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Fourier-Mukai transforms and stability conditions on abelian threefoldsPiyaratne, Hathurusinghege Dulip Bandara January 2014 (has links)
Construction of Bridgeland stability conditions on a given Calabi-Yau threefold is an important problem and this thesis realizes the rst known examples of such stability conditions. More precisely, we construct a dense family of stability conditions on the derived category of coherent sheaves on a principally polarized abelian threefold X with Picard rank one. In particular, we show that the conjectural construction proposed by Bayer, Macr and Toda gives rise to Bridgeland stability conditions on X. First we reduce the requirement of the Bogomolov-Gieseker type inequalities to a smaller class of tilt stable objects which are essentially minimal objects of the conjectural stability condition hearts for a given smooth projective threefold. Then we use the Fourier-Mukai theory to prove the strong Bogomolov-Gieseker type inequalities for these minimal objects of X. This is done by showing any Fourier-Mukai transform of X gives an equivalence of abelian categories which are double tilts of coherent sheaves.
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Extending the Geometric Tools of Heterotic String Compactification and DualitiesKarkheiran, Mohsen 15 June 2020 (has links)
In this work, we extend the well-known spectral cover construction first developed by Friedman, Morgan, and Witten to describe more general vector bundles on elliptically fibered Calabi-Yau geometries. In particular, we consider the case in which the Calabi-Yau fibration is not in Weierstrass form but can rather contain fibral divisors or multiple sections (i.e., a higher rank Mordell-Weil group). In these cases, general vector bundles defined over such Calabi-Yau manifolds cannot be described by ordinary spectral data. To accomplish this, we employ well-established tools from the mathematics literature of Fourier-Mukai functors. We also generalize existing tools for explicitly computing Fourier-Mukai transforms of stable bundles on elliptic Calabi-Yau manifolds. As an example of these new tools, we produce novel examples of chirality changing small instanton transitions. Next, we provide a geometric formalism that can substantially increase the understood regimes of heterotic/F-theory duality.
We consider heterotic target space dual (0,2) GLSMs on elliptically fibered Calabi-Yau manifolds. In this context, each half of the ``dual" heterotic theories must, in turn, have an F-theory dual. Moreover, the apparent relationship between two heterotic compactifications seen in (0,2) heterotic target space dual pairs should, in principle, induce some putative correspondence between the dual F-theory geometries. It has previously been conjectured in the literature that (0,2) target space duality might manifest in F-theory as multiple $K3$-fibrations of the same elliptically fibered Calabi-Yau manifold. We investigate this conjecture in the context of both 6-dimensional and 4-dimensional effective theories and demonstrate that in general, (0,2) target space duality cannot be explained by such a simple phenomenon alone. In all cases, we provide evidence that non-geometric data in F-theory must play at least some role in the induced F-theory correspondence while leaving the full determination of the putative new F-theory duality to the future work. Finally, we consider F-theory over elliptically fibered manifolds, with a general conic base. Such manifolds are quite standard in F-theory sense, but our goal is to explore the extent of the heterotic/F-theory duality over such manifolds. We consider heterotic target space dual (0,2) GLSMs on elliptically fibered Calabi-Yau manifolds. In this context, each half of the ``dual" heterotic theories must, in turn, have an F-theory dual. Moreover, the apparent relationship between two heterotic compactifications seen in (0,2) heterotic target space dual pairs should, in principle, induce some putative correspondence between the dual F-theory geometries. It has previously been conjectured in the literature that (0,2) target space duality might manifest in F-theory as multiple $K3$-fibrations of the same elliptically fibered Calabi-Yau manifold. We investigate this conjecture in the context of both 6-dimensional and 4-dimensional effective theories and demonstrate that in general, (0,2) target space duality cannot be explained by such a simple phenomenon alone. In all cases, we provide evidence that non-geometric data in F-theory must play at least some role in the induced F-theory correspondence while leaving the full determination of the putative new F-theory duality to the future work. Finally, we consider F-theory over elliptically fibered manifolds, with a general conic base. Such manifolds are quite standard in F-theory sense, but our goal is to explore the extent of the heterotic/F-theory duality over such manifolds. / Doctor of Philosophy / String theory is the only physical theory that can lead to self-consistent, effective quantum gravity theories. However, quantum mechanics restricts the dimension of the effective spacetime to ten (and eleven) dimensions. Hence, to study the consequences of string theory in four dimensions, one needs to assume the extra six dimensions are curled into small compact dimensions.
