Derived category and cohomology of resolutions of singularities: examples from representation theory.
Permanent URL:
http://hdl.handle.net/2047/d20005066
Weyman, Jerzy, 1955- (Committee member)
Laredo, Valerio Toledano (Committee member)
Bezrukavnikov, Roman, 1973- (Committee member)
Marian, Alina (Committee member)
In the first part of this thesis, we describe noncommutative desingularizations of determinantal varieties, determinantal varieties defined by minors of generic symmetric matrices, and pfaffian varieties defined by pfaffians of generic anti-symmetric matrices. For maximal minors of square matrices and symmetric matrices, this gives a non-commutative crepant resolution. Along the way, we describe a method to calculate the quiver with relations for any non-commutative desingularizations coming from exceptional collections over partial flag varieties.
In the second part of this thesis, we study t-structures coming from noncommutative symplectic resolutions. A localization theorem for the cyclotomic rational Cherednik algebra Hc = Hc((ℤ/l)n ⋊ Sn) over a field of positive characteristic has been proved by Bezrukavnikov, Finkelberg and Ginzburg. Localizations with different parameters give different t-structures on the derived category of coherent sheaves on the Hilbert scheme of points on a surface. In this short note, we concentrate on the comparison between different t-structures coming from different localizations. When n = 2, we show an explicit construction of tilting bundles that generates these t-structures. These t-structures are controlled by a real variation of stability conditions, a notion related to Bridgeland stability conditions. We also show its relation to the topology of Hilbert schemes and irreducible representations of Hc.
In the last part of this thesis, we study Chern numbers of varieties related by flops. For this purpose, we define the algebraic elliptic cohomology theory coming from Krichever's elliptic genus as an oriented cohomology theory on smooth varieties over an arbitrary perfect field. We show that in the algebraic cobordism ring with rational coefficients, the ideal generated by differences of classical flops coincides with the kernel of Krichever's elliptic genus. This generalizes a theorem of B. Totaro in the complex analytic setting.
cohomology rings
Chern numbers
Mathematics
Noncommutative algebras -- Mathematical models
Rings (Algebra) -- Mathematical models
Arithmetical algebraic geometry -- Mathematical models
Algebraic topology -- Mathematical models
Homology theory -- Mathematical models
Derived categories (Mathematics)
Determinantal varieties -- Mathematical models
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