If \(\pi: \tilde{X} \rightarrow X\) is a monoidal transformation with center \(P\), show that \(H^1\left(\tilde{X}, \Omega_{\tilde{X}}\right) \cong H^1\left(X, \Omega_X\right) \oplus k\). This gives another proof of (5.8).
Hints: Use the projection formula (III, Ex. 8.3) and (III, Ex. 8.1) to show that \(H^i\left(X, \Omega_X\right) \cong H^i\left(\tilde{X}, \pi^* \Omega_X\right)\) for each \(i\). Next use the exact sequence \[ 0 \rightarrow \pi^* \Omega_X \rightarrow \Omega_{\tilde{X}} \rightarrow \Omega_{\tilde{X} / X} \rightarrow 0 \] and a local calculation with coordinates to show that there is a natural isomorphism \(\Omega_{\tilde{X} / X} \cong \Omega_E\), where \(E\) is the exceptional curve. Now use the cohomology sequence of the above sequence (you will need every term) and Serre duality to get the result.
Solution
Let \[ \pi:\widetilde X=\operatorname{Bl}_P X\longrightarrow X \] be the blowup, and let \[ E=\pi^{-1}(P)\cong\PP^1 \] be its exceptional curve.
Put \[ \mathcal A=\pi^*\Omega_X, \qquad \mathcal B=\Omega_{\widetilde X}. \]
1. For every \(i\ge0\) there is a natural isomorphism\[ \boxed{ H^i(\widetilde X,\mathcal A) \cong H^i(X,\Omega_X).} \]
For a monoidal transformation of a nonsingular surface at a point, \[ \pi_*\OO_{\widetilde X}\cong\OO_X \] and the higher direct images \[ R^q\pi_*\OO_{\widetilde X} \] vanish for \(q>0\) by the blowup-cohomology theorem V.3.4; the same input underlies P-AGH531PABLOWUP.
Because \(\Omega_X\) is locally free, the projection formula gives \[ R^q\pi_*(\pi^*\Omega_X) \cong \Omega_X\otimes R^q\pi_*\OO_{\widetilde X}. \] This is the higher-direct-image projection formula Exercise III.8.3. Thus \[ \pi_*\mathcal A\cong\Omega_X \] and \[ R^q\pi_*\mathcal A=0 \qquad(q>0). \] Exercise III.8.1 P-AGH381DEGENLERAY therefore gives the displayed cohomology isomorphisms.
2. There is an exact sequence\[ \boxed{ 0 \longrightarrow \pi^*\Omega_X \longrightarrow \Omega_{\widetilde X} \longrightarrow \Omega_E \longrightarrow 0.} \]
The usual sequence of relative differentials gives a right-exact sequence \[ \pi^*\Omega_X \longrightarrow \Omega_{\widetilde X} \longrightarrow \Omega_{\widetilde X/X} \longrightarrow 0. \] The first map is injective because it is an isomorphism away from the exceptional divisor and \(\pi^*\Omega_X\) is locally free, hence torsion-free. It remains to identify the quotient.
Choose regular parameters \(x,y\) at \(P\). On the blowup chart \[ U_x=\Spec k[x,t], \qquad y=xt, \] the map on differentials is \[ dx\longmapsto dx, \qquad dy\longmapsto t\,dx+x\,dt. \] Hence the cokernel is generated by \(dt\) with the single relation \[ x\,dt=0. \] Since \(E\cap U_x\) is \(x=0\), this cokernel is \[ \OO_{E\cap U_x}\,dt = \Omega_E|_{E\cap U_x}. \]
On the other chart \[ U_y=\Spec k[s,y], \qquad x=sy, \] the cokernel is similarly \[ \OO_{E\cap U_y}\,ds. \] On the overlap one has \[ t=s^{-1}, \qquad dt=-s^{-2}ds, \] which is exactly the transition rule for the cotangent sheaf of \(E\cong\PP^1\). Thus the local cokernels glue to \[ \Omega_{\widetilde X/X}\cong\Omega_E, \] giving the asserted short exact sequence.