Upon this ``compactification," it has been shown (mainly in the 1990s) that different classes of string theories can have equivalent four-dimensional physics. Such classes are called dual. The advantage of these dualities is that often they can map perturbative and non-perturbative limits of these theories.
The goal of this dissertation is to explore and extend the geometric limitations of the duality between heterotic string theory and F-theory. One of the main tools in this particular duality is the Fourier-Mukai transformation. In particular, we consider Fourier-Mukai transformations over non-standard geometries. As an application, we study the F-theory dual of a heterotic/heterotic duality known as target space duality. As another side application, we derive new types of small instanton transitions in heterotic strings. In the end, we consider F-theory compactified over particular manifolds that if we consider them as a geometry dual to a heterotic string, can lead to unexpected consequences.
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The Mukai conjecture for log Fano manifolds / ログ・ファノ多様体に関する向井予想Fujita, Kento 24 March 2014 (has links)
京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18048号 / 理博第3926号 / 新制||理||1566(附属図書館) / 30906 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 森 重文, 教授 玉川 安騎男, 教授 向井 茂 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM
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Theta-duality in abelian varieties and the bicanonical map of irregular varietiesLahoz Vilalta, Marti 18 May 2010 (has links)
The first goal of this Thesis is to contribute to the study of principally polarized abelian varieties (ppav), especially to the
Schottky and the Torelli problems. Ppav admit a duality theory analogous to that of projective spaces, where the role
played by hyperplanes in projective spaces is played by divisors representing the principal polarization. Thus, given a
subvariety Y of a ppav, we can define its thetadual
T(Y) as the set of divisors representing the principal polarization
that contain this subvariety. This set admits a natural schematic structure (as defined by Pareschi and Popa).
Jacobian and Prym varieties are classical examples of ppav constructed from curves. Besides, they are interesting
because some properties of the curves involved in their construction are reflected in their geometry or in the geometry
of some special subvarieties. For example, in the case of Jacobians we have the BrillNoether
loci Wd ( W1 corresponds
to the AbelJacobi
curve) and in the case of Pryms we have the AbelPrym
curve C.
In chapter III, we study the schematic structure of the thetadual
of the BrillNoether
loci Wd and the AbelPrym
curve.
In the first case, we obtain with different methods, the result of Pareschi and Popa T(Wd)= Wgd1.
In the case of the
AbelPrym
curve C, we get that T(C)=V², where V² is the second PrymBrillNoether
locus with the schematic structure
defined by Welters.
Pareschi and Popa have proved a result for ppavs analogous to the Castelnuovo Lemma for projective spaces. That is,
if (A,Θ) is a ppav of dimension g, then g+2 distinct points in general position with respect to Θ, but in special position
with respect to 2Θ, have to be contained in a curve of minimal degree in A, i.e. an AbelJacobi
curve. In particular,
they obtain a Schottky result because A has to be a Jacobian variety and a Torelli result, because the curve is the
intersection of all the divisors in |2Θ| that contain the g+2 points. In chapter IV, as Eisenbud and Harris have done in
the projective Castelnuovo Lemma, we extend this result to possibly nonreduced
finite schemes.
The second goal of this thesis is the study of varieties of general type. Almost by definition, pluricanonical maps are the
essential tool to study them. One of the main problems in this area is to find geometric or numerical conditions to
guarantee that the mth
pluricanonical map (for low m) induces a birational equivalence with its image.
The classification of surfaces whose bicanonical map is nonbirational
has attracted considerable interest among
algebraic geometers. In chapter V, we give a sufficient numerical condition for the birationality of the bicanonical map of
irregular varieties of arbitrary dimension.
We also prove that, if X is a primitive variety, then it only admits very special fibrations to other irregular varieties. For
primitive varieties we get that the following are equivalent:
X
is birational to a divisor Θ in an indecomposable ppav,
the
irregularity q(X) > dim X and the bicanonical map is nonbirational.
