3,610 research outputs found

### Symmetries of Riemann surfaces and magnetic monopoles

This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bringâs curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bringâs curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions

### Categorical Torelli theorems for Fano threefolds

The derived category Db(X) of a variety contains a lot of information about X. If
X and XâČ are Fano, then an equivalence Db(X) â Db(XâČ) implies that X and XâČ
are isomorphic. For prime Fano threefolds X (of Picard rank 1, index 1, and genus
g â„ 6) the derived category decomposes semiorthogonally as âšKu(X), E, OX â©,
where E is a certain vector bundle on X. Therefore one can ask whether less
data (in particular the Kuznetsov component Ku(X)) than the whole of Db(X)
determines X isomorphically (or at least birationally).
In this thesis, we focus on this question in the case of ordinary GushelâMukai
threefolds (genus 6 prime Fano threefolds). We show that Ku(X) determines the
birational class of X which proves a conjecture of KuznetsovâPerry in dimension
3. We also prove a refined categorical Torelli theorem for oridnary GushelâMukai
threefolds. In other words, we show that Ku(X) along with the data of the vector
bundle E is enough to determine X up to isomorphism

### Slopes of modular forms and geometry of eigencurves

Under a stronger genericity condition, we prove the local analogue of ghost
conjecture of Bergdall and Pollack. As applications, we deduce in this case (a)
a folklore conjecture of Breuil--Buzzard--Emerton on the crystalline slopes of
Kisin's crystabelian deformation spaces, (b) Gouvea's
$\lfloor\frac{k-1}{p+1}\rfloor$-conjecture on slopes of modular forms, and (c)
the finiteness of irreducible components of the eigencurve. In addition,
applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce
as corollaries in the reducible and strongly generic case, (d) Gouvea--Mazur
conjecture, (e) a variant of Gouvea's conjecture on slope distributions, and
(f) a refined version of Coleman's spectral halo conjecture.Comment: 97 pages; comments are welcom

### Categorical entropies on symplectic manifolds

In this paper, being motivated by symplectic topology, we study categorical
entropy. Specifically, we prove inequalities between categorical entropies of
functors on a category and its localization. We apply the inequalities to
symplectic topology to prove equalities between categorical entropies on
wrapped, partially wrapped, and compact Fukaya categories if the functors are
induced by the same compactly supported symplectic automorphisms. We also
provide a practical way to compute the categorical entropy of symplectic
automorphisms by using Lagrangian Floer theory if their domains satisfy a type
of Floer-theoretical duality. Our main examples of symplectic manifolds
satisfying the duality conditions are the plumbings of cotangent bundles of
sphere along a tree. Moreover, for symplectic automorphisms of Penner type, we
prove that our computation of categorical entropy becomes a computation by
simple linear algebra.Comment: 38 page

### Hecke Actions on Loops and Periods of Iterated Shimura Integrals

In this paper we show that the action of the classical Hecke operators T_N,
N>0, act on the free abelian groups generated by the conjugacy classes of the
modular group SL_2(Z) and the conjugacy classes of its profinite completion. We
show that this action induces a dual action on the ring of class functions of a
certain relative unipotent completion of the modular group. This ring contains
all iterated integrals of modular forms that are constant on conjugacy classes.
It possesses a natural mixed Hodge structure and, after tensoring with Q_ell$,
a natural action of the absolute Galois group. Each Hecke operator preserves
this mixed Hodge structure and commutes with the action of the absolute Galois
group. Unlike in the classical case, the algebra generated by these Hecke
operators is not commutative. The appendix by Pham Tiep is not included. It can
be found at arXiv:2303.02807.Comment: 92 pages; this version has an appendix by Pham Tiep. It is not
included and can be found at arXiv:2303.02807. In addition, there are various
corrections and improvements, as well as some new material in Section 1

### Perfect even modules and the even filtration

Inspired by the work of Hahn-Raksit-Wilson, we introduce a variant of the
even filtration which is naturally defined on $\mathbf{E}_{1}$-rings and their
modules. We show that our variant satisfies flat descent and so agrees with the
Hahn-Raksit-Wilson filtration on ring spectra of arithmetic interest, showing
that various "motivic" filtrations are in fact invariants of the
$\mathbf{E}_{1}$-structure alone. We prove that our filtration can be
calculated via appropriate resolutions in modules and apply it to the study of
even cohomology of connective $\mathbf{E}_{1}$-rings, proving vanishing above
the Milnor line, base-change formulas, and explicitly calculating cohomology in
low weights