3. The cohomology of the quotient is\[ \boxed{ H^0(E,\Omega_E)=0, \qquad H^1(E,\Omega_E)\cong k, \qquad H^i(E,\Omega_E)=0\ (i\ge2).} \]
Since \[ E\cong\PP^1, \] one has \[ \Omega_E\cong\OO_{\PP^1}(-2). \] Therefore there are no global sections: \[ H^0(E,\Omega_E)=0. \] Serre duality on \(E\) gives \[ H^1(E,\Omega_E) \cong \dualof{H^0(E,\OO_E)} \cong k. \] Finally a coherent sheaf on the one-dimensional projective scheme \(E\) has no cohomology in degrees at least two.
4. The map\[ H^0(\widetilde X,\mathcal A) \longrightarrow H^0(\widetilde X,\mathcal B) \]is an isomorphism.
3. Hence the first map is both injective and surjective.
5. The map\[ \boxed{ H^2(\widetilde X,\mathcal A) \xrightarrow{\sim} H^2(\widetilde X,\mathcal B)} \]is an isomorphism.
The end of the long exact sequence of step 2 contains \[ H^1(E,\Omega_E) \longrightarrow H^2(\widetilde X,\mathcal A) \longrightarrow H^2(\widetilde X,\mathcal B) \longrightarrow H^2(E,\Omega_E)=0. \] Thus the displayed map on \(H^2\) is surjective. We show its source and target have the same finite dimension.
By step 1, \[ h^2(\widetilde X,\mathcal A) = h^2(X,\Omega_X). \] For a nonsingular projective surface \(S\), Serre duality and the rank-two identity \[ (\Omega_S)^\vee\otimes\omega_S \cong \Omega_S \] give \[ h^2(S,\Omega_S)=h^0(S,\Omega_S). \] Therefore \[ h^2(\widetilde X,\mathcal A) = h^0(X,\Omega_X). \] Step 1 in degree zero and step 4 give \[ h^0(X,\Omega_X) = h^0(\widetilde X,\mathcal A) = h^0(\widetilde X,\mathcal B). \] Applying Serre duality on \(\widetilde X\) gives \[ h^0(\widetilde X,\mathcal B) = h^2(\widetilde X,\mathcal B). \] Hence \[ h^2(\widetilde X,\mathcal A) = h^2(\widetilde X,\mathcal B). \] A surjective linear map between finite-dimensional vector spaces of equal dimension is an isomorphism.
6. The long exact sequence of step 2 reduces in degree one to a short exact sequence\[ \boxed{ 0 \longrightarrow H^1(X,\Omega_X) \longrightarrow H^1(\widetilde X,\Omega_{\widetilde X}) \longrightarrow k \longrightarrow 0.} \]
Using step 3, the middle part of the long exact sequence is \[ 0 \longrightarrow H^1(\widetilde X,\mathcal A) \longrightarrow H^1(\widetilde X,\mathcal B) \longrightarrow H^1(E,\Omega_E) \xrightarrow{\delta} H^2(\widetilde X,\mathcal A). \] Step 5 says the following map \[ H^2(\widetilde X,\mathcal A) \longrightarrow H^2(\widetilde X,\mathcal B) \] is injective. Exactness therefore forces \[ \operatorname{im}\delta=0. \] Thus the map \[ H^1(\widetilde X,\mathcal B) \longrightarrow H^1(E,\Omega_E) \] is surjective.
Now apply step 1 to identify \[ H^1(\widetilde X,\mathcal A) \cong H^1(X,\Omega_X) \] and step 3 to identify \[ H^1(E,\Omega_E)\cong k. \] This gives the asserted short exact sequence.
7. Consequently\[ \boxed{ H^1(\widetilde X,\Omega_{\widetilde X}) \cong H^1(X,\Omega_X)\oplus k.} \]
All groups in step 6 are vector spaces over \(k\). Every short exact sequence of \(k\)-vector spaces splits. Choosing a lift in \(H^1(\widetilde X,\Omega_{\widetilde X})\) of \(1\in k\) therefore gives a splitting and the displayed isomorphism.
The splitting need not be canonical; the assertion of the exercise is the vector-space isomorphism.
8. Q.E.D.
Step 1 identifies the cohomology of the pulled-back cotangent bundle, steps <1>2–<1>3 identify the relative differential contribution with the one-dimensional group \(H^1(E,\Omega_E)\), and steps <1>4–<1>6 show that its connecting map to degree two vanishes. Step 7 gives the required direct sum.