# Are Gromov-Witten invariants birational invariants?

Let $$X$$ and $$Y$$ be two smooth complex projective varieties. Then they are also symplectic manifolds. We know Gromov-Witten (GW) invariants are symplectic invariants. That means if there exists a a bijective symplectomorphism $$f: X \to Y$$, then GW invariants of $$X$$ and $$Y$$ are the same.

Now suppose $$f: X \to Y$$ is an algebraic isomorphism, or more generally, a birational map, then $$f$$ may not be a symplectomorphism. In this case, are GW invariants of $$X$$ and $$Y$$ different in general?

On a complex projective variety, Gromov-Witten invariants can be interpreted as virtual counts of curves, so they are biregular invariants.

However, they are not birational invariant in general. The behaviour of Gromov-Witten invariants under an arbitrary birational modification is in fact rather subtle.

For more details and examples you can have a look at Section 1.4 of the paper

D. Abramovich, J. Wise Birational invariance in logarithmic Gromov-Witten theory, Compos. Math. 154, No. 3, 595-620 (2018). ZBL1420.14124.

• By definition, GW invariants are only invariant under complex deformations. Why are they biregular invariants? – Yuhang Chen Nov 13 '19 at 13:27
• Biregular maps (= algebraic isomorphisms) do not change virtual curve counting, because biregularly equivalent varieties contain the same curves. – Francesco Polizzi Nov 13 '19 at 13:33
• But I doubt whether isomorphic varieties $X$ and $Y$ must have same "virtual counting of curves". It's not obvious by looking at the definition of GW invariants of $X$, i.e., the integration of cohomology classes over the virtual fundamental class of the moduli stack of stable maps into $X$. – Yuhang Chen Nov 13 '19 at 13:42
• I am not sure that I understand your concern. At the end of of the story, you are simply (virtually) counting how many curves there are that intersect $n$ chosen submanifolds of $X$. This number is clearly invariant under any biregular map. The counting is only virtual, as there can be non-integer contributions given by the stabilizers at the orbifold points of the moduli stack of stable maps, but again these contributions are biregularly invariant. – Francesco Polizzi Nov 13 '19 at 13:52
• I guess my concern is mainly about the virtual fundamental class. I haven't looked into the details of the definition of the virtual fundamental class. So I am not sure if different algebraic descriptions (i.e., different polynomial equations) of a projective variety will affect anything. – Yuhang Chen Nov 13 '19 at 14:05