Presentation about the paper with the same title http://arxiv.org/abs/1012.3121
Abstract:
In [Ma1] S. Ma established a bijection between Fourier--Mukai partners of a K3 surface and cusps of the K\"ahler moduli space. The K\"ahler moduli space can be described as a quotient of Bridgeland's stability manifold. We study the relation between stability conditions σ near to a cusp and the associated Fourier--Mukai partner Y in the following ways. (1) We compare the heart of σ to the heart of coherent sheaves on Y. (2) We construct Y as moduli space of σ-stable objects.
An appendix is devoted to the group of auto-equivalences of the derived category which respect the component Stab†(X) of the stability manifold
Information geometry: Dualistic manifold structures and their usesFrank Nielsen
Information geometry: Dualistic manifold structures and their uses
by Frank Nielsen
Talk given at ICML GIMLI2018
http://gimli.cc/2018/
See tutorial at:
https://arxiv.org/abs/1808.08271
``An elementary introduction to information geometry''
Information geometry: Dualistic manifold structures and their usesFrank Nielsen
Information geometry: Dualistic manifold structures and their uses
by Frank Nielsen
Talk given at ICML GIMLI2018
http://gimli.cc/2018/
See tutorial at:
https://arxiv.org/abs/1808.08271
``An elementary introduction to information geometry''
The second Fundamental Theorem of Calculus makes calculating definite integrals a problem of antidifferentiation!
(the slideshow has extra examples based on what happened in class)
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
Homogeneous Components of a CDH Fuzzy SpaceIJECEIAES
We prove that fuzzy homogeneous components of a CDH fuzzy topological space (X,T) are clopen and also they are CDH topological subspaces of its 0-cut topological space (X,T0).
RW-CLOSED MAPS AND RW-OPEN MAPS IN TOPOLOGICAL SPACESEditor IJCATR
In this paper we introduce rw-closed map from a topological space X to a topological space Y as the image
of every closed set is rw-closed and also we prove that the composition of two rw-closed maps need not be rw-closed
map. We also obtain some properties of rw-closed maps.
The second Fundamental Theorem of Calculus makes calculating definite integrals a problem of antidifferentiation!
(the slideshow has extra examples based on what happened in class)
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
Homogeneous Components of a CDH Fuzzy SpaceIJECEIAES
We prove that fuzzy homogeneous components of a CDH fuzzy topological space (X,T) are clopen and also they are CDH topological subspaces of its 0-cut topological space (X,T0).
RW-CLOSED MAPS AND RW-OPEN MAPS IN TOPOLOGICAL SPACESEditor IJCATR
In this paper we introduce rw-closed map from a topological space X to a topological space Y as the image
of every closed set is rw-closed and also we prove that the composition of two rw-closed maps need not be rw-closed
map. We also obtain some properties of rw-closed maps.
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein SpacesSEENET-MTP
Balkan Workshop BW2013
Beyond the Standard Models
25 – 29 April, 2013, Vrnjačka Banja, Serbia
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein Spaces
Notes on intersection theory written for a seminar in Bonn in 2010.
Following Fulton's book the following topics are covered:
- Motivation of intersection theory
- Cones and Segre Classes
- Chern Classes
- Gauss-Bonet Formula
- Segre classes under birational morphisms
- Flat pull back
A tutorial on the Frobenious Theorem, one of the most important results in differential geometry, with emphasis in its use in nonlinear control theory. All results are accompanied by proofs, but for a more thorough and detailed presentation refer to the book of A. Isidori.
Latency is a key indicator of service quality, and important to measure and track. However, measuring latency correctly is not easy. In contrast to familiar metrics like CPU utilization or request counts, the "latency" of a service is not easily expressed in numbers. Percentile metrics have become a popular means to measure the request latency, but have several shortcomings, especially when it comes to aggregation. The situation is particularly dire if we want to use them to specify Service Level Objectives (SLOs) that quantify the performance over a longer time horizons. In the talk we will explain these pitfalls, and suggest three practical methods how to implement effective Latency SLOs.
