Geometric and numerical invariants of derived categories of sheaves

<h1>Geometric and numerical invariants of derived categories of sheaves</h1><br>

Instructor: Luigi Lombardi<br>
Ph.D. Course (15 hours)<br>


<h2>Course description:</h2>
In algebraic Geometry derived categories of sheaves on projective varieties are objects of homological nature that encode several properties 
of the underlying variety.  <br>
One of the main conjectures relating derived categories to birational geometry is the so-called DK-hypothesis. 
This asks whether K-equivalent varieties have the same derived category.  <br>
In this course we will survey this problem. We will first introduce the general background regarding triangulaterd categories, 
derived functors,  <br> 
Fourier-Mukai transforms and semiorthogonal decompositions, and then discuss Bridgeland's result for the 3-dimensional case.  <br>
Time permitting we will also discuss the problem of the invariance of the number of linearly independent holomorphic one-forms.


<br> <br>
<h2>Schedule:</h2>
December 12th, 2023, 2pm - 4pm<br>
December 14th, 20223, 10:30am - 12:30pm<br>
December 18th, 2023, 8:30am - 10:30am<br>
December 20th, 2023, 10:30am - 12:30pm<br>
January 10th, 2024, 4pm - 6pm<br>
January 12th, 2024, 1:30pm - 3:30pm<br>
January 16th, 2024, 10:30am - 12:30pm<br>
January 19th, 2024, 1:30pm - 2:30pm<br>
<br>
<h2>Room:</h2>
Aula Dottorato, Department of Mathematics

<br><br>
<h2>Bibliography:</h2>


T. Bridgeland, Flops and derived categories, Invent. Math., 147 (2002), 613-632 <br>
S. I. Gelfand and Yu. I. Manin, Methods of homological algebra, Second edition, Springer <br>
R. Hartshorne. Algebraic Geometry, Springer<br>
D. Huybrechts, Fourier-Mukai transforms in Algebraic Geometry, Oxford Math. Monographs <br>
M. Kashiwara and P. Schapira, Sheaves on manifolds, Springer, GL vol. 292, 1990 <br>
M. Popa, Ch. Schnell. Derived invariance of the number of holomorphic 1-forms and vector fields. Ann. Sci. Ec. Nor. Sup., Serie 4, Vol. 44 (2011), no. 3, 527-536