# Tangent bundle of the incidence variety and its Chern class

I am trying to learn about Thom polynomials and I often find arguments in the literature that just make no real sense to me. Maybe it is due to my lack of knowledge in $$K$$ - theory. I apologize for the lengthy post, but maybe someone knows these things well.

Let $$P := \mathbb{C}P^n$$, $$N := \mathbb{C}P^{n*}$$ and set $$X := P\times N$$. Denote by $$p_1:X\rightarrow P$$ and $$p_2: X\rightarrow N$$ the natural projections. Let $$V = \mathbb{V}(F)\subset P$$ be a hypersurface of degree $$d$$ and $$H = \{(x,a^*)\in P\times N:\text{ }a^*(x) = 0\}$$ be the incidence variety of points and lines in $$P$$. In fact, $$H$$ can be realized as the zero-set (zero-scheme) of a section of the line bundle $$p_1^*\mathcal{O}_P(1)\otimes p_2^*\mathcal{O}_N(1)$$ over $$X$$.

Also, the polynomial $$F$$ defines a section of the line bundle $$p_1^*\mathcal{O}_P(d)$$ over $$X$$. Hence, there is an induced section of the rank $$2$$ vector bundle $$E =p_1^*\mathcal{O}_P(d)\oplus (p_1^*\mathcal{O}_P(1)\otimes p_2^*\mathcal{O}_N(1))$$ over $$X$$. Its zero-set (zero-scheme) is the subvariety $$M :=\{ (x,a^*)\in V\times N:\text{ }a^*(x) =0\}$$, i.e. the incidence variety of points in $$V$$ and lines in $$P$$.

Define $$f:M\rightarrow N$$ by composing $$p_2:X\rightarrow N$$ with the inclusion $$i:M\hookrightarrow X$$, i.e. $$f = p_2\circ i$$. I am interested in computing the total Chern class $$c(f^*TN-TM)$$, where $$f^*TM-TN$$ is the virtual bundle living in the $$K$$ - group of $$M$$.

The claim is: $$TM = i^*(TX-E)\text{ and hence } f^{*}TN - TM = i^{*}(E - p_1^*TP).$$

It appears to me that these are just standard calculations in $$K$$ - groups but I don't get it.

I now have some idea on how to prove these relations. Consider a real vector bundle $$\pi:E\rightarrow X$$ of rank $$k$$, where $$E$$ and $$X$$ are $$C^\infty$$ manifolds and let $$X$$ be $$n$$,-,dimensional. If $$s$$ is a section transverse to the zero - section, then its zero set $$Z(s) = \{x\in X|\text{ }s(x)=0_x\in E_x\}$$ is a submanifold of $$X$$. What's more, on local bundle charts $$U\subset X$$, the restriction $$s|U$$ is a submersion in points $$x\in U$$ with $$s(x) = 0$$, meaning $$\text{im}(ds(x))= \mathbb{R}^k$$.
Let $$M := Z(s)$$, then the tangent space $$T_xM$$ can be identified with $$\text{ker}(ds(x))$$ and the normal space is isomorphic to $$\mathbb{R}^k$$, which follows from the canonical exact sequence $$0\rightarrow T_xM = \text{ker}(ds(x))\rightarrow T_xX\rightarrow N_xM = T_xX/T_xM\rightarrow 0.$$ Hence, $$TX = TM \oplus i^*E$$, where $$i:M\hookrightarrow X$$ is the inclusion. This proves $$TM = TX-i^*E\in K_0(X)$$ The second identity follows from $$TX = TP\oplus TN$$.
Although the computation above is valid for real vector bundles, I think we can interpret any complex vector bundle as a real vector bundle of twice the rank. Also $$\mathbb{C}P^n$$ is a $$2n$$ - dimensional real manifold...