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Results tagged with sg.symplectic-geometry
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user 7460
Hamiltonian systems, symplectic flows, classical integrable systems
3
votes
Accepted
Are Gromov-Witten invariants birational invariants?
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 …
- 66.3k
7
votes
Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?
I think Verbisky proves a refined form of the Mirror Conjecture for hyperkaehler manifolds, not the conjecture in the strict form. This is explained at page 3 of the paper that you link.
In fact, th …
- 66.3k
2
votes
Accepted
Computing the coefficients of the polynomial $\dim H^0(X,L^k)$ in non-smooth case
Sometimes one can at least compute $\chi(\mathcal{O}_X(X, \, L^k))$.
The formula that one expects will be of the form "ordinary Riemann-Roch formula plus correction terms depending on the singularit …
- 66.3k
41
votes
Accepted
What is a Lagrangian submanifold intuitively?
Lagrangian submanifolds arise naturally in Hamiltonian Mechanics, because of the classical Arnold-Liouville theorem. Let me state it here:
Theorem (Arnold-Liouville). Let $(M, \omega, H)$ be an inte …
- 66.3k
4
votes
Calculating the decomposition of a vector bundle over rational curve
Since you know the explicit equation of the conic, you can compute everything by using Macaulay2.
The following script should be clear:
i1 : k=ZZ/32003;
i2 : ringP1=k[x, y];
i3 : ringP4=k[z1, z2, …
- 66.3k
8
votes
Accepted
Hyperbolicity for algebraic varieties and relation to curves on them
Take a very general hypersurface $j \colon X \hookrightarrow \mathbb{P}^n$ of sufficiently high degree, $3 \leq n \leq 4$. Then $X$ is Kobayashi hyperbolic (Kobayashi actually conjectured that this is …
- 66.3k
10
votes
Accepted
Why are the following varieties symplectomorphic?
In order to obtain an explicit description of the diffeomorphism, one can use the following argument.
Take $B=\mathbb{C}^{n-1}$, with coordinates $t_1, \ldots, t_{n-1}$, and consider the complex spac …
- 66.3k