, we describe a theory of quasi-coherent modules in Banach
algebraic geometry. We prove descent theorems for quasi-coherent modules in
various analytic and arithmetic contexts.Comment: improved versio

### Meromorphic Line Bundles and Holomorphic Gerbes

We show that any complex manifold that has a divisor whose normalization has
non-zero first Betti number either has a non-trivial holomorphic gerbe which
does not trivialize meromorphicly or a meromorphic line bundle not equivalent
to any holomorphic line bundle. Similarly, higher Betti numbers of divisors
correspond to higher gerbes or meromorphic gerbes. We give several new
examples.Comment: 14 page

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