Costello's pushforward formula: errata and generalization

Abstract

Costello's pushforward formula relates virtual fundamental classes of virtually birational algebraic stacks. Its original formulation omits a necessary hypothesis, whose addition is not sufficient to correct the proof. We supply a substitute for Costello's notion of pure degree and prove the pushforward formula with this definition.We also show the hypotheses of the corrected pushforward formula are satisfied in a variety of its applications. Some adjustments to the original proofs are required in several cases, including the original one.

The Log Product Formula in Quantum K-theory

“…Costello's original pushfoward formula [Cos06, theorem 5•0•1] is incorrect as stated; see [HW21]. We prove a K • -theoretic version of Costello's corrected pushforward formula in pure degree one: THEOREM 2•7 ("Costello's pushforward formula in K • -theory").…”

“…Nevertheless, we prove a log version of Costello's formula in K • -theory for pure degree one (cf. [Her19,theorem 4•1], [HW21]). THEOREM 0•4 (= 2.7).…”

A note on quantum K-theory of root constructions

“…There are many situations in cohomological Gromov-Witten theory in which "virtually birational" maps occur, see [18] for a detailed list. In addition, we note that the morphism u in [12,Lemma 4.16] is virtually birational.…”

Gromov–Witten theory and invariants of matroids

“…It is therefore both proper and birational, and identifies fundamental classes under pushforward. Since the vertical maps carry the same perfect obstruction theory, it now follows from the Costello-Herr-Wise comparison theorem that the fine and saturated virtual fundamental class pushes forward to the virtual fundamental class on the space of fine maps [11,26]. Finally, the evaluation maps from K (X ) fs to the strata of X and the forgetful map to the moduli stack of prestable curves factor through the space M 0,r (X , β) of ordinary stable maps, and then by [49, Theorem 1.1], they necessarily factor through the space K (X ) fine as well.…”