Monitoring systems will get smarter in order to keep up with the demands of tomorrow's IT architectures. Features like anomaly detection, root cause analysis, and forecasting tools will be critical components of this next level of monitoring. At the same time, the data that monitoring systems ingest is ever increasing in amount and velocity.
This session covers architectural models for advanced online analytics. We argue that stateful online computations provide a means to realize machine learning on high-velocity data. We show how alerting systems, event engines, stream aggregators, and time-series databases interact to support smart, scalable, and resilient monitoring solutions.
Heinrich Hartmann is the Chief Data Scientist at Circonus. He is driving the development of analytics methods that transform monitoring data into actionable information as part of the Circonus monitoring platform. In his prior life, Heinrich pursued an academic career as a mathematician (PhD in Bonn, Oxford). Later he transitioned into computer science and worked as consultant for a number of different companies and research institutions.
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
What is greenhouse gasses and how many gasses are there to affect the Earth.
Cusps of the Kähler moduli space and stability conditions on K3 surfaces
1. FourierMukai partners and stability conditions on K3
surfaces
Heinrich Hartmann
University of Bonn
24.2.2011
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 1 / 19
2. Outline
1 Moduli spaces of sheaves on K3 surfaces
2 Stability conditions and the Kähler moduli space
3 Geometric interpretations of Ma's result
4 Future research plans
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 2 / 19
3. Moduli spaces on K3 surfaces are well behaved
Let X be a K3 surface and let N(X) = K(Coh(X))/rad(χ) be the
numerical K-group, endowed with the pairing (_._) = −χ(_, _).
Theorem (Mukai)
Let v ∈ N(X) be a vector with v.v = 0 and v.N(X) = Z. Then there
exists an ample class h ∈ NS(X) such that:
1 The moduli space M = Mh(v) is again a K3 surface.
In particular M is ne, smooth, compact and two-dimensional.
2 The FourierMukai functor is an equivalence:
ΦU : Db(M)
∼
−→ Db(X).
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 3 / 19
4. Ma interpretation of FourierMukai partners
Let X be a K3 surface. Recently, Shouhei Ma gave a surprising
interpretation of the set of FourierMukai partners of X:
Theorem (Ma)
There is a canonical bijection between
K3 surfaces Y
with Db(Y ) ∼= Db(X)
←→
standard cusps of the
Kähler moduli space KM(X)
.
There is a version for non-standard cusps and K3 surfaces twisted by
a Brauer class.
The proof uses deep theorems due to Mukai and Orlov to reduce the
statement to lattice theory.
Is there a geometric reason for this correspondence?
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 4 / 19
5. Construction of the Kähler moduli space
Consider the period domain
D(X) = { [z] ∈ P(N(X)C) | z.z = 0, z.¯z 0 } .
We and dene the Kähler moduli space to be
KM(X) = D(X)+
/Γ,
where Γ is the image of Aut(Db(X)) in O(N(X)).
The BailyBorel compactication KM(X) ⊂ KM(X) is a normal
projective variety. The complement KM(X) KM(X) consists of
components of dimension 0 and 1 which are in bijection to Bi/Γ, where
Bi = { I ⊂ N(X) | primitive, isotropic, rk(I) = i + 1 }
for i = 0, 1 respectively.
Boundary components of dimension 0 are called cusps.
We call a cusp [I] ∈ KM(X) standard if I.N(X) = Z.
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 5 / 19
6. Picture of the Kähler moduli space
Figure: Kähler moduli space with cusps, associated K3 surfaces and two dierent
degenerating paths.
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 6 / 19
7. Bridgeland stability conditions
Let D be a C-linear, triangulated category.
Denition (Bridgeland)
A stability condition σ on D consists of:
a heart of a bounded t-structure A ⊂ D and
a vector z ∈ N(D)C, called central charge.
satisfying the following properties:
1 For all E ∈ A, E = 0 the complex number (z.[E]) = r exp(iπφ)
satises r 0 and φ ∈ (0, 1].