When X is a primitive variety of general type and q(X) = dim X we prove, under certain conditions over the Stein
factorization of the Albanese map, that the only possibility for the bicanonical map being nonbirational
is that X is a
double cover branched along a divisor in |2Θ|. These results extend to arbitrary dimension, wellknown
theorems in the
case of surfaces and curves. / El primer objectiu d'aquesta tesi és contribuir a l'estudi de les varietats abelianes principalment polaritzades (vapp),
especialment als problemes de Schottky i Torelli. Les vapp admeten una teoria de dualitat anàloga a la dualitat dels
espais projectius, on el paper que juguen els hiperplans de l'espai projectiu és substituït pels divisors que representen
la polarització principal. Així doncs, donada una subvarietat Y d'una vapp, podem definir el seu thetadual
T(Y) com el
conjunt dels divisors que representen la polarització principal i contenen aquesta subvarietat. Aquest conjunt admet
una estructura esquemàtica natural (tal i com la defineixen Pareschi i Popa).
Les varietats Jacobianes i de Prym són exemples clàssics de vapp construïdes a partir de corbes. A més, són
interessants perquè certes propietats de les corbes involucrades es veuen reflectides en elles o en algunes
subvarietats especials. Per exemple, en el cas de les Jacobianes tenim els llocs de BrillNoether
Wd ( W1 correspon a
la corba d'AbelJacobi)
i en el cas de les Pryms tenim la corba d'AbelPrym
C.
Al capítol III de la tesi s'estudia l'estructura esquemàtica del thetadual
dels llocs de BrillNoether
Wd i de la corba
d'AbelPrym.
En el primer cas, es reobté amb uns altres mètodes, el resultat de Pareschi i Popa T(Wd)= Wgd1.
En el
cas de la corba d'AbelPrym
C, s'obté que T(C)=V², onV² és el segon lloc de PrymBrillNoether
amb l'estructura
esquemàtica definida per Welters.
Pareschi i Popa han demostrat un resultat anàleg per les vapp al Lemma de Castelnuovo pels espais projectius. És a
dir, si (A,Θ) és una vapp de dimensió g, aleshores g+2 punts en posició general respecte Θ, però en posició especial
respecte 2Θ, han d'estar continguts en una corba de grau minimal a A, i.e. una corba d'AbelJacobi.
En particular,
s'obté un resultat de Schottky ja que A ha de ser una Jacobiana i un resultat de Torelli, ja que la corba és la intersecció
de tots els divisors de |2Θ| que contenen els g+2 punts. Al capítol IV, tal i com Eisenbud i Harris van fer en el cas
projectiu, s'estén aquest resultat a esquemes finits possiblement no reduïts.
El segon objectiu d'aquesta tesi és contribuir a l'estudi de les varietats de tipus general. Pràcticament per definició, les
aplicacions pluricanòniques són essencials pel seu estudi. Un dels problemes principals de l'àrea és donar condicions
geomètriques o numèriques per assegurar que la mèsima
aplicació pluricanònica (per m baix) indueix una
equivalència biracional amb la imatge.
La classificació de les superfícies que tenen l'aplicació bicanònica no biracional ha atret l'atenció de molts geòmetres
algebraics. Al capítol V, es dóna un criteri numèric suficient per assegurar la biracionalitat de l'aplicació bicanònica de
les varietats irregulars de dimensió arbitrària.
També es demostra que si X és una varietat primitiva, aleshores només admet fibracions molt especials a altres
varietats irregulars. Per aquestes varietats s'obté que és equivalent que X sigui biracional a un divisor Θ en una vapp
indescomponible, a què la irregularitat q(X) > dim X i l'aplicació bicanònica sigui no biracional. Quan X és una varietat
primitiva de tipus general i q(X) = dim X es demostra sota certes condicions de la descomposició de Stein del morfisme
d'Albanese, que l'única possibilitat per tal que l'aplicació bicanònica sigui no biracional és que X sigui un recobriment
doble sobre una vapp ramificat al llarg d'un divisor a |2Θ|. Aquest resultats estenen a dimensió arbitrària, teoremes ben
coneguts en el cas de superfícies i corbes.
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Equivariant Moduli Theory on K3 SurfacesChen, Yuhang 08 September 2022 (has links)
No description available.
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