### Bridgeland Stability Conditions and the Hilbert Scheme of Skew Lines in Projective Space

Bridgeland stability conditions are powerful tools for studying derived categories, with several applications to algebraic geometry. They were introduced by Bridgeland in 2002 [Bri07], who was motivated by Douglasâ work on Î -stability of D-branes [Dou02] in the context of string theory. Bridgeland showed that the set Stab(D) of stability conditions on a triangulated category D is a complex manifold, a result of extreme importance and central to all mathematical applications of this field of study. But in order to use this concept of stability conditions in string theory (as intended by Bridgeland), one needs to prove the existence of stability conditions on the bounded derived category Db(X) of a compact Calabi-Yau threefold X. This task is far from easy, as it took more than a decade before the first example was produced for the smooth quintic threefold by Li in [Li18]. This achievement came into fruition thanks to the extensive amount of work in the domain over this period of time, where the existence of stability conditions was progressively established for arbitrary smooth projective varieties of dimension one [Bri07, Oka06, Mac07], dimension two [Bri08, AB13], and then some dimension three cases (see Section 1.3).
One of the main applications of stability conditions on Db(X) (for an arbitrary variety X) is to study the geometry of moduli spaces of coherent sheaves over X with some Chern character v via the strategy known as âwall crossingâ. In loose terms, a âwallâ is a codimension one submanifold of Stab(Db(X)) such that by changing stability conditions along a continuous path in Stab(Db(X)) that goes through the wall causes the moduli space of sheaves over X with Chern character v to transform. When X is of dimension two, we have a solid control over wall crossing thanks to BayerâMacr`Ä± [BM11], who provided a full understanding of how moduli spaces of sheaves change as we cross walls, as well as knowing the exact geometrical relationship these walls have with the underlying surface. In addition the precise structure of the walls is known and there are effective techniques to detect them. This thorough picture of wall crossing in dimension two is demonstrated through various complete studies of moduli spaces of sheaves over surfaces [AB13, ABCH13, Mea12].
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### Deformation theory of G-valued pseudocharacters and symplectic determinant laws

We give an introduction to the theory of pseudorepresentations of Taylor, Rouquier, Chenevier and
Lafforgue. We refer to Taylorâs and Rouquierâs pseudorepresentations as pseudocharacters. They are
very closely related, the main difference being that Taylorâs pseudocharacters are defined for a group,
where as Rouquierâs pseudocharacters are defined for algebras. Chenevierâs pseudorepresentations are
so-called polynomial laws and will be called determinant laws. Lafforgueâs pseudorepresentations are a
generalization of Taylorâs pseudocharacters to other reductive groups G, in that the corresponding notion
of representation is that of a G-valued representation of a group. We refer to them as G-pseudocharacters.
We survey the known comparison theorems, notably Emersonâs bijection between Chenevierâs determinant
laws and Lafforgueâs GL(n)-pseudocharacters and the bijection with Taylorâs pseudocharacters away from
small characteristics.
We show, that duals of determinant laws exist and are compatible with duals of representations. Analogously,
we obtain that tensor products of determinant laws exist and are compatible with tensor products
of representations. Further the tensor product of Lafforgueâs pseudocharacters agrees with the tensor
product of Taylorâs pseudocharacters.
We generalize some of the results of [Che14] to general reductive groups, in particular we show that
the (pseudo)deformation space of a continuous Lafforgue G-pseudocharacter of a topologically finitely
generated profinite group Î with values in a finite field (of characteristic p) is noetherian. We also show,
that for specific groups G it is sufficient, that Î satisfies Mazurâs condition ÎŠ_p.
One further goal of this thesis was to generalize parts of [BIP21] to other reductive groups. Let F/Qp
be a finite extension. In order to carry this out for the symplectic groups Sp2d, we obtain a simple and
concrete stratification of the special fiber of the pseudodeformation space of a residual G-pseudocharater
of Gal(F) into obstructed subloci Xdec(Î), Xpair(Î), Xspcl(Î) of dimension smaller than the expected dimension
n(2n + 1)[F : Qp].
We also prove that Lafforgueâs G-pseudocharacters over algebraically closed fields for possibly nonconnected
reductive groups G come from a semisimple representation. We introduce a formal scheme
and a rigid analytic space of all G-pseudocharacters by a functorial description and show, building on
our results of noetherianity of pseudodeformation spaces, that both are representable and admit a decomposition
as a disjoint sum indexed by continuous pseudocharacters with values in a finite field up to
conjugacy and Frobenius automorphisms.
At last, in joint work with Mohamed Moakher, we give a new definition of determinant laws for symplectic
groups, which is based on adding a âPfaffian polynomial lawâ to a determinant law which is invariant under
an involution. We prove the expected basic properties in that we show that symplectic determinant laws
over algebraically closed fields are in bijection with conjugacy classes of semisimple representation and
that Cayley-Hamilton lifts of absolutely irreducible symplectic determinant laws to henselian local rings
are in bijection with conjugacy classes of representations. We also give a comparison map with Lafforgueâs
pseudocharacters and show that it is an isomorphism over reduced rings

### Derived Categories and Fourier-Mukai Transforms

openThe body of this thesis unfolds in detail the theory of derived categories in order to exploit and explore the geometry of the derived category of coherent sheaves on a projective variety, and inherently of the variety itself. By tracing the development of Fourier-Mukai transforms, we elucidate some of the different modalities and criteria for establishing derived equivalences between two varieties. In particular we delve into some of the results that bridge the geometry of a variety with that of its derived category---among them---we discuss the Bondal-Orlov reconstruction theorem and the derived equivalence between an abelian variety and its dual

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