The universal Euler characteristic for varieties of characteristic zero

By Franziska Bittner. Compositio Mathematica, 2001

268 citations recorded by the source corpus.

Using the weak factorization theorem, we give a simple presentation for the value group of the universal Euler characteristic with compact support for varieties of characteristic zero and describe the value group of the universal Euler characteristic of pairs. This gives a new proof for the existence of natural Euler characteristics with values in the Grothendieck group of Chow motives. A generalization of the presentation to the relative setting allows us to define duality and the six operations.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8157ea7909af3b0646cc62a3bf7b5a728a373ae4

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In Mathematical Physics

Citations in the corpus

An introduction to motivic Hall algebras (2010, unknown): “Some good references are [2,17].”
Stability conditions and curve counting invariants on Calabi–Yau 3-folds (2011, background): “The structure of the group K(Var/C) is studied in [7].”
Chow motives versus non-commutative motives (2011, methodology): “for every smooth projective C-varietyX , combined with Bittner-Looijenga’s presentation of K0(VarC) (see [8, 30]), allow us then to conclude that μχc factors through μNC via the ring homomorphism (9.”
The motivic nearby fiber and degeneration of stable rationality (2017, methodology): “2.5.1] (see also the discussion at the end of §2.5 in [NP17]). Alternatively, one can use Bittner’s presentation of the Grothendieck ring in terms of smooth and proper varieties and blow-up relations [Bi04]. One uses the formula (3.1.2) as the deﬁnition of the motivic volume and applies Weak Factori”
HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES (2005, methodology): “1 given in [11] is of similar nature.”
Characteristic classes of mixed Hodge modules (2009, background): “: Y →X] →f!(1Y ), with can(L) = 1pt. The advantage of M0(var/X) compared to K0(var/X) is the fact, that it has an induced duality involution D: M0(var/X) →M0(var/X) characterized uniquely by (compare [8]): D([f : M →X]) = L−m ·[f : M →X] for f : M →X a proper morphism with M smooth and pure m-dimens”
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008, background): “e resulting category contains the element L = [P1 k ]−[Spec(k)]. Adding formally the inverse L−1 we obtain the ring which we denote by Motcoh(Spec(k)). It is an easy corollary of Bittner theorem (see [6]) that the natural map Mot(Spec(k)) →Motcoh(Spec(k)) which assigns to a smooth projective variety”
Motivic invariants of arc-symmetric sets and blow-Nash equivalence (2003, unknown): “In § 2, we give conditions, inspired by a result of Bittner [Bit04], on an invariant deﬁned on connected components of compact non-singular real algebraic varieties such that it extends to an additive invariant on the constructible category of arc-symmetric sets.”
Arc spaces, motivic integration and stringy invariants (2004, background): “V → V is the blowing-up with centre Z and exceptional variety E [ Bi1 ].”
MOTIVIC INVARIANTS OF ARTIN STACKS AND ‘STACK FUNCTIONS’ (2005, methodology): “The ring K0(VarK) for charK = 0 is studied by Bittner [1] and is clearly very large, and K0(StaK) is even larger.”