2 Existence of Hader-Narasimhan ltrations.
3 Local niteness.
An object E ∈ A is called σ-stable if for all sub-objects F ⊂ E in A
φ(F) φ(E).
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 7 / 19
8. Stability conditions on K3 surfaces
Let X be a K3 surface and D = Db(X).
Theorem (Bridgeland)
The set of stability conditions Stab(D) on D has the structure of a
complex manifold. The map σ = (A, z) → z induces a Galois-cover
π : Stab†
(X) −→ P+
0 (X) ⊂
open
N(X)C,
where Stab†(X) is the connected component of Stab(D), containing the
stability conditions σX(ω, β). Moreover
Deck(π) ∼= Aut†
0(D)
is the group of auto-equivalences respecting the component Stab†(X) and
acting trivially on H∗(X, Z).
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 8 / 19
9. Stability conditions and the Kähler moduli space.
We get the following diagram
Stab†(X)
Gl+
2 (R)
π // P+
0 (X)
Gl+
2 (R)
Stab†(X)/Gl
+
2 (R) //
Aut†(D)
D+
0 (X)
Γ
Aut†(D) Stab†(X)/Gl
+
2 (R)
π // KM0(X),
where KM0(X) ⊂ KM(X) is a special open subset, and Aut†(D) ⊂ Aut(D)
is the subgroup of auto-equivalences which respect the distinguished
component.
Fact: π is an isomorphism.
This fact has been stated by Bridgeland and Ma before. However, it seems
to depend on the following results.
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 9 / 19
10. Equivalences respecting the distinguished component
Theorem
The following equivalences respect the distinguished component.
For a ne, compact, two-dimensional moduli space of Gieseker-stable
sheaves Mh(v), the FourierMukai equivalence induced by the
universal family.
The spherical twists along Gieseker-stable spherical vector bundles.
The spherical twists along OC(k) for a (−2)-curve C ⊂ X and k ∈ Z.
This allows us to show the following strengthening of a result of
[HuybrechtsMacriStellari].
Corollary
The map Aut†(Db(X)) −→ Γ ⊂ O(N(X)) is surjective.
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 10 / 19
11. Main question
There is a canonical map
¯π : Stab†
(X) −→ KM(X).
What is the relation between stability conditions σ with ¯π(σ) near
to a cusp and the associated K3 surface Y ?
1 How is the heart of σ related to the heart Coh(Y )?
2 Can we construct Y as a moduli space of σ-stable objects?
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 11 / 19
12. Cusps and hearts
We can always nd a degeneration σ(t) of stability conditions, such that
the hearts converge to Coh(Y ):
Theorem
Let [I] ∈ KM(X) be a standard cusp and Y the K3 surface associated to
[I] via Ma's theorem. Then there exists a path σ(t) ∈ Stab†(X), t 0 and
an equivalence Φ : Db(Y )
∼
−→ D such that
1 lim
t→∞
π(σ(t)) = [I] ∈ KM(X) and
2 lim
t→∞
A(σ(t)) = Φ(Coh(Y )) as subcategories of D.
There are many other hearts that can occur as limits!
How can we classify all of them?
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 12 / 19
13. Linear degenerations
We dene a class of pahts γ(t) ∈ KM(X) called linear degeneration to
a cusp [I].
The prototypical example of a linear degeneration is ¯π(σY (β, tω)),
where σY (β, ω) is an explicit stability condition associated to
β, ω ∈ NS(X)R with ω ample, dened by Bridgeland.
Proposition
Let [v] ∈ KM(X) be a standard cusp and γ(t) ∈ KM(X) be a linear
degeneration to [v], then γ(t) is a geodesic converging to [v].
Conjecture
Every geodesic converging to [v] is a linear degeneration.
True in the Picard-rank one case
True if one works with BorelSerre compactication, c.f. [Borel-Ji].
