Abstracts - Fall 2019 

Speaker:  Nivedita Bhaskhar

Title: Reduced Whitehead groups of algebras over p-adic curves

Abstract: Any central simple algebra A over a field K is a form of a matrix algebra. Further A/K comes equipped with a reduced norm map which is obtained by twisting the determinant function. Every element in the  commutator subgroup [A*, A*] has reduced norm 1 and hence lies in SL_1(A), the group of reduced norm one elements of A. Whether the reverse inclusion holds was formulated as a question in 1943 by Tannaka and Artin in terms of the triviality of the  reduced Whitehead group SK_1(A) := SL_1(A)/[A*,A*]. 

Platonov negatively settled the Tannaka-Artin question by giving a counter example over a cohomological dimension (cd) 4 base field. In the same paper however, the triviality of SK_1(A) was shown for all algebras over cd at most 2 fields. In this talk, we investigate the situation for l-torsion algebras over a class of cd 3 fields of some arithmetic flavour, namely function fields of p-adic curves where l is any prime not equal to p. We partially answer a question of Suslin by proving the triviality of the reduced Whitehead group for these algebras. The proof relies on the techniques of patching as developed by Harbater-Hartmann-Krashen and exploits the arithmetic of these fields.

Speaker:  Stefano Filipazzi 

Title: On the connectedness principle and dual complexes for generalized pairs

Abstract: Let (X,B) be a pair (a variety with an effective Q-divisor), and let f: X -> S be a contraction with -(K_X+B) nef over S. A conjecture, known as the Shokurov-Koll'ar connectedness principle, predicts that f^{-1}(s) intersect Nklt(X,B) has at most two connected components, where s is an arbitrary point in S and Nklt(X,B) denotes the non-klt locus of (X,B). The conjecture is known in some cases, namely when -(K_X+B) is big over S, and when it is Q-trivial over S. In this talk, we discuss a proof of the full conjecture and extend it to the case of generalized pairs. Then we apply it to the study of the dual complex of generalized log Calabi-Yau pairs. This is joint work with Roberto Svaldi.

Speaker:  Michail Savvas

Title: Almost Perfect Obstruction Theory and K-theoretic Donaldson-Thomas Invariants

Abstract: Perfect obstruction theories are a fundamental ingredient used to define enumerative invariants associated to moduli problems. In this talk, we introduce the notion of an almost perfect obstruction theory on a Deligne-Mumford stack and show that it gives rise to a virtual structure sheaf in its K-theory. Several moduli problems of interest without a perfect obstruction theory admit almost perfect obstruction theories. This enables us to define K-theoretic enumerative invariants, including classical and generalized K-theoretic Donaldson-Thomas invariants of sheaves andcomplexes on Calabi-Yau threefolds. Based on joint work with Young-Hoon Kiem.

Speaker:  Stefan Schreieder 

Title: Stably irrational hypersurfaces of small slopes

Abstract: We show that over any uncountable field of characteristic different from two, a very general hypersurface of dimension n>2 and degree at least log_2(n)+2 is not stably rational. This improves earlier results of Koll\'ar and Totaro, which produced a linear lower bound on the degree.