K3 Seminar Spring 2019: Difference between revisions
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| bgcolor="#BCD2EE" | Abstract: I will describe Chow theoretic correspondences as a motivation to derived correspondences. We will then define integral functors on derived categories. The dual abelian variety will be given as a moduli space in terms of its functor of points, leading us to a definition of the universal Poincaré bundle on $A \times \hat{A}$. We will look at the integral transform from $D(A)$ to $D(\hat{A})$ induced by the Poincaré bundle. Cohomology of the Poincaré bundle will be stated and used for the computation of the $K$-theoretic Fourier-Mukai transform on elliptic curves. With the help of the base change theorem, we will describe the Fourier-Mukai duals of homogeneous line bundles on $A$. For an elliptic curve $E$, we will establish the equivalence between | | bgcolor="#BCD2EE" | Abstract: I will describe Chow theoretic correspondences as a motivation to derived correspondences. We will then define integral functors on derived categories. The dual abelian variety will be given as a moduli space in terms of its functor of points, leading us to a definition of the universal Poincaré bundle on $A \times \hat{A}$. We will look at the integral transform from $D(A)$ to $D(\hat{A})$ induced by the Poincaré bundle. Cohomology of the Poincaré bundle will be stated and used for the computation of the $K$-theoretic Fourier-Mukai transform on elliptic curves. With the help of the base change theorem, we will describe the Fourier-Mukai duals of homogeneous line bundles on $A$. For an elliptic curve $E$, we will establish the equivalence between | ||
1. the abelian category of semistable bundles of slope 0 on $E | 1. the abelian category of semistable bundles of slope 0 on $E$, and | ||
2. the abelian category of coherent torsion sheaves on &E | 2. the abelian category of coherent torsion sheaves on &E$. | ||
Simple and indecomposable objects of these categories will be described (with the help of the structure theorem of PIDs) and we will deduce Atiyah’s classification of the indecomposable vector bundles of degree 0. | Simple and indecomposable objects of these categories will be described (with the help of the structure theorem of PIDs) and we will deduce Atiyah’s classification of the indecomposable vector bundles of degree 0. | ||
Revision as of 21:43, 6 February 2019
When: Thursday 5-7 pm
Where: Van Vleck TBA
Schedule
| Date | Speaker | Title |
| March 7 | Mao Li | Basics of K3 Surfaces and the Grothendieck-Riemann-Roch theorem |
| March 14 | Shengyuan Huang | Elliptic K3 Surfaces |
| March 28 | Zheng Lu | Moduli of Stable Sheaves on a K3 Surface |
| April 4 | Canberk Irimagzi | Fourier-Mukai Transforms |
| April 11 | David Wagner | Cohomology of Complex K3 Surfaces and the Global Torelli Theorem |
| April 25 | TBA | Derived Categories of K3 Surfaces |
March 7
| Mao Li |
| Title: Basics of K3 Surfaces and the Grothendieck-Riemann-Roch theorem |
| Abstract: |
March 14
| Shengyuan Huang |
| Title: Elliptic K3 Surfaces |
| Abstract: |
March 28
| Zheng Lu |
| Title: Moduli of Stable Sheaves on a K3 Surface |
| Abstract: |
April 4
| Canberk Irimagzi |
| Title: Fourier-Mukai Transforms |
| Abstract: I will describe Chow theoretic correspondences as a motivation to derived correspondences. We will then define integral functors on derived categories. The dual abelian variety will be given as a moduli space in terms of its functor of points, leading us to a definition of the universal Poincaré bundle on $A \times \hat{A}$. We will look at the integral transform from $D(A)$ to $D(\hat{A})$ induced by the Poincaré bundle. Cohomology of the Poincaré bundle will be stated and used for the computation of the $K$-theoretic Fourier-Mukai transform on elliptic curves. With the help of the base change theorem, we will describe the Fourier-Mukai duals of homogeneous line bundles on $A$. For an elliptic curve $E$, we will establish the equivalence between
1. the abelian category of semistable bundles of slope 0 on $E$, and 2. the abelian category of coherent torsion sheaves on &E$. Simple and indecomposable objects of these categories will be described (with the help of the structure theorem of PIDs) and we will deduce Atiyah’s classification of the indecomposable vector bundles of degree 0. |
April 11
| David Wagner |
| Title: Cohomology of Complex K3 Surfaces and the Global Torelli Theorem |
| Abstract: |
April 25
| TBA |
| Title: Derived Categories of K3 Surfaces |
| Abstract: |
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