# geometry – Symplectic Hodge Star and Koszul differential

Let $$M = text{Spec}(R)$$ be a symplectic affine variety of dimension $$n$$ with the symplectic form $$omega$$. There is the symplectic Hodge star $$star: Omega^k_{M} to Omega^{2n-k}_{M}$$ given by the condition that for any $$alpha, beta in Omega^k$$ we should have $$beta wedge alpha = Lambda^k G(beta, alpha) cdot v_n$$ where $$G(-, -): TM^* times TM^* to R$$ is a bivector field corresponding to $$omega$$ and $$v_n = omega^n/n!$$. I want to understand the first nontrivial (for me) case: $$M = T^* mathbb{A}^2$$. In this case $$omega = dx_1 dx_2 + dx_3 dx_4$$ and $$v_2 = dx_1 dx_2 dx_3 dx_4$$. Also, $$G = frac{partial}{partial_{x_1}} frac{partial}{partial_{x_2}} + frac{partial}{partial_{x_3}} frac{partial}{partial_{x_4}}.$$ It seems to me that we have the following

$$star(f) = f dx_1 dx_2 dx_3 dx_4$$

$$star(alpha) = alpha wedge omega$$ where $$alpha in Omega^1_{M}$$

$$star(zeta) = -zeta$$ for $$zeta in Omega^{2}_{M}$$

$$star(eta) = i_{G} eta$$ where $$eta in Omega^{3}_{M}$$ and $$i_{G}$$ is the contraction with $$G$$.

OK, then let us see how the Koszul differential works and it is the place where I met a contradiction in math (of course, the problem is with me but I cannot figure it out).

$$textbf{Setup.}$$ The Koszul differential $$B: Omega^{k}_{M} to Omega^{k-1}_{M}$$ has two interpretations, the first one is $$B = (i_G, d)$$. The second is $$B = (-1)^{k+1} star d star.$$

$$textbf{My misunderstanding.}$$ Let $$x in Omega^3_{M}$$. The first definition gives $$Bx = i_G dx – d i_{G}x$$. The second definition gives $$Bx = star d star x = star d i_{G} x = -d i_{G} x$$. As we see, we should have $$i_{G} dx=0$$ which is not the case, for example, for $$x_1 dx_2 dx_3 dx_4.$$

$$textbf{My quesion.}$$ Where is the mistake? Or, can we write down the star isomorphism in the general case? I think it is related to the multiplication by $$omega^{n-k}$$, but mostly I’m interested how to write down the formula for $$star: Omega^{k}_{M} to Omega^{2n-k}_{M}$$ for $$k>n$$. Is there any reference?