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 13 / 19
14. Classication of linear degenerations
Theorem
Let [I] be a standard cusp of KM(X). Let σ(t) ∈ Stab†(X) be a path in
the stability manifold such that ¯π(σ(t)) ∈ KM(X) is a linear degeneration
to [I]. Let Y be the K3 surface associated to [v] by Ma. Then there exist
1 a derived equivalence Φ : Db(Y )
∼
−→ D,
2 classes β ∈ NS(Y )R, ω ∈ Amp(Y ) and
3 a path g(t) ∈ Gl
+
2 (R)
such that
σ(t) = Φ∗(σ∗
Y (β, t ω) · g(t))
for all t 0.
Moreover, the hearts of σ(t) · g(t)−1 are independent of t for t 0 and
can be explicitly described as a tilt of Coh(Y ).
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 14 / 19
15. Moduli of complexes on K3 surfaces
Let D be the derived category of a K3 surface X. For a stability condition
in the sense of Bridgeland σ ∈ Stab(D) and v ∈ N(D) we consider the
following moduli-space of semi-stable objects
Mσ
(v) = { E ∈ D | E σ-semi-stable, [E] = v } /even shifts.
This space has the structure of an Artin-stack of nite type due to results
by Lieblich and Toda. We prove the following result.
Theorem
If v ∈ N(X) is a vector with v.v = 0, v.N(X) = Z and σ ∈ Stab†(X) is
v-general stability condition, then:
1 The moduli space Mσ(v) is represented by a K3 surface X.
2 The universal family U ∈ Db(M × X) induces a derived equivalence
ΦU : Db(M)
∼
−→ Db(X).
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 15 / 19
16. Immediate open questions:
Are all geodesics to cusps linear degenerations?
Borel and Ji show, that our linear degenerations are the
EDM-geodesics in the BorelSerre compactication of D(X)/Γ.
Study bers of the morphism
D(X)/Γ
BS
−→ D(X)/Γ
BB
= KM(X).
Is the stability manifold connected?
Do all auto-equivalences of Db(X) preserve the component Stab†(X)?
Open cases are:
Unstable spherical vector bundles
Moduli spaces of simple bundles
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 16 / 19
17. Cusps on abelian surfaces
Can Ma's theorem be generalized to other (CalabiYau)-varieties?
Abelian surfaces A are the rst test case.
The stability manifold has been described by Bridgeland:
Stab†
(A) −→ P+
(A)
is the universal cover.
The auto-equivalences of abelian varieties known by Orlov and
Polishchuk
0 −→ Z ⊕ (A × ˆA) −→ Aut(Db(A)) −→ U(A × ˆA) −→ 0,
where U(A × ˆA) ⊂ Aut(A × ˆA) is a certain explicit subgroup.
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 17 / 19
18. Automorphic functions on the stability manifold
The Kähler moduli space
KM0(X) = Aut†
(X) Stab†
(X)/Gl
+
2 (R)
is a quasi-projective variety. Sections of an ample line bundle give rise
to automorphic functions on the stability manifold.
Use DonaldsonThomas/Joyce invariants DTα(v) to construct
interesting functions on the stability manifold. (c.f. Toda,
MellitOkada)
Study Fourier-expansion of these functions at various cusps.
Already interesting in Picard-rank one case, where
KM(X) ∼= H/Γ+
0 (n)
is a Fricke modular curve.
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 18 / 19
19. Periods and Fano manifolds
In an other work we computed the mirror map
φ : KM(X)
∼
−→ CM(Y )
between the Kähler moduli space of a generic quartic X ⊂ P3 and the
complex deformation space of the mirror K3 surface (Dwork pencil).
We have KM(X) ∼= H/Γ+
0 (2) and CM(Y ) ∼= P1 {0, 1, ∞}, therefore
φ gives rise to a modular function ˜φ : H −→ C.
The function ˜φ is explicitly given as a quotient of solutions to the
PicardFuchs equation ˜φ = W1/W2.
By Mirror symmetry for the Fano manifold P3 the the PicardFuchs
equation for Y equals the Quantum dierential equation for P3.
The solutions W1, W2 can be constructed directly in terms of
GromovWitten invariants. (KatzarkovKontsevichPantev, Iritani)
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 19 / 19