We consider virtual pullbacks in K-theory, and show that they are bivariant classes satisfy certain functoriality. As applications to K-theoretic counting invariants, we include proofs of a localization formula for schemes degeneration Donaldson–Thomas theory.

The paper is Part III of our ongoing project to study a case Crepant Transformation Conjecture: K-equivalence Conjecture for ordinary flops. In this we prove the invariance quantum rings general flops, whose local models are certain non-split toric bundles over arbitrary smooth base. An essential ingredient in proof splitting principle, which reduces statement Gromov--Witten theory on split bundles.

On two subspaces of the Bruhat-Tits tree, effective actions are calculated. The limits these field theories found to be same conformal theory over p-adic numbers when taken boundary tree. Their relations version AdS/CFT also discussed.

Witten invariants of V × W with those and in the case log structure on is trivial.

We consider virtual pullbacks in $K$-theory, and show that they are bivariant classes satisfy certain functoriality. As applications to $K$-theoretic counting invariants, we include proofs of a localization formula for schemes degeneration Donaldson-Thomas theory.

Let $X$ be a Calabi-Yau 4-fold and $D$ smooth connected divisor on it. We consider tautological bundles of $L=\mathcal{O}_X(D)$ Hilbert schemes points their counting invariants defined by integrating the Euler classes against virtual classes. relate these to Maulik-Nekrasov-Okounkov-Pandharipande's pullback technique confirm conjecture Cao-Kool. The same strategy is also applied obtain formula for one dimensional stable sheaves. This in turn gives nontrivial identity primary descendent as...

We give an effective algorithm to compute the Euler characteristics $\chi (\overline {\mathcal {M}}_{1,n}, \bigotimes _{i=1}^n L_i^{ d_i})$. This work is a sequel 1997 of first author. In addition, we simple proof Pandharipande's vanishing theorem $H^j {M}}_{0,n}, d_i})=0$ for $j \ge 1, d_i 0$.

The purpose of this short article is to prove a product formula relating the log Gromov-Witten invariants $V \times W$ with those $V$ and $W$ in case structure on trivial.

We give an effective algorithm to compute the Euler characteristics $χ(\mbar_{1,n}, \otimes_{i=1}^n L_i^{d_i})$. In addition, we a simple proof of Pandharipande's vanishing theorem $H^j (\mbar_{0,n}, L_i^{d_i})=0$ for $j \ge 1, d_i \ge0$.