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Register a new userVirtual excess intersection theory
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Authors
Adeel A. Khan
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We prove a K-theoretic excess intersection formula for derived Artin stacks. When restricted to classical schemes, it gives a refinement and new proof of R. Thomasons formula.
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We establish an arithmetic intersection theory in the framework of Arakelov geometry over adelic curves. To each projective scheme over an adelic curve, we associate a multi-homogenous form on the group of adelic Cartier divisors, which can be written as an integral of local intersection numbers along the adelic curve. The integrability of the local intersection number is justified by using the theory of resultants.
These are some notes on the basic properties of algebraic K-theory and G-theory of derived algebraic spaces and stacks, and the theory of fundamental classes in this setting.
We prove a topological invariance statement for the Morel-Voevodsky motivic homotopy category, up to inverting exponential characteristics of residue fields. This implies in particular that SH[1/p] of characteristic p>0 schemes is invariant under passing to perfections. Among other applications we prove Grothendieck-Verdier duality in this context.
We develop a version of Hodge theory for a large class of smooth cohomologically proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant $K$-theory of $X$ with respect to a maximal compact subgroup $M subset G$. The result is a natural pure Hodge structure of weight $n$ on $K^n_M(X^{an})$. We also treat categories of matrix factorizations for equivariant Landau-Ginzburg models.
We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated bivariant theory in the sense of Fulton-MacPherson. We import the tools of Fultons intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess intersections, self-intersections, and blow-ups. We also develop a theory of Euler classes of vector bundles in this setting. For the Milnor-Witt spectrum recently constructed by Deglise-Fasel, we get a bivariant theory extending the Chow-Witt groups of Barge-Morel, in the same way the higher Chow groups extend the classical Chow groups. As another application we prove a motivic Gauss-Bonnet formula, computing Euler characteristics in the motivic homotopy category.
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